tag:theconversation.com,2011:/id/topics/pi-day-25710/articlesPi Day – The Conversation2023-03-08T15:17:16Ztag:theconversation.com,2011:article/2000462023-03-08T15:17:16Z2023-03-08T15:17:16ZPi gets all the fanfare, but other numbers also deserve their own math holidays<figure><img src="https://images.theconversation.com/files/514054/original/file-20230307-2837-xqlq9k.jpg?ixlib=rb-1.1.0&rect=281%2C209%2C3413%2C2449&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">One mathematical constant describes the population growth rate of a bunch of rabbits.</span> <span class="attribution"><a class="source" href="https://www.gettyimages.com/detail/photo/rabbits-on-field-royalty-free-image/1146008449">Supalerk Laipawat/EyeEm via Getty Images</a></span></figcaption></figure><p>March 14 is celebrated as Pi Day because the date, when written as 3/14, matches the start of the decimal expansion 3.14159… of the most famous mathematical constant.</p>
<p>By itself, pi is simply a number, one among countless others between 3 and 4. What makes it famous is that it’s built into every circle you see – circumference equals pi times diameter – not to mention a range of other, unrelated contexts in nature, from the <a href="https://mathworld.wolfram.com/NormalDistribution.html">bell curve</a> distribution to <a href="https://www.scientificamerican.com/article/pi-in-the-sky-general-relativity-passes-the-ratios-test/">general relativity</a>.</p>
<p>The true reason to celebrate Pi Day is that mathematics, which is a purely abstract subject, turns out to describe our universe so well. My book “<a href="https://wwnorton.com/books/9781324007036">The Big Bang of Numbers</a>” explores how remarkably hardwired into our reality math is. Perhaps the most striking evidence comes from mathematical constants: those rare numbers, including pi, that break out of the pack by appearing so frequently – and often, unexpectedly – in natural phenomena and related equations, that <a href="https://www.manilsuri.com/">mathematicians like me</a> exalt them with special names and symbols. </p>
<p>So, what other <a href="https://www.cambridge.org/us/academic/subjects/mathematics/recreational-mathematics/mathematical-constants?format=HB&isbn=9780521818056">mathematical constants</a> are worth celebrating? Here are my proposals to start filling out the rest of the calendar.</p>
<h2>The Golden Ratio</h2>
<p>For January, I nominate the <a href="https://www.britannica.com/science/golden-ratio">Golden Ratio</a>, phi. Two quantities are said to be in this ratio if dividing the larger by the smaller quantity gives the same answer as dividing the sum of the two quantities by the larger quantity. Phi equals 1.618…, and since there’s no Jan. 61, we could celebrate it on Jan. 6.</p>
<p><a href="https://www.penguinrandomhouse.com/books/102878/the-golden-ratio-by-mario-livio/">First calculated by Euclid</a>, this ratio was popularized by Italian mathematician Luca Pacioli, who wrote a <a href="https://www.maa.org/press/periodicals/convergence/mathematical-treasure-luca-pacioli-s-divina-proportione">book in 1509</a> extravagantly extolling its aesthetic properties. Supposedly, Leonardo da Vinci, who drew 60 drawings for this book, <a href="https://monalisa.org/2012/09/12/leonardo-and-mathematics-in-his-paintings/">incorporated it into the dimensions of Mona Lisa’s features</a>, a choice some claim is responsible for her beauty.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/512253/original/file-20230224-2421-b4cons.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="a rectangle over Mona Lisa's face labels the vertical and horizontal ratio" src="https://images.theconversation.com/files/512253/original/file-20230224-2421-b4cons.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/512253/original/file-20230224-2421-b4cons.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=301&fit=crop&dpr=1 600w, https://images.theconversation.com/files/512253/original/file-20230224-2421-b4cons.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=301&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/512253/original/file-20230224-2421-b4cons.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=301&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/512253/original/file-20230224-2421-b4cons.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=378&fit=crop&dpr=1 754w, https://images.theconversation.com/files/512253/original/file-20230224-2421-b4cons.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=378&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/512253/original/file-20230224-2421-b4cons.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=378&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The vertical and horizontal measures of Mona Lisa’s face fit the Golden Ratio.</span>
<span class="attribution"><a class="source" href="https://wwnorton.com/books/9781324007036">'The Big Bang of Numbers'</a></span>
</figcaption>
</figure>
<p>The first inkling that phi occurs in nature came from another Italian, Fibonacci, while <a href="https://plus.maths.org/content/life-and-numbers-fibonacci">studying how rabbits multiply</a>. A common reproductive assumption was that each pair of rabbits begets another pair every month. Start with a single rabbit pair, and successive populations will then follow the sequence 1, 2, 4, 8, 16, 32, 64, 128, 256 and so on – that is, get multiplied by a monthly “growth ratio” of 2.</p>
<p>What Fibonacci observed, though, was that rabbits spent the first cycle reaching sexual maturity and only began reproducing after that. A single pair now gives the new, slower progression 1, 1, 2, 3, 5, 8, 13, 21, 34… instead. This is the <a href="https://mathworld.wolfram.com/FibonacciNumber.html">famous sequence</a> named after Fibonacci; notice that each population turns out to be the sum of its two predecessors.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/512255/original/file-20230224-2346-mamx70.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="diagram of how many rabbits you'll have month by month" src="https://images.theconversation.com/files/512255/original/file-20230224-2346-mamx70.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/512255/original/file-20230224-2346-mamx70.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=420&fit=crop&dpr=1 600w, https://images.theconversation.com/files/512255/original/file-20230224-2346-mamx70.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=420&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/512255/original/file-20230224-2346-mamx70.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=420&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/512255/original/file-20230224-2346-mamx70.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=528&fit=crop&dpr=1 754w, https://images.theconversation.com/files/512255/original/file-20230224-2346-mamx70.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=528&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/512255/original/file-20230224-2346-mamx70.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=528&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Fibonacci’s rabbits don’t really double their population each generation – their growth ratio actually approaches the 1.618… of phi.</span>
<span class="attribution"><a class="source" href="https://wwnorton.com/books/9781324007036">'The Big Bang of Numbers'</a></span>
</figcaption>
</figure>
<p>How does phi show up amid all these randy rabbits? Well, progressing through the sequence, you see that each number is about 1.6 times the previous one. In fact, this growth ratio keeps getting closer and closer to 1.618…. For instance, 21 equals about 1.615 times 13, and 34 equals about 1.619 times 21. This means the rabbits settle down to reproducing with a growth ratio that is no longer 2, but rather, gets closer and closer to the Golden Ratio.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/512256/original/file-20230224-2023-hria2d.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="'petals' on the base of a pine cone spiral outward from the center in 13 lines" src="https://images.theconversation.com/files/512256/original/file-20230224-2023-hria2d.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/512256/original/file-20230224-2023-hria2d.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=240&fit=crop&dpr=1 600w, https://images.theconversation.com/files/512256/original/file-20230224-2023-hria2d.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=240&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/512256/original/file-20230224-2023-hria2d.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=240&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/512256/original/file-20230224-2023-hria2d.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=302&fit=crop&dpr=1 754w, https://images.theconversation.com/files/512256/original/file-20230224-2023-hria2d.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=302&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/512256/original/file-20230224-2023-hria2d.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=302&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The number of spirals in a pine cone is usually a Fibonacci number.</span>
<span class="attribution"><a class="source" href="https://wwnorton.com/books/9781324007036">'The Big Bang of Numbers'</a></span>
</figcaption>
</figure>
<p>Actual rabbits are unlikely to follow this rule precisely. For one, they have the unfortunate tendency to get eaten by predators. But the <a href="https://www.britannica.com/science/Fibonacci-number">Fibonacci numbers</a> – like 5, 8, 13 and so on – <a href="https://www.youtube.com/watch?v=ahXIMUkSXX0">show up extensively in nature</a>, like in the number of spirals you might see in a typical pine cone. And yes, phi itself makes a few appearances as well, perhaps most notably in the way <a href="https://www.jstor.org/stable/1743115">leaves arrange themselves around a stem</a> to maximize exposure to sunlight.</p>
<h2>The constant ‘e’</h2>
<p>February offers another blockbuster constant, <a href="https://rdcu.be/c6V6z">Euler’s number e</a>, which has the value 2.718…. So mark next Feb. 7 for the shindig.</p>
<p>To understand e, consider “doubling” growth again, but now in terms of the “population” of dollars in your bank account. By some miracle, your money in this example is earning you 100% interest, compounded each year. Each $1 invested becomes $2 at year’s end.</p>
<p>Suppose, however, the interest is compounded semiannually. Then 50% of the interest is credited midyear, giving you $1.50. You get the remaining 50% interest on this $1.50 at the end of the year, which works out to $0.75, giving you $2.25 ($1.50 + $0.75). So your investment gets multiplied by 2.25, rather than 2.</p>
<p>What if a war broke out between banks, each offering to compound the same 100% interest over shorter and more frequent intervals? Would the sky be the limit in terms of your payout? The answer is no. You could raise your growth ratio from 2 to about 2.718 – more precisely, to e – but <a href="https://www.quercusbooks.co.uk/titles/tony-crilly/50-maths-ideas-you-really-need-to-know/9781848667419/">no higher</a>. Although you get more frequent credits, they have progressively diminishing returns.</p>
<p><iframe id="jClwn" class="tc-infographic-datawrapper" src="https://datawrapper.dwcdn.net/jClwn/2/" height="400px" width="100%" style="border: none" frameborder="0"></iframe></p>
<p>In the late 17th century, the <a href="https://www.stevenstrogatz.com/books/infinite-powers">discovery of calculus</a> led to a quantum leap in people’s ability to grapple with the universe. Math could now analyze anything that changed – which extended its domain to most phenomena in nature. The constant e is famous because of its <a href="https://mathworld.wolfram.com/e.html">iconic role in calculus</a>: It turns out to be the most natural growth factor to track change. Consequently, it shows up in laws describing many natural processes - from <a href="https://www.nature.com/scitable/knowledge/library/how-populations-grow-the-exponential-and-logistic-13240157/">population growth</a> to <a href="https://doi.org/10.1103/PhysRev.44.654">radioactive decay</a>.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/AAir4vcxRPU?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">The constant e is a big part of calculus – and turns up in all kinds of natural phenomena.</span></figcaption>
</figure>
<p>Next on our calendar of mathematical constants would come pi, of course, for March. My nominee for April is <a href="https://mathworld.wolfram.com/FeigenbaumConstant.html">Feigenbaum’s constant delta</a>, which equals 4.669… and measures how quickly growth processes spin off into chaos. </p>
<p>I’ll wait for my first batch to achieve official holiday status before going any further – happy to consider any candidates <a href="https://www.manilsuri.com/about">you want to nominate</a>.</p><img src="https://counter.theconversation.com/content/200046/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Manil Suri does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Pi gets a lot of attention this time of year, but there are plenty of other mathematical constants just as deserving of recognition.Manil Suri, Professor of Mathematics and Statistics, University of Maryland, Baltimore CountyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1788302022-03-11T16:02:34Z2022-03-11T16:02:34ZPi day: a brief history of our fascination with this magical number, from pies to ‘piems’<figure><img src="https://images.theconversation.com/files/450999/original/file-20220309-25-1gg0u7a.jpg?ixlib=rb-1.1.0&rect=0%2C0%2C5326%2C3476&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/pi-day-cherry-pie-homemade-traditional-1297494592">Oksana Mizina/Shutterstock</a></span></figcaption></figure><p>Imagine a cup of tea. Wrap a piece of string around the circumference of the cup, and measure the length of the string. Then, lay your spoon on top of the cup, making sure it lies across the centre of the cup, and measure the length from side to side – the diameter. Finally, divide the circumference by the diameter, and record the result. Next time you eat soup, repeat the process with the bowl. </p>
<p>You will discover that the ratios of the circumference to the diameter in both cases are remarkably close to each other. If you decide to experiment with other circular shapes, you will find that no matter how large or small the objects are, as long as they are round, the ratios will all be very close to 3.14. You just stumbled upon a universal law of circular objects.</p>
<p>The Greek letter pi (π) <a href="https://www.scientificamerican.com/article/what-is-pi-and-how-did-it-originate/">was introduced</a> in 1706 to denote that constant ratio between the circumference of a circle to its diameter. But the fascination with the number pi goes back millennia.</p>
<p>While pi exists through the constancy of the result of dividing circumference by diameter for all circles, it’s important to note that this constancy is not quite as universal as the ancient Greeks thought. For circles drawn on curved surfaces, such as the spherical surface of Earth, the division is <a href="https://physics.illinois.edu/news/34508">not constant at all</a>, and pi ceases to exist. </p>
<p>Flat geometry, also known as Euclidean geometry, is the universe of mathematical objects where pi exists. The ancient Greeks only studied flat geometry and so for them the constant pi was truly a universal wonder whose precise value they sought to pinpoint. Being just a simple ratio, how hard could it be?</p>
<p>Archimedes <a href="https://www.pbs.org/wgbh/nova/physics/approximating-pi.html">placed</a> pi between 223/71 and 22/7, so between 3.140 and 3.142, while Ptolemy found the <a href="https://mathshistory.st-andrews.ac.uk/HistTopics/Pi_through_the_ages/">first approximation</a> correct to three decimal places: 3.141. Improvements to seven decimal places were achieved by Chinese mathematicians in the 5th century AD, based on a new technique discovered in the 3rd century by mathematician and writer <a href="https://medium.com/@joshuafitzgerald/this-is-not-a-circle-the-extraordinary-algorithm-of-liu-hui-20a90fecd06e">Liu Hui</a>. A millennium would pass before further significant advances led the 14th century Indian mathematician <a href="https://mathshistory.st-andrews.ac.uk/Biographies/Madhava/">Madhava of Sangamagrama</a> to reach 11 decimal places. </p>
<p>Progress accelerated with better analytical tools. With modern computers, as of August 2021, the record stands at <a href="https://www.theguardian.com/science/2021/aug/16/swiss-researchers-calculate-pi-to-new-record-of-628tn-figures">62.8 trillion digits</a>. </p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/3-14-essential-reads-about-for-pi-day-74022">3.14 essential reads about π for Pi Day</a>
</strong>
</em>
</p>
<hr>
<p>As the knowledge of the digits of pi expanded, people tried to detect a pattern. A simple rule to describe all digits in one go, or to pinpoint pi, as the ancient Greeks had hoped to do. However, in the 1760s, the French-Swiss mathematician <a href="https://www.mathscareers.org.uk/pi-johann-lambert/">Johann Heinrich Lambert</a> proved that the decimal expansion of pi does not follow any simple rule for its digits; pi is irrational, meaning that its decimal expansion does not repeat or terminate. </p>
<p>The number pi is simple to define and captures a fundamental geometric fact. At the same time, computing it was a challenge for some of the best mathematicians who ever lived. In a sense, it can never be fully captured by simple computations. These factors contribute to the allure of pi and are perhaps a source of its ongoing influence on our culture.</p>
<figure class="align-center ">
<img alt="Pi written out on a blackboard." src="https://images.theconversation.com/files/451013/original/file-20220309-23-1w7tnov.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/451013/original/file-20220309-23-1w7tnov.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/451013/original/file-20220309-23-1w7tnov.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/451013/original/file-20220309-23-1w7tnov.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/451013/original/file-20220309-23-1w7tnov.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/451013/original/file-20220309-23-1w7tnov.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/451013/original/file-20220309-23-1w7tnov.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">There’s no simple pattern to pi.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/number-pi-written-chalk-on-blackboard-1601840431">Kuki Ladron de Guevara/Shutterstock</a></span>
</figcaption>
</figure>
<h2>Pies and poems</h2>
<p>The impact of pi on society can be gleaned from the fact it has its very own day, and its presence in poetry, among other cultural manifestations. </p>
<p><a href="https://www.piday.org/">Pi Day</a> falls on March 14, or 3/14 according to the American dating system. It’s celebrated by the recitation of pi’s digits and the enjoyment of round pies. </p>
<p>As profound as eating pies is, pi has generated an entire literary style. Consider the following poem:</p>
<blockquote>
<p>Pie</p>
<p>I wish I could determine pi</p>
<p>Eureka, cried the great inventor</p>
<p>Christmas pudding, Christmas pie</p>
<p>Is the problem’s very centre.</p>
</blockquote>
<p>If you count the letters in each word, you’ll get 3.14159265358979323846, which is pi correct to 20 decimal places. This is an example of a “piem”. There are many more piems in English, as well as in various other languages. A piem can be both a mnemonic device (a linguistic tool to help us remember something) and an artistic endeavour. </p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/pi-and-its-part-in-the-most-beautiful-formula-in-mathematics-56067">Pi and its part in the most beautiful formula in mathematics</a>
</strong>
</em>
</p>
<hr>
<p>This literary style, where the number of letters in consecutive words is dictated by the decimal expansion of pi, is called “<a href="http://www.cadaeic.net/pilish.htm">Pilish</a>”, and has been around since the early 1900s. Beyond short poems, Pilish has given rise to longer pieces of prose, and even <a href="http://www.cadaeic.net/notawake.htm">an entire novel</a>. Author Michael Keith indicates the first 10,000 digits of pi through a book, titled <a href="https://www.amazon.co.uk/Not-Wake-embodying-digits-decimals-ebook/dp/B0077QIOE4">Not A Wake</a> (as you can see, its title encodes 3.14). </p>
<p>If you’re feeling creative and want to test your Pilish, here is a <a href="https://valhallaconsulting.com.au/PilishChecker.html">Pilish checker</a> you can use. Alternatively, eating pie isn’t a bad way to celebrate all things fascinating about pi.</p><img src="https://counter.theconversation.com/content/178830/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Ittay Weiss does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Pi has spawned its own literary style, where the number of letters in consecutive words is dictated by the decimal expansion of pi.Ittay Weiss, Senior Lecturer in Mathematics, University of PortsmouthLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1483992021-03-11T19:50:35Z2021-03-11T19:50:35ZPi Day: Celebrating the life-changing role of math programs in prisons<figure><img src="https://images.theconversation.com/files/388184/original/file-20210308-17-ydqq7u.jpg?ixlib=rb-1.1.0&rect=26%2C0%2C4466%2C2748&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">University study programs in prisons can increase inmates' chances of success after release.</span> <span class="attribution"><span class="source">(Shutterstock)</span></span></figcaption></figure><p>March 14 is recognized as <a href="https://www.piday.org/">global Pi Day</a>. Pi — represented by the Greek letter π — is the ratio of the circumference of a circle to its diameter, which is approximately 3.14159 (hence the 14th day of the third month 3/14). </p>
<p>In 2017, Pi Day was celebrated at the <a href="https://www.doc.wa.gov/news/2017/03172017.htm">Monroe Correctional Center</a> in Washington state, where prisoners who were math enthusiasts were joined by mathematicians. It was the first event organized by the Prison Mathematics Project (PMP), an organization started by Christopher Havens to share his passion for math with fellow inmates.</p>
<p>This year, the PMP’s Pi Day celebrations will be <a href="https://www.prisonmathproject.org/">virtual</a> with mathematicians from all over the world coming together and sharing their knowledge and passion with a growing community of prisoners who love mathematics.</p>
<h2>Prison maths</h2>
<p>Christopher Havens was serving a 25-year sentence for murder when he was sent into solitary confinement. That’s when his life began to change. Havens started solving simple math problems handed to him by a volunteer, and discovered a gift for mathematics. He reached out to mathematicians for membership, and Havens was mentored first by my mother and then my father, who are both mathematicians. And in February 2020, Havens published his first <a href="https://doi.org/10.1007/s40993-020-0187-5">mathematics article</a>. </p>
<p>I wrote about <a href="https://theconversation.com/an-inmates-love-for-math-leads-to-new-discoveries-130123">Havens’ incredible story</a> for <em>The Conversation</em> in May 2020. And I’m writing today to let you know what happened after that. </p>
<p>The article went viral: It was republished on hundreds of websites, and Havens and I were interviewed by <a href="https://globalnews.ca/video/7065529/prisoner-finds-second-chance-through-math">Global News</a> and the <a href="https://www.cbc.ca/radio/thecurrent/the-current-for-oct-15-2020-1.5763055/how-a-convicted-killer-s-passion-for-math-inspired-him-to-change-his-life-and-others-1.5753905">CBC</a>. </p>
<p>In addition to all this media attention, we received dozens of emails from people wanting to help. Many were mathematicians. Some of them started mentoring Havens and are now <a href="https://gogetfunding.com/support-christopher-havens-journey-to-become-a-mathematician-outside-prison/">supporting him in other ways</a>. </p>
<h2>Going national</h2>
<p>One of the people who reached out was Walker Blackwell, who wrote to Havens saying he wanted to help him transform the PMP into a national non-profit organization to help inmates all over the United States. Havens was enthused, until he found out Blackwell’s age: 15. </p>
<p>Havens’ initial hesitation changed as he considered Blackwell’s age:</p>
<blockquote>
<p>You can imagine how somebody in my situation might be hesitant. Immediately I thought of whether his parents knew that their son was reaching out to a man in prison for the worst possible crime imaginable. This whole scenario made me remember how it feels being in a position where people judge you or make decisions based on circumstance. Being incarcerated, people often do not take me seriously. I imagined this to be precisely what Walker was going through with his age. And so right then, not only did he inspire me, but I related to him — the only thing left to do was to make sure his parents were aware of our communications.</p>
</blockquote>
<p>Blackwell’s parents were aware, and the collaboration started. </p>
<p>The PMP now has an official <a href="https://www.prisonmathproject.org/">website</a>. Its goal is to pair prisoners interested in mathematics with volunteer academic mathematicians. The volunteers put together a “math pack” for the inmate and correspond with them, like Havens did with my mother.</p>
<p>The PMP now has 20 volunteers and 15 prisoners already matched. The <a href="https://www.maa.org/">Mathematical Association of America</a> has pledged to distribute their flagship popular math magazine, <a href="https://www.maa.org/press/periodicals/math-horizons"><em>Math Horizons</em></a>, to inmates participating in the program. Havens hopes to be able to negotiate free versions of mathematics software for the PMP.</p>
<p>Havens also started a local project with the help of Kristen Morgan, the associate dean of corrections education for his prison. After the earlier article prompted an administrator with the Department of Corrections to reach out to Morgan, she decided to start a program with Havens called “The Social Mathematics Experiment.”</p>
<p>In this new program, inmates and staff meet together weekly to discuss anything from technical math problems to more general topics about education and self-rehabilitation. Learning math becomes a spark for personal transformation, more than just an academic achievement. </p>
<h2>The value of prison education</h2>
<p>There’s <a href="https://issuu.com/uclapubaffairs/docs/correctional_education_as_a_crime_c">tons</a> of <a href="https://www.rand.org/pubs/research_reports/RR266.html">evidence</a> about <a href="http://www.unodc.org/pdf/criminal_justice/Handbook_on_Crime_Prevention_Guidelines_-_Making_them_work.pdf">how</a> <a href="https://prisonstudiesproject.org/why-prison-education-programs/">education</a> in prison <a href="https://www.americanprogress.org/issues/education-k-12/news/2018/03/02/447321/education-opportunities-prison-key-reducing-crime/">decreases</a> <a href="https://www.jstor.org/stable/23282764">recidivism</a>. And yet, when Havens arrived in his current prison, he could not get into a degree program because of a policy stating that only people closer to being released would be able to enter. </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/XBVMjVXAMYw?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">A look at how completing a college degree means a drastically lower risk of returning to prison.</span></figcaption>
</figure>
<p>In my home province of Québec, <a href="https://www.securitepublique.gouv.qc.ca/fileadmin/Documents/services_correctionnels/publications/plan_action_2010-2013.pdf">80 per cent of prisoners did not finish high school</a>. While the Québec government has offered some form of education to their prisoners in the last two decades, the means put into the programs are simply not enough to cover the <a href="https://www.cmv-educare.com/wp-content/uploads/2011/06/%C3%89tude-exploratoire-%C3%A9tablissements-de-d%C3%A9tention-rapport-final.pdf">needs</a>. And the situation is not any better in the <a href="http://www.intersectionalanalyst.com/intersectional-analyst/2017/7/20/everything-you-were-never-taught-about-canadas-prison-systems">rest of Canada</a>.</p>
<p>One of the few Canadian programs that I am aware of that links post-secondary educational institutions and prisons is <a href="http://wallstobridges.ca/">Walls to Bridges</a>, launched by Wilfrid Laurier University in 2011. This program sees for-credit courses held inside prisons and attended by both regular “outside” students and inmates, creating bridges that become transformative for everyone involved. So far, courses are mostly taught by professors based in the humanities and social sciences. </p>
<p>Havens’ story is a call to all of us in sciences and engineering to become more aware of the transformative power of our own disciplines. While I hope that our governments take concrete steps to make more educational programs available in prisons, I also wish that more people would become involved.</p><img src="https://counter.theconversation.com/content/148399/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Marta Cerruti does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Christopher Havens is a prison inmate serving time for murder. He’s also a mathematics whiz who’s advocating for more math in prison as a way to improve the chances of prisoners after release.Marta Cerruti, Associate Professor, Materials Engineering, McGill UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/556882016-03-14T10:06:19Z2016-03-14T10:06:19ZPi pops up where you don’t expect it<figure><img src="https://images.theconversation.com/files/114522/original/image-20160309-13712-g9bqus.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Pi is at the center of all circles.</span> <span class="attribution"><a class="source" href="https://upload.wikimedia.org/wikipedia/commons/8/8c/Matheon2.jpg">Holger Motzkau</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>Happy Pi Day, where we celebrate the world’s most famous number. The exact value of π=3.14159… has fascinated people since ancient times, and mathematicians have computed <em>trillions</em> of digits. But why do we care? Would it actually matter if somebody got the 11,137,423,895,285th digit wrong?</p>
<p>Probably not. The world would keep on turning (with a circumference of 2πr). What matters about π isn’t so much the actual value as the <em>idea</em>, and the fact that π seems to crop up in lots of unexpected places.</p>
<p>Let’s start with the expected places. If a circle has radius r, then the circumference is 2πr. So if a circle has radius of one foot, and you walk around the circle in one-foot steps, then it will take you 2π = 6.28319… steps to go all the way around. Six steps isn’t nearly enough, and after seven you will have overshot. And since the value of π is irrational, no multiple of the circumference will be an even number of steps. No matter how many times you take a one-foot step, you’ll never come back exactly to your starting point.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/114518/original/image-20160309-13712-1om653l.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114518/original/image-20160309-13712-1om653l.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114518/original/image-20160309-13712-1om653l.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=650&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114518/original/image-20160309-13712-1om653l.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=650&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114518/original/image-20160309-13712-1om653l.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=650&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114518/original/image-20160309-13712-1om653l.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=817&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114518/original/image-20160309-13712-1om653l.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=817&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114518/original/image-20160309-13712-1om653l.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=817&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Calculating the area of a circle with wedges.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:CircleArea.svg">Jim.belk</a></span>
</figcaption>
</figure>
<p>From the circumference of a circle we get the area. Cut a pizza into an even number of slices, alternately colored yellow and blue. Lay all the blue slices pointing up, and all the yellow slices pointing down. Since each color accounts for half the circumference of the circle, the result is approximately a strip of height r and width πr, or area πr<sup>2</sup>. The more slices we have, the better the approximation is, so the exact area must be <em>exactly</em> πr<sup>2</sup>. </p>
<h2>Pi in other places</h2>
<p>You don’t just get π in circular motion. You get π in <em>any</em> oscillation. When a mass bobs on a spring, or a pendulum swings back and forth, the position behaves just like one coordinate of a particle going around a circle. </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/9r0HexjGRE4?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Simple harmonic motion is another view of circular motion.</span></figcaption>
</figure>
<p>If your maximum displacement is one meter and your maximum speed is one meter/second, it’s just like going around a circle of radius one meter at one meter/second, and your period of oscillation will be exactly 2π seconds.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/114722/original/image-20160310-26261-66iv16.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114722/original/image-20160310-26261-66iv16.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114722/original/image-20160310-26261-66iv16.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=480&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114722/original/image-20160310-26261-66iv16.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=480&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114722/original/image-20160310-26261-66iv16.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=480&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114722/original/image-20160310-26261-66iv16.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=603&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114722/original/image-20160310-26261-66iv16.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=603&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114722/original/image-20160310-26261-66iv16.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=603&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The area of the space under the normal-distribution curve is the square root of pi.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:E%5E(-x%5E2).svg">Autopilot</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>Pi also crops up in probability. The function
f(x)=e<sup>-x²</sup>, where e=2.71828… is Euler’s number, describes the most common probability distribution seen in the real world, governing everything from SAT scores to locations of darts thrown at a target. The area under this curve is exactly the square root of π. </p>
<p>How did π get into it?! The two-dimensional function f(x)f(y) <a href="https://www.google.com/#q=z+%3D+exp(-(x%5E2%2By%5E2">stays the same if you rotate the coordinate axes</a>. Round things relate to circles, and circles involve π. </p>
<p>Another place we see π is in the calendar. A normal 365-day year is just over 10,000,000π seconds. Does that have something to do with the Earth going around the sun in a nearly circular orbit? Actually, no. It’s just coincidence, thanks to our arbitrarily dividing each day into 24 hours, each hour into 60 minutes, and each minute into 60 seconds.</p>
<p>What’s <em>not</em> coincidence is how the length of the day varies with the seasons. If you plot the hours of daylight as a function of the date, starting at next week’s equinox, you get the same sine curve that describes the position of a pendulum or one coordinate of circular motion.</p>
<h2>Advanced appearances of π</h2>
<p>More examples of π come up in calculus, especially in
infinite series like <br>
1 - (<sup>1</sup>⁄<sub>3</sub>) + (<sup>1</sup>⁄<sub>5</sub>) - (<sup>1</sup>⁄<sub>7</sub>) + (<sup>1</sup>⁄<sub>9</sub>) + ⋯ = π/4<br>
and <br>
1<sup>2</sup> + (<sup>1</sup>⁄<sub>2</sub>)<sup>2</sup> + (<sup>1</sup>⁄<sub>3</sub>)<sup>2</sup> + (<sup>1</sup>⁄<sub>4</sub>)<sup>2</sup> + (<sup>1</sup>⁄<sub>5</sub>)<sup>2</sup> + ⋯ = π<sup>2</sup>/6<br>
(The first comes from the <a href="http://mathworld.wolfram.com/TaylorSeries.html">Taylor series</a> of the arctangent of 1, and the second from the <a href="http://mathworld.wolfram.com/FourierSeries.html">Fourier series</a> of a sawtooth function.) </p>
<p>Also from calculus comes Euler’s <a href="https://www.math.toronto.edu/mathnet/questionCorner/epii.html">mysterious equation</a>
<br>
e<sup>iπ</sup> + 1 = 0
<br>
relating the five most important numbers in mathematics: 0, 1, i, π, and e, where i is the (imaginary!) square root of -1.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/114520/original/image-20160309-13712-hc11ya.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114520/original/image-20160309-13712-hc11ya.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114520/original/image-20160309-13712-hc11ya.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114520/original/image-20160309-13712-hc11ya.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114520/original/image-20160309-13712-hc11ya.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114520/original/image-20160309-13712-hc11ya.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114520/original/image-20160309-13712-hc11ya.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114520/original/image-20160309-13712-hc11ya.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A graph of the exponential function y=e^x.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Exp.svg">Peter John Acklam</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>At first this looks like nonsense. How can you possibly take a number like e to an imaginary power?! Stay with me. The rate of change of the exponential function f(x)=e<sup>x</sup> is equal to the value of the function itself. To the left of the figure, where the function is small, it’s barely changing. To the right, where the function is big, it’s changing rapidly. Likewise, the rate of change of any function of the form f(x)=e<sup>ax</sup> is proportional to e<sup>ax</sup>.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/114521/original/image-20160309-13693-16ccfxu.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114521/original/image-20160309-13693-16ccfxu.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114521/original/image-20160309-13693-16ccfxu.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=582&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114521/original/image-20160309-13693-16ccfxu.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=582&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114521/original/image-20160309-13693-16ccfxu.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=582&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114521/original/image-20160309-13693-16ccfxu.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=732&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114521/original/image-20160309-13693-16ccfxu.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=732&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114521/original/image-20160309-13693-16ccfxu.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=732&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The relationship between an angle, its sine, cosine and a circle.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Sin-cos-defn-I.png">345Kai</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>We can then <em>define</em> f(x)= e<sup>ix</sup> to be a complex function whose rate of change is i times the function itself, and whose value at 0 is 1. This turns out to be a combination of the trigonometric functions that describe
circular motion, namely cos(x) + i sin(x). Since going a distance π takes you halfway around the unit circle, cos(π)=-1 and sin(π)=0, so e<sup>iπ</sup>=-1. </p>
<p>Finally, some people prefer to work with τ=2π=6.28… instead of π. Since going a distance 2π takes you all the way around the circle, they would write that e<sup>iτ</sup> = +1. If you find that confusing, take a few months to think about it. Then you can celebrate June 28 by baking <em>two</em> pies.</p><img src="https://counter.theconversation.com/content/55688/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Lorenzo Sadun has received funding from the National Science Foundation. </span></em></p>We know pi appears when we talk about circles. But it appears in many other places, too. Why, pi, why?Lorenzo Sadun, Professor of Mathematics, The University of Texas at AustinLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/559172016-03-14T06:22:11Z2016-03-14T06:22:11ZHow a farm boy from Wales gave the world pi<figure><img src="https://images.theconversation.com/files/114842/original/image-20160311-11299-1u4nzyh.jpeg?ixlib=rb-1.1.0&rect=0%2C0%2C2000%2C1230&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Maths pi-oneer</span> <span class="attribution"><span class="source">William Hogarth/National Portrait Gallery</span></span></figcaption></figure><p>One of the most important numbers in maths might today be named after the Greek letter π or “pi”, but the convention of representing it this way actually doesn’t come from Greece at all. It comes from the pen of an 18th century farmer’s son and largely self-taught mathematician from the small island of Anglesey in Wales. The Welsh Government has even renamed <a href="https://theconversation.com/prepared-for-pi-day-this-year-its-a-once-in-a-century-celebration-38576">Pi Day</a> (on March 14 or 3/14, which matches the first three digits of pi, 3.14) as “<a href="http://www.walesonline.co.uk/special-features/pi-day-cymru-involved-wales-8818765">Pi Day Cymru</a>”.</p>
<p>The importance of the number we now call pi has been known about since <a href="https://www.math.rutgers.edu/%7Echerlin/History/Papers2000/wilson.html">ancient Egyptian times</a>. It <a href="https://theconversation.com/pi-day-is-silly-but-itself-is-fascinating-and-universal-37948">allows you to calculate</a> the circumference and area of a circle from its diameter (and vice versa). But it’s also a number that crops up across all <a href="https://cosmologyandspace.wordpress.com/2015/03/13/on-pi-day-how-scientists-use-this-number/">scientific disciplines</a> from cosmology to thermodynamics. Yet even after mathematicians worked out how to calculate pi accurately to over 100 decimal places at the start of the 18th century, we didn’t have an agreed symbol for the number.</p>
<h2>From accountant to maths pioneer</h2>
<p>This all changed thanks to <a href="http://www.historytoday.com/patricia-rothman/william-jones-and-his-circle-man-who-invented-pi">William Jones</a> who was born in 1674 in the parish of Llanfihangel Tre’r Beirdd. After attending a charity school, Jones landed a job as a merchant’s accountant and then as a maths teacher on a warship, before publishing A New Compendium of the Whole Art of Navigation, his first book in 1702 on the mathematics of navigation. On his return to Britain he began to teach maths in London, possibly starting by holding classes in coffee shops for a small fee.</p>
<p>Shortly afterwards he published <a href="https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics"><em>Synopsis palmariorum matheseos</em></a>, a summary of the current state of the art developments in mathematics which reflected his own particular interests. In it is the first recorded use of the symbol π as the number that gives the ratio of a circle’s circumference to its diameter.</p>
<p>We typically think of this number as being about 3.14, but Jones rightly suspected that the digits after its decimal point were infinite and non-repeating. This meant it could never be “expressed in numbers”, as he put it. That was why he recognised the number needed its own symbol. It is <a href="http://www.historytoday.com/patricia-rothman/william-jones-and-his-circle-man-who-invented-pi">commonly thought</a> that he chose pi either because it is the first letter of the word for periphery (περιφέρεια) or because it is the first letter of the word for perimeter (περίμετρος), or both.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/114843/original/image-20160311-11288-qj1qsk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/114843/original/image-20160311-11288-qj1qsk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=286&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114843/original/image-20160311-11288-qj1qsk.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=286&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114843/original/image-20160311-11288-qj1qsk.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=286&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114843/original/image-20160311-11288-qj1qsk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=360&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114843/original/image-20160311-11288-qj1qsk.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=360&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114843/original/image-20160311-11288-qj1qsk.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=360&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Finding pi.</span>
<span class="attribution"><span class="source">_Synopsis palmariorum matheseos_</span></span>
</figcaption>
</figure>
<p>In the pages of his <em>Synopsis</em>, Jones also showed his familiarity with the notion of an infinite series and how it could help <a href="http://www.geom.uiuc.edu/%7Ehuberty/math5337/groupe/expresspi.html">calculate pi</a> far more accurately than was possible just by drawing and measuring circles. An infinite series is the total of all the numbers in a sequence that goes on forever, for example ½ + ¼ + ⅛ + and so on. Adding an infinite sequence of ever-smaller fractions like this can bring you closer and closer to a number with an infinite number of digits after the decimal point - just like pi. So by defining the right sequence, mathematicians were able to calculate pi to an increasing number of decimal places.</p>
<p>Infinite series also assist our understanding of rational numbers, more commonly referred to as fractions. Irrational numbers are the ones, <a href="http://theconversation.com/pi-might-look-random-but-its-full-of-hidden-patterns-55994">like pi</a>, that can’t be written as a fraction, which is why Jones decided it needed its own symbol. What he wasn’t able to do was prove with maths that the digits of pi definitely were infinite and non-repeating and so that the number was truly irrational. This would eventually be achieved in 1768 by the French mathematician <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Lambert.html%5D">Johann Heinrich Lambert</a>. Jones dipped his toes into the subject and showed an intuitive grasp of the complexity of pi but lacked the analytical tools to enable him to develop his ideas further.</p>
<h2>Scientific success</h2>
<p>Despite this - and his obscure background - Jones’s book was a success and led him to become an <a href="http://www.historytoday.com/patricia-rothman/william-jones-and-his-circle-man-who-invented-pi">important and influential</a> member of the scientific establishment. He was noticed and befriended by two of Britain’s foremost mathematicians - Edmund Halley and Sir Isaac Newton - and was elected a fellow of the Royal Society in 1711. He later became the editor and publisher of many of Newton’s manuscripts and built up an extraordinary library that was one of the greatest collections of books on science and mathematics ever known, and only recently fully dispersed.</p>
<p>Despite this success, the use of the symbol π spread slowly at first. It was popularised in 1737 by the Swiss mathematician Leonhard Euler (1707–83), one of the most eminent mathematicians of the 18th century, who likely came across Jones’ work while studying Newton at the University of Basel. His endorsement of the symbol in his own work ensured that it received wide publicity, yet even then the symbol wasn’t adopted universally until as late as 1934. Today π is instantly recognised worldwide but few know that its history can be traced back to a small village in the heart of Anglesey.</p><img src="https://counter.theconversation.com/content/55917/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Gareth Ffowc Roberts does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>This Pi Day we should celebrate William Jones, the 18th century Welsh farm boy who named the mysterious number.Gareth Ffowc Roberts, Emeritus Professor of Education, Bangor UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/559942016-03-14T06:22:10Z2016-03-14T06:22:10ZPi might look random but it’s full of hidden patterns<figure><img src="https://images.theconversation.com/files/114837/original/image-20160311-11274-hcx64e.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>After thousands of years of trying, mathematicians are still working out the number known as pi or “π”. We typically think of pi as approximately 3.14 but the <a href="http://www.numberworld.org/y-cruncher/">most successful attempt</a> to calculate it more precisely worked out its value to over 13 trillion digits after the decimal point. We have known since the 18th century that we will never be able to calculate all the digits of pi because it is an irrational number, one that continues forever without any repeating pattern.</p>
<p>In 1888, the logician John Venn, who also invented the Venn diagram, attempted to visually show that the digits of pi were random by drawing a graph showing the first 707 decimal places. He assigned a compass point to the digits 0 to 7 and then drew lines to show the path indicated by each digit.</p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"577410531256115200"}"></div></p>
<p>Venn did this work using pen and paper but this is still used today with modern technology to create even more detailed and beautiful patterns.</p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"365016946850267136"}"></div></p>
<p>But, despite the endless string of unpredictable digits that make up pi, it’s not what we call a truly random number. And it actually contains all sorts of surprising patterns.</p>
<h2>Normal not random</h2>
<p>The reason we can’t call pi random is because the digits it comprises are precisely determined and fixed. For example, the second decimal place in pi is always 4. So you can’t ask what the probability would be of a different number taking this position. It isn’t randomly positioned. </p>
<p>But we can ask the related question: “<a href="http://pi314.at/math/normal.html">Is pi a normal number</a>?” A decimal number is said to be normal when every sequence of possible digits is equally likely to appear in it, making the numbers look random even if they technically aren’t. By looking at the digits of pi and applying statistical tests you can try to determine if it is normal. From the tests performed so far, it is still an open question whether pi is normal or not.</p>
<p>For example in 2003, <a href="http://www.super-computing.org">Yasumasa Kanada</a> published the distribution of the number of times different digits appear in the first trillion digits of pi:</p>
<pre class="highlight plaintext"><code> Digit Occurrences
0 99,999,485,134
1 99,999,945,664
2 100,000,480,057
3 99,999,787,805
4 100,000,357,857
5 99,999,671,008
6 99,999,807,503
7 99,999,818,723
8 100,000,791,469
9 99,999,854,780
Total 1,000,000,000,000
</code></pre>
<p>His results imply that these digits seem to be fairly evenly distributed, but it is not enough to prove that all of pi would be normal. </p>
<h2>Every sequence</h2>
<p>We need to remember the surprising fact that if pi was normal then any finite sequence of digits you could name could be found in it. For example, at position 768 in the pi digits there are six 9s in succession. The chance of this happening if pi is normal and every sequence of <em>n</em> digits is equally likely to occur, is 0.08%.</p>
<p>This block of nines is famously called the “Feynman Point” after the Nobel Prize-winner Richard Feynman. He once jokingly claimed that if he had to recite pi digits he would name them up to this point and then say “<a href="http://www.feynman.com/science/qed-lectures-in-new-zealand/">and so on</a>”.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/goIOWcnagP0?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
</figure>
<p>Other interesting sequences of digits have also been found. At position 17,387,594,880 you find the sequence 0123456789, and surprisingly earlier at position 60 you find these ten digits in a scrambled order.</p>
<p>Pi-hunters search for dates of birth and other significant personal numbers in pi asking the question: “Where do I occur in the pi digits?” If you want to test to see where your own special numbers are in pi, then you can do so by using the free online software called <a href="http://www.angio.net/pi/">Pi birthdays</a>.</p><img src="https://counter.theconversation.com/content/55994/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Steve Humble does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Mathematicians have long been revealing the beauty in the one of nature’s most mysterious numbers.Steve Humble, Mathematics Education Primary and Secondary PGCE, Newcastle UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/560672016-03-13T19:15:49Z2016-03-13T19:15:49ZPi and its part in the most beautiful formula in mathematics<figure><img src="https://images.theconversation.com/files/114546/original/image-20160309-4478-zq6qhh.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Pi has an interesting relationship with some other unique constants in mathematics.</span> <span class="attribution"><span class="source">Shutterstock/Shawn Hempel</span></span></figcaption></figure><p><a href="http://www.piday.org/">Pi Day</a> is upon us again, for those who note today’s date in the format 3/14 (March 14). But rather than talk about Pi Day itself, as <a href="https://theconversation.com/prepared-for-pi-day-this-year-its-a-once-in-a-century-celebration-38576">I did last year</a>, this year I want to talk about Pi and mathematical notions of <a href="http://www.huffingtonpost.com/david-h-bailey/why-mathematics-matters_b_4794617.html">beauty</a>.</p>
<p>How better to do so than to talk about the 18th century European scholar <a href="http://www.biography.com/people/leonhard-euler-21342391">Leonard Euler’s</a> famous formula:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/114596/original/image-20160310-31880-wlggpb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/114596/original/image-20160310-31880-wlggpb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=95&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114596/original/image-20160310-31880-wlggpb.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=95&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114596/original/image-20160310-31880-wlggpb.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=95&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114596/original/image-20160310-31880-wlggpb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=119&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114596/original/image-20160310-31880-wlggpb.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=119&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114596/original/image-20160310-31880-wlggpb.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=119&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Euler’s beautiful formula. Note that e is the base for the natural logarithm and i is the symbol for the square root of -1, explained later.</span>
<span class="attribution"><span class="source">The Conversation</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Often described as “<a href="http://www.gresham.ac.uk/lectures-and-events/the-most-beautiful-formula-in-mathematics">the most beautiful formula in mathematics</a>”, Euler seems never to have actually written it down – naming conventions in mathematics are a bit dodgy. Rather, it is a special case of Euler’s discovery that exponential growth and circular motion are equivalent, given by the following formula:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/114599/original/image-20160310-26261-hxpsgc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/114599/original/image-20160310-26261-hxpsgc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=95&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114599/original/image-20160310-26261-hxpsgc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=95&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114599/original/image-20160310-26261-hxpsgc.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=95&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114599/original/image-20160310-26261-hxpsgc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=119&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114599/original/image-20160310-26261-hxpsgc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=119&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114599/original/image-20160310-26261-hxpsgc.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=119&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">This formula is often referred to as cis, combining cos and sin together, and θ is the Greek symbol Theta.</span>
<span class="attribution"><span class="source">The Conversation</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>The US theoretical physicist <a href="http://www.livescience.com/51399-eulers-identity.html">Richard Feynman called</a> this “the most remarkable formula in mathematics”.</p>
<p>Ed Sandifer, a founder of the <a href="http://www.eulersociety.org/">Euler Society</a>, has a lovely <a href="http://eulerarchive.maa.org/hedi/HEDI-2007-08.pdf">2007 article discussing in detail</a> Euler’s approaches – over 40 years – to show how the formula (above) worked.</p>
<p>I shall try to get the story of this formula across with very few more symbols.</p>
<h2>What the formula does</h2>
<p>Euler’s formula involves five fundamental constants: 0, 1, i, e, and Pi, and on adding equality, addition and exponentiation, combines them into a seven-symbol word in a mysterious and useful way.</p>
<p>Equivalently, it can also be written:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/114602/original/image-20160310-26261-dpog3q.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/114602/original/image-20160310-26261-dpog3q.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=95&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114602/original/image-20160310-26261-dpog3q.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=95&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114602/original/image-20160310-26261-dpog3q.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=95&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114602/original/image-20160310-26261-dpog3q.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=119&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114602/original/image-20160310-26261-dpog3q.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=119&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114602/original/image-20160310-26261-dpog3q.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=119&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Euler’s formula again, rewritten.</span>
<span class="attribution"><span class="source">The Conversation</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>This is even more succinct and introduces negative numbers.</p>
<p>It is a common feature of mathematics that discoveries are often first used and only later understood. As the 18th century French mathematician <a href="http://www.britannica.com/biography/Jean-Le-Rond-dAlembert">Jean d'Alembert</a> wrote: “algebra is generous, she often gives more than we ask”.</p>
<p>Let me discuss the 2,000-year history of the building blocks of Euler’s formula. You don’t have to understand the actual mathematics, just gain an appreciation of the origin of the various elements of the formula and how they combine so neatly together.</p>
<h2>Equality (=)</h2>
<p>The “=” symbol is attributed to the Welsh scientist <a href="http://www.britannica.com/biography/Robert-Recorde">Robert Recorde</a> in 1557.</p>
<p>Arguments about the meaning of equality in mathematics have mirrored and driven discussions about definite descriptions in philosophy more generally.</p>
<p>The British logician <a href="http://www.britannica.com/biography/Bertrand-Russell">Bertrand Russell</a>’s famous example is Venus, described as the morning star and as the evening star. An over-discussed example in mathematics is whether 0.99999999… and 1 are equal. They are and they aren’t.</p>
<h2>Zero (0)</h2>
<p>Notions of nothingness and the void or infinity go back much further, but the Greeks and others had not discovered rules to manipulate with “0”.</p>
<p>A mathematically tractable notion of <a href="http://yaleglobal.yale.edu/about/zero.jsp">zero</a> is attributed to the great Indian thinker <a href="http://www.britannica.com/biography/Brahmagupta">Brahmagupta</a> around 650CE.</p>
<p>When married with the other Indian discovery of positional notation, calculation became much more accessible. This ability did not come fully to Europe until the 15th century and later.</p>
<h2>1</h2>
<p>Without “1” there would be no advanced <a href="http://www.britannica.com/topic/Peano-axioms">arithmetic</a>. With “0” and “1”, we also have binary notation and modern digital computers. What the US theoretical physicist John Archibald Wheeler called “<a href="http://www.scientificamerican.com/article/pioneering-physicist-john-wheeler-dies/">it from bit</a>”.</p>
<p>This leads to modern group theory, algebra, cryptography and much, much else.</p>
<h2>i</h2>
<p>The use of imaginary numbers also dates largely from the 16th-17th century. The French philosopher and mathematician <a href="http://www.britannica.com/biography/Rene-Descartes">Rene Decartes</a> used the term disparagingly.</p>
<p>Mathematical concepts we now take for granted sometimes took centuries to be adopted and understood. No wonder school children rebel.</p>
<p>It took Euler and then German mathematician <a href="http://www.britannica.com/biography/Carl-Friedrich-Gauss">Carl Friedrich Gauss</a> to truly exploit imaginary numbers and make the word “imaginary” have a positive mathematical connotation.</p>
<p>Defining “i” as the square root of -1 has the wonderful consequence that a polynomial of degree n has n (complex) roots.</p>
<p>For example x<sup>4</sup>-1 = (x+1) (x-1) (x-i) (x+i), it has four roots. This leads to what is now called complex analysis.</p>
<p>Most of modern mathematics and mathematical physics (such as quantum theory) could not be done without complex analysis.</p>
<h2>Pi (π)</h2>
<p>Pi originates as the area of the circle of radius one or the circumference of a circle of diameter one.</p>
<p>The great Greek mathematician <a href="http://www.britannica.com/biography/Archimedes">Archimedes of Syracuse</a> (287-c212 BCE) used this idea to provide the approximation of 22/7 for Pi (3.141592…).</p>
<p>Euler discovered the modern definition which takes Pi/2 as the smallest positive zero of the cosine function defined by what is known as a <a href="http://www.britannica.com/topic/Taylor-series">Taylor series</a>. This is a bit complicated but if you just think of the series as a very large <a href="http://www.mathsisfun.com/algebra/polynomials.html">polynomial</a> you will get the idea.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/114607/original/image-20160310-26256-1g88ypq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/114607/original/image-20160310-26256-1g88ypq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=138&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114607/original/image-20160310-26256-1g88ypq.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=138&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114607/original/image-20160310-26256-1g88ypq.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=138&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114607/original/image-20160310-26256-1g88ypq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=173&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114607/original/image-20160310-26256-1g88ypq.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=173&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114607/original/image-20160310-26256-1g88ypq.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=173&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Cis as two Taylor series.</span>
<span class="attribution"><span class="source">The Conversation</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Here n! = 1 x 2 x … x n is called the factorial of n. This was another 17th-century discovery.</p>
<h2>e</h2>
<p>The constant “e” originated in the 17th century as the base of the natural logarithm, and to three decimal places is 2.718…, though like Pi, it’s a <a href="http://www.britannica.com/topic/transcendental-number">transcendental number</a> and continues without repeating to countless decimal places.</p>
<p><a href="http://www.maa.org/press/periodicals/convergence/euler-the-master-of-us-all">Euler, the master</a> of us all – who named both “pi” and “e” – realised that e<sup>x</sup> also had a dandy Taylor series:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/114601/original/image-20160310-26256-12t0i5u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/114601/original/image-20160310-26256-12t0i5u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=182&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114601/original/image-20160310-26256-12t0i5u.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=182&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114601/original/image-20160310-26256-12t0i5u.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=182&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114601/original/image-20160310-26256-12t0i5u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=228&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114601/original/image-20160310-26256-12t0i5u.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=228&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114601/original/image-20160310-26256-12t0i5u.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=228&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The exponential function.</span>
<span class="attribution"><span class="source">The Conversation</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Then setting theta (θ) equal to one, gives an efficient formula for e. </p>
<p>Now we know all the building blocks all we need do in the second equation (above) is set Theta to be Pi, and with a little trigonometry, knowing that sin (π) = 0 and cos (π) = -1, then reducing the formula step by step, out pops the original beautiful formula.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/114892/original/image-20160313-11274-1cmkqbs.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/114892/original/image-20160313-11274-1cmkqbs.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=300&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114892/original/image-20160313-11274-1cmkqbs.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=300&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114892/original/image-20160313-11274-1cmkqbs.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=300&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114892/original/image-20160313-11274-1cmkqbs.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=377&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114892/original/image-20160313-11274-1cmkqbs.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=377&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114892/original/image-20160313-11274-1cmkqbs.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=377&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">One Pi is entered into the formula and the calculations are made, a little mathematical juggling of various elements on either side of the = sign gives us the final beautiful formula.</span>
<span class="attribution"><span class="source">The Conversation</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<h2>What is mathematical beauty?</h2>
<p>As you can see, to view the formula as beautiful it is necessary to understand the elements, at least roughly.</p>
<p>Bertrand Russell in his History of Western Philosophy
<a href="http://www.goodreads.com/quotes/647862-mathematics-rightly-viewed-possesses-not-only-truth-but-supreme-beauty-a">put it so</a>:</p>
<blockquote>
<p>Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.</p>
</blockquote>
<p>Most mathematicians would agree that to be beautiful a formula must be unexpected, concise and useful – in the rarefied sense that professional mathematicians recognise.</p>
<p>When forced to, most mathematicians will list Archimedes, Gauss and Euler among the top five mathematical thinkers of all time. The other two are <a href="http://www.britannica.com/biography/Isaac-Newton">Isaac Newton</a> (for calculus and mechanics) and <a href="http://www.britannica.com/biography/Bernhard-Riemann">Bernhard Riemann</a> (for the Riemann hypothesis and Riemannian geometry).</p>
<p>With three of these brilliant thinkers and fundamental constants engaged, it is no wonder that Euler’s formula is lionised as it is, as the most beautiful formula in mathematics.</p><img src="https://counter.theconversation.com/content/56067/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Jonathan Borwein (Jon) receives funding from the Australian Research Council.</span></em></p>On international Pi Day it’s time to look at Pi’s position in unique formula that’s praised much for its beauty in uniting several mathematical constants.Jonathan Borwein (Jon), Laureate Professor of Mathematics, University of NewcastleLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/557442016-03-11T17:22:29Z2016-03-11T17:22:29ZThe search for the value of pi<figure><img src="https://images.theconversation.com/files/114523/original/image-20160309-13712-1jxaozq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">This "pi plate" shows some of the progress toward finding all the digits of pi.</span> <span class="attribution"><a class="source" href="https://upload.wikimedia.org/wikipedia/commons/d/d2/Pi_plate.jpg">Piledhigheranddeeper</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>The number represented by pi (π) is used in calculations whenever something round (or nearly so) is involved, such as for circles, spheres, cylinders, cones and ellipses. Its value is necessary to compute many important quantities about these shapes, such as understanding the relationship between a circle’s radius and its circumference and area (circumference=2πr; area=πr<sup>2</sup>).</p>
<p>Pi also appears in the calculations to determine the area of an ellipse and in finding the radius, surface area and volume of a sphere.</p>
<p>Our world contains many round and near-round objects; finding the exact value of pi helps us build, manufacture and work with them more accurately.</p>
<p>Historically, people had only very coarse estimations of pi (such as 3, or 3.12, or 3.16), and while they knew these were estimates, they had no idea how far off they might be. </p>
<p>The search for the accurate value of pi led not only to more accuracy, but also to the development of new concepts and techniques, such as limits and iterative algorithms, which then became fundamental to new areas of mathematics.</p>
<h2>Finding the actual value of pi</h2>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/114531/original/image-20160309-13689-10j7dgu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114531/original/image-20160309-13689-10j7dgu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114531/original/image-20160309-13689-10j7dgu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=737&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114531/original/image-20160309-13689-10j7dgu.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=737&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114531/original/image-20160309-13689-10j7dgu.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=737&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114531/original/image-20160309-13689-10j7dgu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=926&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114531/original/image-20160309-13689-10j7dgu.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=926&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114531/original/image-20160309-13689-10j7dgu.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=926&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Archimedes.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Archimedes1.jpg">André Thévet (1584)</a></span>
</figcaption>
</figure>
<p>Between 3,000 and 4,000 years ago, people used trial-and-error approximations of pi, without doing any math or considering potential errors. The earliest written approximations of pi are <a href="http://www.exploratorium.edu/pi/history_of_pi/">3.125 in Babylon</a> (1900-1600 B.C.) and <a href="http://www.ualr.edu/lasmoller/pi.html">3.1605 in ancient Egypt</a> (1650 B.C.). Both approximations start with 3.1 – pretty close to the actual value, but still relatively far off.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/114533/original/image-20160309-13704-lrnh10.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114533/original/image-20160309-13704-lrnh10.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114533/original/image-20160309-13704-lrnh10.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=200&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114533/original/image-20160309-13704-lrnh10.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=200&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114533/original/image-20160309-13704-lrnh10.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=200&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114533/original/image-20160309-13704-lrnh10.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=252&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114533/original/image-20160309-13704-lrnh10.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=252&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114533/original/image-20160309-13704-lrnh10.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=252&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Archimedes’ method of calculating pi involved polygons with more and more sides.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Archimedes_pi.svg">Leszek Krupinski</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>The first rigorous approach to finding the true value of pi was based on geometrical approximations. Around 250 B.C., the Greek mathematician Archimedes drew polygons both around the outside and within the interior of circles. Measuring the perimeters of those gave upper and lower bounds of the range containing pi. He started with hexagons; by using polygons with more and more sides, he ultimately calculated three accurate digits of pi: 3.14. Around A.D. 150, Greek-Roman scientist Ptolemy used this method to calculate a value of 3.1416.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/114534/original/image-20160309-13693-ipve24.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114534/original/image-20160309-13693-ipve24.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114534/original/image-20160309-13693-ipve24.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114534/original/image-20160309-13693-ipve24.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114534/original/image-20160309-13693-ipve24.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114534/original/image-20160309-13693-ipve24.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114534/original/image-20160309-13693-ipve24.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114534/original/image-20160309-13693-ipve24.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Liu Hui’s method of calculating pi also used polygons, but in a slightly different way.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Liuhui_Pi_Inequality.svg">Gisling and Pbroks13</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>Independently, around A.D. 265, Chinese mathematician Liu Hui created another simple polygon-based iterative algorithm. He proposed a very fast and efficient approximation method, which gave four accurate digits. Later, around A.D. 480, Zu Chongzhi adopted Liu Hui’s method and achieved seven digits of accuracy. This record held for another 800 years. </p>
<p>In 1630, Austrian astronomer Christoph Grienberger arrived at 38 digits, which is the most accurate approximation manually achieved using polygonal algorithms.</p>
<h2>Moving beyond polygons</h2>
<p>The development of infinite series techniques in the 16th and 17th centuries greatly enhanced people’s ability to approximate pi more efficiently. An infinite series is the sum (or much less commonly, product) of the terms of an infinite sequence, such as ½, ¼, 1/8, 1/16, … 1/(2<sup>n</sup>). The first written description of an infinite series that could be used to compute pi was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji around 1500 A.D., the proof of which was presented around 1530 A.D. </p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/114535/original/image-20160309-13734-1wgyx0o.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114535/original/image-20160309-13734-1wgyx0o.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114535/original/image-20160309-13734-1wgyx0o.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=745&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114535/original/image-20160309-13734-1wgyx0o.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=745&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114535/original/image-20160309-13734-1wgyx0o.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=745&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114535/original/image-20160309-13734-1wgyx0o.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=936&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114535/original/image-20160309-13734-1wgyx0o.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=936&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114535/original/image-20160309-13734-1wgyx0o.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=936&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Sir Isaac Newton.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Sir_Isaac_Newton_(1642-1727)._Oil_painting_by_a_follower_of_Wellcome_L0016625.jpg">Wellcome Trust</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>In 1665, English mathematician and physicist Isaac Newton used infinite series to compute pi to 15 digits using calculus he and German mathematician Gottfried Wilhelm Leibniz discovered. After that, the record kept being broken. It reached 71 digits in 1699, 100 digits in 1706, and 620 digits in 1956 – the best approximation achieved without the aid of a calculator or computer.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/114536/original/image-20160309-13737-1igjx6f.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114536/original/image-20160309-13737-1igjx6f.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114536/original/image-20160309-13737-1igjx6f.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=729&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114536/original/image-20160309-13737-1igjx6f.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=729&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114536/original/image-20160309-13737-1igjx6f.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=729&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114536/original/image-20160309-13737-1igjx6f.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=916&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114536/original/image-20160309-13737-1igjx6f.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=916&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114536/original/image-20160309-13737-1igjx6f.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=916&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Carl Louis Ferdinand von Lindemann.</span>
</figcaption>
</figure>
<p>In tandem with these calculations, mathematicians were researching other characteristics of pi. Swiss mathematician Johann Heinrich Lambert (1728-1777) first proved that pi is an irrational number – it has an infinite number of digits that never enter a repeating pattern. In 1882, German mathematician Ferdinand von Lindemann proved that pi cannot be expressed in a rational algebraic equation (<a href="http://sprott.physics.wisc.edu/pickover/trans.html">such as pi²=10</a> or 9pi<sup>4</sup> - 240pi<sup>2</sup> + 1492 = 0).</p>
<h2>Toward even more digits of pi</h2>
<p>Bursts of calculations of even more digits of pi followed the adoption of iterative algorithms, which repeatedly build an updated value by using a calculation performed on the previous value. A simple example of an iterative algorithm allows you to approximate the square root of 2 as follows, using the formula (x+2/x)/2:</p>
<ul>
<li>(2+2/2)/2 = 1.5</li>
<li>(<strong>1.5</strong>+2/<strong>1.5</strong>)/2 = 1.4167</li>
<li>(<strong>1.4167</strong>+2/<strong>1.4167</strong>)/2 = 1.4142, which is a very close approximation already.</li>
</ul>
<p>Advances toward more digits of pi came with the use of a Machin-like algorithm (a generalization of English mathematician <a href="http://mathworld.wolfram.com/Machin-LikeFormulas.html">John Machin’s formula</a> developed in 1706) and the <a href="http://mathfaculty.fullerton.edu/mathews/n2003/GaussianQuadMod.html">Gauss-Legendre algorithm</a> (late 18th century) in electronic computers (invented mid-20th century). In 1946, ENIAC, the first electronic general-purpose computer, calculated 2,037 digits of pi in 70 hours. The most recent calculation found <a href="http://www.numberworld.org/y-cruncher/">more than 13 trillion digits of pi</a> in 208 days!</p>
<p>It has been widely accepted that for most numerical calculations involving pi, a dozen digits provides sufficient precision. According to mathematicians <a href="http://www.springer.com/us/book/9783642567353?wt_mc=GoogleBooks.GoogleBooks.3.EN&token=gbgen#otherversion=9783540665724">Jörg Arndt and Christoph Haenel</a>, 39 digits are sufficient to perform most cosmological calculations, because that’s the accuracy necessary to calculate the circumference of the observable universe to within one atom’s diameter. Thereafter, more digits of pi are not of practical use in calculations; rather, today’s pursuit of more digits of pi is about testing supercomputers and numerical analysis algorithms. </p>
<h2>Calculating pi by yourself</h2>
<p>There are also fun and simple methods for estimating the value of pi. One of the best-known is a method called “<a href="http://www.eveandersson.com/pi/monte-carlo-circle">Monte Carlo</a>.” </p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/114488/original/image-20160309-13717-1c27vzw.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114488/original/image-20160309-13717-1c27vzw.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114488/original/image-20160309-13717-1c27vzw.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114488/original/image-20160309-13717-1c27vzw.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114488/original/image-20160309-13717-1c27vzw.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114488/original/image-20160309-13717-1c27vzw.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114488/original/image-20160309-13717-1c27vzw.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114488/original/image-20160309-13717-1c27vzw.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A square with inscribed circle.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Square-circle.svg">Deweirdifier</a></span>
</figcaption>
</figure>
<p>The method is fairly simple. To try it at home, draw a circle and a square around it (as at left) on a piece of paper. Imagine the square’s sides are of length 2, so its area is 4; the circle’s diameter is therefore 2, and its area is pi. The ratio between their areas is pi/4, or about 0.7854.</p>
<p>Now pick up a pen, close your eyes and put dots on the square at random. If you do this enough times, and your efforts are truly random, eventually the percentage of times your dot landed inside the circle will approach 78.54% – or 0.7854.</p>
<p>Now you’ve joined the ranks of mathematicians who have calculated pi through the ages.</p><img src="https://counter.theconversation.com/content/55744/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Xiaojing Ye does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>On the occasion of Pi Day, a look at the history of calculating the actual, and increasingly exact, value of pi (π).Xiaojing Ye, Assistant Professor of Mathematics and Statistics, Georgia State UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/379482015-03-13T10:04:18Z2015-03-13T10:04:18ZPi Day is silly, but π itself is fascinating and universal<figure><img src="https://images.theconversation.com/files/74673/original/image-20150312-13499-179byai.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Would anyone like a slice of my π pie?</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/the_pdub/6837660458">Tarehna Wicker</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA</a></span></figcaption></figure><p>Math students everywhere will be eating pies in class this week in celebration of what is known as Pi Day, the 14th day of the 3rd month. </p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/74670/original/image-20150312-13508-1j5dq3l.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/74670/original/image-20150312-13508-1j5dq3l.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/74670/original/image-20150312-13508-1j5dq3l.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=731&fit=crop&dpr=1 600w, https://images.theconversation.com/files/74670/original/image-20150312-13508-1j5dq3l.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=731&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/74670/original/image-20150312-13508-1j5dq3l.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=731&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/74670/original/image-20150312-13508-1j5dq3l.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=918&fit=crop&dpr=1 754w, https://images.theconversation.com/files/74670/original/image-20150312-13508-1j5dq3l.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=918&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/74670/original/image-20150312-13508-1j5dq3l.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=918&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Larry Shaw initiated the first Pi Day in 1988.</span>
<span class="attribution"><a class="source" href="http://commons.wikimedia.org/wiki/File:Prince-of-pi.jpg">Ron Hipschman</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>The symbol π (pronounced paɪ in English) is the sixteenth letter of the Greek alphabet and is used in mathematics to stand for a real number of special significance. When π is written in decimal notation, it begins 3.14, suggesting the date 3/14. In fact, the decimal expansion of π begins 3.1415, so Pi Day 2015, whose date was abbreviated as 3/14/15, was said to be of special significance, a once-per-century coincidence. (The same was said about the following year, on 3/14/16, since 3.1416 is a closer approximation to π than is 3.1415.)</p>
<p>Besides a reason to enjoy baked goods while feeling mathematically in-the-know, just what is π anyway?</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/74678/original/image-20150312-13526-1hulh2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/74678/original/image-20150312-13526-1hulh2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/74678/original/image-20150312-13526-1hulh2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=487&fit=crop&dpr=1 600w, https://images.theconversation.com/files/74678/original/image-20150312-13526-1hulh2.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=487&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/74678/original/image-20150312-13526-1hulh2.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=487&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/74678/original/image-20150312-13526-1hulh2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=612&fit=crop&dpr=1 754w, https://images.theconversation.com/files/74678/original/image-20150312-13526-1hulh2.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=612&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/74678/original/image-20150312-13526-1hulh2.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=612&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A circle’s measurements define π.</span>
<span class="attribution"><a class="source" href="http://www.education.rec.ri.cmu.edu/roboticscurriculum/multimedia/mathcirc.shtml">Robotics Academy</a></span>
</figcaption>
</figure>
<p>It’s defined to be the ratio between the circumference of a circle and the diameter of that circle. This ratio is the same for any size circle, so it’s intrinsically attached to the idea of circularity. The circle is a fundamental shape, so it’s natural to wonder about this fundamental ratio. People have been doing so going back at least to the <a href="https://numberwarrior.wordpress.com/2008/12/03/on-the-ancient-babylonian-value-for-pi/">ancient Babylonians</a>.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/74661/original/image-20150312-13517-ugl65i.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/74661/original/image-20150312-13517-ugl65i.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/74661/original/image-20150312-13517-ugl65i.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=295&fit=crop&dpr=1 600w, https://images.theconversation.com/files/74661/original/image-20150312-13517-ugl65i.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=295&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/74661/original/image-20150312-13517-ugl65i.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=295&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/74661/original/image-20150312-13517-ugl65i.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=371&fit=crop&dpr=1 754w, https://images.theconversation.com/files/74661/original/image-20150312-13517-ugl65i.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=371&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/74661/original/image-20150312-13517-ugl65i.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=371&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The hexagon’s perimeter is shorter than the circle’s, while the square’s is longer.</span>
<span class="attribution"><a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>You can see that π is greater than 3 if you look at a hexagon inscribed within a circle. The perimeter of the hexagon is shorter than the circumference of the circle, and yet the ratio of the hexagon’s perimeter to the circle’s diameter is 3. And you can see that π is less than 4 if you look at the square that circumscribes a circle. The square’s perimeter is longer than the circle’s circumference, and yet the ratio of this perimeter to the diameter of the circle is 4. So π is somewhere in there between 3 and 4. OK, but what number <em>is</em> it?</p>
<p>A little experimentation with a measuring tape and a dinner plate suggests that π might be 22/7, a number whose decimal expansion begins 3.14. But it turns out that 22/7 is approximately 3.1429, while even 2,250 years ago <a href="http://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html">Archimedes</a> knew that π is approximately 3.1416. The fraction 355/113 is much closer to π but still not exactly equal to it. </p>
<h2>Fractionally closer?</h2>
<p>So this raises the question: is there some other fraction out there that equals π, not merely approximately but exactly? The answer is no. In 1761, Swiss mathematician <a href="http://www.pi314.net/eng/lambert.php">Johann Lambert</a> proved that no fraction exactly equals π. This implies that its decimal expansion is never-ending, with no repeated pattern.</p>
<p>The German mathematician <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Lindemann.html">Ferdinand Lindemann</a> proved in 1882 that π is in fact <em>transcendental</em>, which means that it does not solve any polynomial equation with integer coefficients. This implies in some sense that there isn’t ever going to be a simple way of describing π arithmetically. Nowadays, machines can compute trillions of decimal digits of π, but that in no way helps us understand what π is exactly. It’s easiest just to say that, to be exact, π is equal to … π.</p>
<p>No one knows whether each of the ten digits – 0 through 9 – appears with equal frequency in the decimal expansion of π, as we would expect if the digits of π were produced by a random digit generator. This illustrates that a strikingly elementary question can be out of reach of modern mathematics. Perhaps in a century mankind will know the answer to this question, but it’s not even clear at this time how to attack it effectively.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/74683/original/image-20150312-13523-4we6ir.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/74683/original/image-20150312-13523-4we6ir.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/74683/original/image-20150312-13523-4we6ir.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/74683/original/image-20150312-13523-4we6ir.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/74683/original/image-20150312-13523-4we6ir.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/74683/original/image-20150312-13523-4we6ir.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/74683/original/image-20150312-13523-4we6ir.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/74683/original/image-20150312-13523-4we6ir.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">You could measure circumference and diameter of these pies to get π.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/djwtwo/5525295088">Dennis Wilkinson</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA</a></span>
</figcaption>
</figure>
<h2>Everything’s coming up π</h2>
<p>What is astonishing about π is that it appears in many different mathematical contexts and across all mathematical areas. It turns out that π is the ratio of the area of a circle to the area of the square built on the radius of the circle. That seems like a coincidence, because π was defined to be a different ratio. But the two ratios are the same. π is also the ratio of the surface area of a sphere to the area of the square built on the diameter of the square. And what about the ratio of the volume of sphere to the volume of the cube built on the sphere’s diameter? That’s π/6.</p>
<p>The area under the bell-shaped curve y=1/(1+x²) is π. But this curve isn’t actually the well-known and universal bell-shaped curve seen in statistics that has the formula y=e<sup>-x²</sup>. The area under that curve is the square root of π! If you drop a pin of length one centimeter on a sheet of lined paper with lines spaced at centimeter intervals, the probability that the pin crosses one of the lines is 2/π. If you choose two whole numbers at random, the probability that they will have no common factor is 6/π².</p>
<p>There are thousands of formulas for π of one sort or another, although it isn’t clear whether any of them will satisfy the desire to know what π is exactly. One such formula is</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/74718/original/image-20150312-13520-18pyzln.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/74718/original/image-20150312-13520-18pyzln.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/74718/original/image-20150312-13520-18pyzln.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=124&fit=crop&dpr=1 600w, https://images.theconversation.com/files/74718/original/image-20150312-13520-18pyzln.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=124&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/74718/original/image-20150312-13520-18pyzln.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=124&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/74718/original/image-20150312-13520-18pyzln.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=156&fit=crop&dpr=1 754w, https://images.theconversation.com/files/74718/original/image-20150312-13520-18pyzln.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=156&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/74718/original/image-20150312-13520-18pyzln.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=156&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Ramanujan’s equation for π.</span>
<span class="attribution"><span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>where the sigma symbol indicates that one must plug in all the whole numbers in place of the symbol “k” in the subsequent formula and add up the resulting infinitely-many fractions. What is remarkable about this expression is that it was discovered by the legendary Indian genius <a href="http://www.cecm.sfu.ca/organics/papers/borwein/paper/html/node15.html#SECTION00080000000000000000">Srinivasan Ramanujan</a> in 1914, working alone. No one knows how Ramanujan came up with this amazing formula. Moreover, his formula wasn’t even shown to be correct until 1985 – and that demonstration used high-speed computers to which Ramanujan had no access.</p>
<h2>π is beyond universal</h2>
<p>The number π is a universal constant that is ubiquitous across mathematics. In fact, it is an understatement to call it “universal,” because π lives not only in this universe but in any conceivable universe. It existed even prior to the Big Bang. It is permanent and unchanging.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/74671/original/image-20150312-13508-96jlmv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/74671/original/image-20150312-13508-96jlmv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/74671/original/image-20150312-13508-96jlmv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=447&fit=crop&dpr=1 600w, https://images.theconversation.com/files/74671/original/image-20150312-13508-96jlmv.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=447&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/74671/original/image-20150312-13508-96jlmv.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=447&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/74671/original/image-20150312-13508-96jlmv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=561&fit=crop&dpr=1 754w, https://images.theconversation.com/files/74671/original/image-20150312-13508-96jlmv.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=561&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/74671/original/image-20150312-13508-96jlmv.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=561&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Math enthusiasts need to cut loose sometimes too.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/vancouverislanduniversity/6982954735">Vancouver Island University</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span>
</figcaption>
</figure>
<p>That’s why the celebration of Pi Day seems so silly. The Gregorian calendar, the decimal system, the Greek alphabet, and pies are relatively modern, human-made inventions, chosen arbitrarily among many equivalent choices. Of course a mood-boosting piece of lemon meringue could be just what many math lovers need in the middle of March at the end of a long winter. But there’s an element of absurdity to celebrating π by noting its connections with these ephemera, which have themselves no connection to π at all, just as absurd as it would be to celebrate Earth Day by eating foods that start with the letter “E.”</p>
<p><em>Editor’s note: This is an updated version of an article originally published on March 13, 2015.</em></p><img src="https://counter.theconversation.com/content/37948/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Daniel Ullman does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>3/14 on the calendar approximates the first three digits of the mathematical constant π. Math nerds will celebrate with baked goods, but π is a deeper, nobler entity.Daniel Ullman, Professor of Mathematics, George Washington UniversityLicensed as Creative Commons – attribution, no derivatives.