tag:theconversation.com,2011:/ca-fr/topics/zero-43571/articlesZero – La Conversation2023-04-20T16:32:15Ztag:theconversation.com,2011:article/2020202023-04-20T16:32:15Z2023-04-20T16:32:15ZShakespeare by numbers: how mathematical breakthroughs influenced the Bard’s plays<p>Mathematical motifs feature in many of Shakespeare’s most memorable scenes. He lived and wrote in <a href="https://www.bl.uk/shakespeare/articles/key-features-of-renaissance-culture">the late 16th century</a>, when <a href="https://www.mhs.ox.ac.uk/staff/saj/texts/mathematicus.htm">new mathematical concepts</a> were transforming perceptions of the world. Part of the role of the theatre was to process the cultural implications of all these changes.</p>
<p>People in Shakespeare’s time were used to the idea of the <a href="https://www.newscientist.com/article/mg22229654-800-shakespeare-did-radical-astronomy-inspire-hamlet/">infinite</a>: the planets, the heavens, the weather. But they were much less used to the inverse idea that the very small (and even nothingness) could be expressed by mathematical axioms. In fact, the first recorded English use of the word “zero” wasn’t <a href="https://www.merriam-webster.com/dictionary/zero#word-history">until 1598</a>. </p>
<p>Thinkers like Italian mathematician <a href="https://www.britannica.com/biography/Fibonacci">Fibonacci</a>, who lived in the 13th century, helped to introduce the concept of zero – known then as a “cipher” – into the mainstream. But it wasn’t until philosopher <a href="https://www.britannica.com/biography/Rene-Descartes">René Descartes </a> and mathematicians <a href="https://www.britannica.com/biography/Isaac-Newton">Sir Isaac Newton</a> and <a href="https://www.britannica.com/biography/Gottfried-Wilhelm-Leibniz">Gottfried Leibniz</a> developed <a href="https://marktomforde.com/academic/miscellaneous/calculus-history/calchistory.html">calculus</a> in the late 16th and early 17th centuries that “zero” started to figure prominently in society. </p>
<p>Moreover, scientist <a href="https://mathshistory.st-andrews.ac.uk/Biographies/Hooke/">Robert Hooke</a> didn’t discover microorganisms until 1665, meaning the idea that life could exist on a micro level remained something of fantasy.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/518478/original/file-20230330-16-xzv2wu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A black and white engraving showing Calvius with mathematical tools. He wears a clock and pointed hat." src="https://images.theconversation.com/files/518478/original/file-20230330-16-xzv2wu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/518478/original/file-20230330-16-xzv2wu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=868&fit=crop&dpr=1 600w, https://images.theconversation.com/files/518478/original/file-20230330-16-xzv2wu.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=868&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/518478/original/file-20230330-16-xzv2wu.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=868&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/518478/original/file-20230330-16-xzv2wu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1091&fit=crop&dpr=1 754w, https://images.theconversation.com/files/518478/original/file-20230330-16-xzv2wu.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1091&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/518478/original/file-20230330-16-xzv2wu.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1091&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="caption">A 16th century engraving of astronomer Christopher Clavius after a painting by Francisco Villamena (1606).</span>
<span class="attribution"><a class="source" href="https://library.si.edu/es/image-gallery/73149">Smithsonian Libraries</a></span>
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<p>With the <a href="http://www.mhs.ox.ac.uk/staff/saj/thesis/introduction.htm">growing influence of neoclassical ideas</a> in England, small, insignificant figures had begun to be used to represent very large concepts. </p>
<p>This was happening both in modes of calculation (which used proportion) and in the practice of writing mathematical symbols. </p>
<p>For example, during the 16th and early 17th centuries, the equals, multiplication, division, root, decimal, and inequality symbols were gradually introduced and standardised.</p>
<p>Alongside this came the work of <a href="https://mathshistory.st-andrews.ac.uk/Biographies/Clavius/">Christopher Clavius</a> – a German Jesuit astronomer who helped Pope Gregory XIII to introduce the Gregorian calendar – and other mathematicians on fractions. Then referred to as <a href="https://nrich.maths.org/2515">“broken numbers”</a>, they <a href="https://jontalle.web.engr.illinois.edu/uploads/298/HistoryMath-Burton.85.pdf">stirred up great angst</a> among those who clung to classical models of number theory. </p>
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<img alt="" src="https://images.theconversation.com/files/520292/original/file-20230411-22-7ego8p.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/520292/original/file-20230411-22-7ego8p.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/520292/original/file-20230411-22-7ego8p.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/520292/original/file-20230411-22-7ego8p.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/520292/original/file-20230411-22-7ego8p.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/520292/original/file-20230411-22-7ego8p.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/520292/original/file-20230411-22-7ego8p.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">
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<p><em>This article is part of our <a href="https://theconversation.com/uk/topics/william-shakespeare-14574">First Folio 400</a> series. These articles mark the 400th anniversary of the publication of the First Folio, the first collected edition of William Shakespeare’s plays.</em></p>
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<p>The struggle to come to terms with the entanglement of the very large and the very small is splendidly displayed in many of Shakespeare’s works. This includes his history play Henry V and tragedy Troilus and Cressida.</p>
<p>The opening chorus of Henry V displays Shakespeare’s interest in proportion and the <a href="https://www.salon.com/2013/08/04/shakespeare_defined_our_concept_of_nothingness/#:%7E:text=How%20do%20we%20write%20three,0%20(zero%20units)%3A%203%2C000.">concept of zero</a> through its repeated “O” and references to contemporary mathematical thought:</p>
<blockquote>
<p>O for a muse of fire, that would ascend / The brightest heaven of invention: / A kingdom for a stage, princes to act, / And monarchs to behold the swelling scene […] / may we cram / Within this wooden O the very casques / That did affright the air at Agincourt? / O pardon: since a crookèd figure may / Attest in little place a million, / And let us, ciphers to this great account, / On your imaginary forces work. </p>
</blockquote>
<p><a href="https://global.oup.com/academic/product/henry-v-the-oxford-shakespeare-9780199536511?cc=gb&lang=en&">Scholars</a> largely agree that Shakespeare’s “crookèd figure” is actually zero. This is despite, of course, the rather obvious objection that zero is the least crooked of all numbers. </p>
<p>In the line “a crookèd figure may / Attest in little place a million”, Shakespeare references 16th century <a href="https://www.newscientist.com/article/mg22229654-800-shakespeare-did-radical-astronomy-inspire-hamlet/">mathematical debates</a> surrounding the idea that the very small is capable of both representing and influencing the very big. In this case, the zero is capable of transforming 100,000 into 1,000,000.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/518476/original/file-20230330-26-6m7kcb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Interior shot of Shakespeare's open air Globe theatre showing its round shape." src="https://images.theconversation.com/files/518476/original/file-20230330-26-6m7kcb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/518476/original/file-20230330-26-6m7kcb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=401&fit=crop&dpr=1 600w, https://images.theconversation.com/files/518476/original/file-20230330-26-6m7kcb.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=401&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/518476/original/file-20230330-26-6m7kcb.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=401&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/518476/original/file-20230330-26-6m7kcb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/518476/original/file-20230330-26-6m7kcb.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/518476/original/file-20230330-26-6m7kcb.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>
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<span class="caption">Use of ‘zero’ or ‘O’ in Shakespeare can also be read as a metaphor for his circular Globe theatre.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/london-uk-april-20-2019-globe-1376701724">Nick Brundle/Shutterstock</a></span>
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<p>In this mathematical analogy, “crookèd figure[s]” can “attest” much greater things. The chorus suggests that by using one’s “imaginary forces”, much greater things may come from the forthcoming stage performances. </p>
<p>This extended metaphor reappears in Shakespeare’s tragicomedy, <a href="https://theconversation.com/the-winters-tale-review-jarring-shakespeares-globe-production-lacks-warmth-202566">The Winter’s Tale</a> when the “<a href="https://www.salon.com/2013/08/04/shakespeare_defined_our_concept_of_nothingness/">cipher</a>” (numbers) transform into many thousands of thank yous:</p>
<blockquote>
<p>Like a cipher, / Yet standing in rich place, I multiply / With one “We thank you” many thousands more / That go before it.</p>
</blockquote>
<p>There is a further, visual metaphor in Henry V’s opening prologue where the chorus asks pardon of an “O” to help them represent many things in the “wooden O” – the <a href="https://www.shakespearesglobe.com/">Globe Theatre</a>. This is perhaps evidence of Shakespeare’s ongoing interest in insignificant figures “attest[ing]” much greater things.</p>
<p>Elsewhere in his work, mathematical metaphors encircle themselves in moments of crisis. In Troilus and Cressida, Shakespeare uses mathematical language to chart the slow motion collapse of <a href="https://www.jstor.org/stable/24778494">Troilus’s mental stability</a> after witnessing his lover Cressida’s flirtation with another man.</p>
<p>For Troilus, Cressida disintegrates into “fractions”, “fragments” and “bits and greasy relics”. To mirror this, Shakespeare’s verse descends into jagged pieces, like the early modern name for fractions: “broken numbers”.</p>
<p>With 2023 marking 400 years since the publication of Shakespeare’s First Folio, it is exciting to see how the Bard’s plays spoke to significant developments in the 16th-century mathematical world.</p>
<p>Shakespeare’s plays registered the 16th-century crisis of classical mathematics in the face of newer ideas. But they also offered space for audiences to come to terms with these new ideas and think differently about the world through the lens of mathematics.</p><img src="https://counter.theconversation.com/content/202020/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Madeleine S. Killacky 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>In the late 16th century, new mathematical concepts were transforming perceptions of the world. Shakespeare’s plays helped audiences to process these changes.Madeleine S. Killacky, PhD Candidate, Medieval Literature, Bangor UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1972232023-01-17T06:07:50Z2023-01-17T06:07:50ZCurious Kids: is there such a thing as nothing?<figure><img src="https://images.theconversation.com/files/503647/original/file-20230109-5923-hy7ptu.jpg?ixlib=rb-1.1.0&rect=0%2C744%2C6989%2C3688&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/astronaut-leaving-earth-searching-new-home-1666242118">oneinchpunch/Shutterstock</a></span></figcaption></figure><p><strong>Is there such a thing as nothing? – Reggie, aged seven, Darlington</strong></p>
<p><strong>Could someone see nothing? What does nothing look like? – Maya, aged nine, Bristol</strong></p>
<p>Imagine you hear a noise outside your window. You think it might be a dog barking, or maybe a child shouting. But when you get up and have a look, there’s no dog or child. “Oh,” you say, “there’s nothing there.”</p>
<p>We often say we’ve “got nothing”, or that there’s “nothing there”. But what we mean is that we haven’t got a particular thing. When you looked outside, lots of things were there – trees, houses, cars and bicycles maybe – but the particular thing you were looking for wasn’t there. </p>
<p>Even if you were looking into a completely empty room, there would still be things there. There’s always air, and <a href="https://forces.si.edu/atmosphere/02_01_02.html">air is made up of molecules</a>, like the oxygen we need to breathe in to keep us alive. </p>
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<img alt="" src="https://images.theconversation.com/files/282267/original/file-20190702-126345-1np1y7m.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/282267/original/file-20190702-126345-1np1y7m.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=293&fit=crop&dpr=1 600w, https://images.theconversation.com/files/282267/original/file-20190702-126345-1np1y7m.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=293&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/282267/original/file-20190702-126345-1np1y7m.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=293&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/282267/original/file-20190702-126345-1np1y7m.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=368&fit=crop&dpr=1 754w, https://images.theconversation.com/files/282267/original/file-20190702-126345-1np1y7m.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=368&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/282267/original/file-20190702-126345-1np1y7m.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=368&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
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<p><em><a href="https://theconversation.com/au/topics/curious-kids-36782">Curious Kids</a> is a series by <a href="https://theconversation.com/uk">The Conversation</a> that gives children the chance to have their questions about the world answered by experts. If you have a question you’d like an expert to answer, send it to <a href="mailto:curiouskids@theconversation.com">curiouskids@theconversation.com</a> and make sure you include the asker’s first name, age and town or city. We won’t be able to answer every question, but we’ll do our very best.</em></p>
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<p>But what about space? There’s no air in space: that’s why astronauts need to wear a space suit that provides <a href="https://www.nasa.gov/audience/forstudents/nasaandyou/home/spacesuits_bkgd_en.html">oxygen to breathe</a> if they go on a “space walk”.</p>
<p>There are actually molecules in space, too, but they are so few and far between that it’s mostly empty space. This is called <a href="https://www.sciencefocus.com/space/is-space-a-perfect-vacuum/">a vacuum</a>. We might think that the emptiness we can see in between stars in the night sky, where there are very few molecules, will be where we find “nothing”.</p>
<p>It turns out, though, when you look into the dark night sky, you are not seeing “nothing”. This empty space is filled with energy, and that’s something even if you cannot hold it in your hand. Energy is what makes things happen.</p>
<figure class="align-center ">
<img alt="Mother and child looking up at Milky Way" src="https://images.theconversation.com/files/503670/original/file-20230109-13-hyqjpv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/503670/original/file-20230109-13-hyqjpv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=613&fit=crop&dpr=1 600w, https://images.theconversation.com/files/503670/original/file-20230109-13-hyqjpv.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=613&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/503670/original/file-20230109-13-hyqjpv.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=613&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/503670/original/file-20230109-13-hyqjpv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=770&fit=crop&dpr=1 754w, https://images.theconversation.com/files/503670/original/file-20230109-13-hyqjpv.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=770&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/503670/original/file-20230109-13-hyqjpv.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=770&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
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<span class="caption">Looking at the universe.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/silhouette-mother-child-sitting-together-hold-1085451182">KIDSADA PHOTO/Shutterstock</a></span>
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<p>Energy never goes away and it is never made out of nothing. When you run, the energy in your motion comes from the energy you got out of the food you ate. When you stop, that energy goes into heat and making a minuscule impact on the motion of the Earth as your feet rub against it when you brake.</p>
<p>Einstein realised that energy and particles (tiny bits of stuff) are two sides of the same coin. The tiniest things we know of – such as particles, which are more than a million billion times smaller than we are – are made from energy. <a href="https://academickids.com/encyclopedia/index.php/Quantum_mechanics">About a century ago</a> scientists realised that these tiny things act differently to bigger things in the universe. Their behaviour cannot be predicted precisely, and it can only be described with something called “<a href="https://www.newscientist.com/definition/quantum-mechanics/">quantum mechanics</a>”.</p>
<h2>Particles and anti-particles</h2>
<p>This also means that the content of vacuum is not precisely zero. Tiny particles in it can meet their exact opposites, “<a href="https://www.bbc.co.uk/news/science-environment-13667475">anti-particles</a>”. When this happens they cancel each other out and vanish, but this leaves behind the energy that had made them in the first place.</p>
<p>The vacuum of space is a “soup” of energy and pairs of particles and anti-particles. The Dutch scientist <a href="https://www.scientificamerican.com/article/something-from-nothing-vacuum-can-yield-flashes-of-light/">Hendrik Casimir</a> suggested a way of proving this in 1948 – and half a century later, we have found out that he was right.</p>
<p>This energy in the vacuum also makes the universe <a href="https://www.space.com/42178-bringing-dark-energy-into-the-light.html">grow bigger</a> with time, like when you blow up a balloon.</p>
<p>There are other things in the vacuum of space, too. Space contains “fields”, which are a way to describe the influence something can have throughout a region of space. For instance, the Earth pulls at the Moon through space by way of a field called “gravity”. </p>
<p>When famous physicist Albert Einstein was figuring out how gravity works, he found that <a href="https://www.sciencenews.org/article/einsteins-genius-changed-sciences-perception-gravity">space actually has shape</a>. Something with a lot of mass, like a star or a planet, bends the space around it, like how a heavy ball held in the middle of an outstretched blanket makes the blanket change shape. If space has shape then surely space cannot be nothing.</p>
<figure class="align-center ">
<img alt="Earth and Moon shown with spacetime bending in green lines" src="https://images.theconversation.com/files/504659/original/file-20230116-5823-nz2wpv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/504659/original/file-20230116-5823-nz2wpv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=338&fit=crop&dpr=1 600w, https://images.theconversation.com/files/504659/original/file-20230116-5823-nz2wpv.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=338&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/504659/original/file-20230116-5823-nz2wpv.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=338&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/504659/original/file-20230116-5823-nz2wpv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=424&fit=crop&dpr=1 754w, https://images.theconversation.com/files/504659/original/file-20230116-5823-nz2wpv.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=424&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/504659/original/file-20230116-5823-nz2wpv.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=424&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Space bending around a planet.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/gravity-field-bend-spacetime-relativity-earth-1330914503">canbedone/Shutterstock</a></span>
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<p>It looks as if we won’t find “nothing” anywhere in our universe. Perhaps the place to look for nothing is beyond or outside the universe. This may be an impossible question, though. The universe is where space is, where stuff and energy are, and where time is. </p>
<h2>Outside the universe?</h2>
<p>Physicist <a href="https://www.newscientist.com/article/2053929-a-brief-history-of-stephen-hawking-a-legacy-of-paradox/">Stephen Hawking</a> explained the universe as having <a href="https://www.nextbigfuture.com/2018/03/hawking-talks-about-no-clear-big-bang-and-no-boundary-to-space-time.html">no boundary</a>, either in space or in time. When you are inside a house, the walls are the boundary and you could think about what’s outside, perhaps even see it through a window. But if the universe has no boundary, then there is no such thing as “outside” (or “before”). We could call it “nothing”, but it would be better to say it’s the absence of anything.</p>
<p>Nothing does exist as an idea. We can use “nothing” to count with: the idea of “zero” was being <a href="https://www.history.com/news/who-invented-the-zero">used 4,000 years ago</a>, and the oval symbol we use today was invented over 1,000 years ago.</p>
<p>Perhaps this is the only place where “nothing” exists: as an idea in our minds.</p><img src="https://counter.theconversation.com/content/197223/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Jacco van Loon 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>Nothing is harder to find than you might think.Jacco van Loon, Astronomer, Keele UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/973162018-06-07T20:28:49Z2018-06-07T20:28:49ZBees join an elite group of species that understands the concept of zero as a number<figure><img src="https://images.theconversation.com/files/221898/original/file-20180606-137295-1owba2y.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Bees live in complex environments, and make lots of decisions every day that are crucial for survival. </span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/honey-bees-kept-bee-box-hive-1099240694?src=73MQBwoOpxtqlXSw7mEAng-1-75">from www.shutterstock.com </a></span></figcaption></figure><p>When it comes to bees, it seems that nothing really does matter. </p>
<p>As shown in a <a href="http://science.sciencemag.org/lookup/doi/10.1126/science.aar4975">paper</a> published today, our research demonstrates that the honeybee can understand the quantitative value of nothing, and place zero in the correct position along a line of sequential numbers. </p>
<p>This is the first evidence showing that an insect brain can understand the concept of zero, and has implications for our understanding of how complex number processing evolved. More broadly, it may help us design better artificial intelligence solutions for operating in complex environments.</p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/want-a-better-camera-just-copy-bees-and-their-extra-light-sensing-eyes-80385">Want a better camera? Just copy bees and their extra light-sensing eyes</a>
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</em>
</p>
<hr>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/KQkP85I2UJM?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Bee brains are tiny - but they do get that zero is a number.</span></figcaption>
</figure>
<h2>What is ‘zero’, anyway?</h2>
<p>There are <a href="https://www.sciencedirect.com/science/article/pii/S1364661316301255">four stages of understanding the concept of zero</a> in human culture, history, psychology and animal learning.</p>
<p><strong>Stage one:</strong> Understanding zero as the absence of something, such as no food on your plate. This first level is likely enabled at an early stage of visual processing. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/221135/original/file-20180531-69490-1ja5sg3.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/221135/original/file-20180531-69490-1ja5sg3.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=370&fit=crop&dpr=1 600w, https://images.theconversation.com/files/221135/original/file-20180531-69490-1ja5sg3.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=370&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/221135/original/file-20180531-69490-1ja5sg3.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=370&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/221135/original/file-20180531-69490-1ja5sg3.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=464&fit=crop&dpr=1 754w, https://images.theconversation.com/files/221135/original/file-20180531-69490-1ja5sg3.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=464&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/221135/original/file-20180531-69490-1ja5sg3.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=464&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Perceiving nothing on your plate requires understanding the absence of information.</span>
<span class="attribution"><span class="source">'Adrian Dyer/RMIT University'</span></span>
</figcaption>
</figure>
<p><strong>Stage two:</strong> Understanding zero as “nothing” vs. “something”, such as the presence or absence of light in a room. “Nothing” is thus treated as a meaningful behavioural category.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/221138/original/file-20180531-69497-2kt6a1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/221138/original/file-20180531-69497-2kt6a1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=218&fit=crop&dpr=1 600w, https://images.theconversation.com/files/221138/original/file-20180531-69497-2kt6a1.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=218&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/221138/original/file-20180531-69497-2kt6a1.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=218&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/221138/original/file-20180531-69497-2kt6a1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=274&fit=crop&dpr=1 754w, https://images.theconversation.com/files/221138/original/file-20180531-69497-2kt6a1.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=274&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/221138/original/file-20180531-69497-2kt6a1.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=274&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Perceiving the absence of information relative to a stimulus like a light is the second stage of understanding zero.</span>
<span class="attribution"><span class="source">'Adrian Dyer/RMIT University'</span></span>
</figcaption>
</figure>
<p><strong>Stage three:</strong> Understanding that zero can have a numeric value and belongs at the low end of the positive number line. For example: 0 < 1 < 2 < 3 etc. (where < means “less than”). </p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/221133/original/file-20180531-69484-zv83tx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/221133/original/file-20180531-69484-zv83tx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=751&fit=crop&dpr=1 600w, https://images.theconversation.com/files/221133/original/file-20180531-69484-zv83tx.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=751&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/221133/original/file-20180531-69484-zv83tx.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=751&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/221133/original/file-20180531-69484-zv83tx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=944&fit=crop&dpr=1 754w, https://images.theconversation.com/files/221133/original/file-20180531-69484-zv83tx.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=944&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/221133/original/file-20180531-69484-zv83tx.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=944&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">By learning to choose less than options, bees learnt over one day to be able to transfer information and place zero at the lower end of all previously experienced numbers.</span>
<span class="attribution"><span class="source">Scarlett Howard, Adrian Dyer and Jair Garcia/RMIT University</span></span>
</figcaption>
</figure>
<p><strong>Stage four:</strong> Understanding that zero can be assigned a symbolic representation which can be used in modern mathematics and calculations, for example: 1 – 1 = 0.</p>
<p>Our new study shows honeybees have achieved stage three of understanding the concept of zero. </p>
<p>The honeybee now joins the elite few species which have demonstrated an understanding of zero to this advanced level. While <a href="https://www.sciencedirect.com/science/article/pii/S0010027700001128">rhesus monkeys</a>, <a href="https://www.hindawi.com/journals/ijz/2011/806589/">vervet monkeys</a>, a single <a href="https://link.springer.com/article/10.1007/s100710100086">chimpanzee</a>, and one <a href="https://homepages.uni-tuebingen.de/andreas.nieder/Nieder%20(2016)%20TICS.pdf">African grey parrot</a> have demonstrated the ability to learn or spontaneously understand the concept of zero, this is the first time that such a high level of cognitive number processing has been observed in an insect.</p>
<h2>Why care about zero?</h2>
<p>The <a href="https://homepages.uni-tuebingen.de/andreas.nieder/Nieder%20(2016)%20TICS.pdf">importance of zero</a> throughout human history is not to be underestimated. </p>
<p><a href="http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Chinese_numerals.html">Chinese counting rods used a blank space</a> to help represent a place holder in values, however zero went unnoticed as a number with a quantitative value for centuries. For example, Roman numerals do not have a symbol for zero. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/221140/original/file-20180531-69501-196wsew.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/221140/original/file-20180531-69501-196wsew.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=218&fit=crop&dpr=1 600w, https://images.theconversation.com/files/221140/original/file-20180531-69501-196wsew.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=218&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/221140/original/file-20180531-69501-196wsew.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=218&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/221140/original/file-20180531-69501-196wsew.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=274&fit=crop&dpr=1 754w, https://images.theconversation.com/files/221140/original/file-20180531-69501-196wsew.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=274&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/221140/original/file-20180531-69501-196wsew.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=274&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Zero in Chinese Rod Calculus around 4th Century B.C.</span>
<span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/0#/media/File:Zero_in_Rod_Calculus.png">Wikimedia Commons</a></span>
</figcaption>
</figure>
<p>It was not until <a href="https://archive.org/details/Brahmasphutasiddhanta_Vol_1">628AD that zero had a written record</a> which noted it as a number in its own right by Indian mathematician Brahma Gupta in his book Brahmasputha Siddhanta. This is the first written record to provide rules to use when doing calculations with zero. </p>
<p>The earliest record of the symbolic zero (0) we are familiar with today is from an <a href="https://www.livehistoryindia.com/amazing-india/2017/04/29/zero-number-one">Indian inscription on the wall</a> of a temple in Gwalior, India (AD 876). Arabic numerals, along with the modern idea of zero, did not reach the West <a href="https://www.britannica.com/topic/Hindu-Arabic-numerals">until 1200 AD</a>. </p>
<p>Interestingly, while it took centuries for the concept of zero to be fully understood and utilised in human culture, honeybees have learnt to apply previous number knowledge to demonstrate an understanding of zero <em>within a day</em> when presented with training to promote numerical cognition.</p>
<h2>How we asked bees about zero</h2>
<p>Bees often forage in <a href="https://theconversation.com/plants-use-advertising-like-strategies-to-attract-bees-with-colour-and-scent-92673">complex environments</a> and have evolved <a href="https://theconversation.com/which-square-is-bigger-honeybees-see-visual-illusions-like-humans-do-87673">visual processing solutions</a> adapted to this life. </p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/which-square-is-bigger-honeybees-see-visual-illusions-like-humans-do-87673">Which square is bigger? Honeybees see visual illusions like humans do</a>
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<hr>
<p>In our research, we tested number processing in bees by individually training them with special apparatus to collect a sugary reward, and learn the rules of “less than” considering the numbers 1 – 6. </p>
<p>An individual bee would need to choose between two numbers each time it returned to the experiment. For example, a bee would be presented with two new numbers (3 vs. 4; 1 vs. 2; 2 vs. 5, etc.) until it had reached, over many learning events, at least 80% accuracy for landing on and thus choosing the lowest number.</p>
<p>Once the bee achieved this, it would be presented with the previously unseen stimulus of “an empty set” representing zero. </p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/221707/original/file-20180605-119885-1mp78sb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/221707/original/file-20180605-119885-1mp78sb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=837&fit=crop&dpr=1 600w, https://images.theconversation.com/files/221707/original/file-20180605-119885-1mp78sb.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=837&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/221707/original/file-20180605-119885-1mp78sb.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=837&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/221707/original/file-20180605-119885-1mp78sb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1052&fit=crop&dpr=1 754w, https://images.theconversation.com/files/221707/original/file-20180605-119885-1mp78sb.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1052&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/221707/original/file-20180605-119885-1mp78sb.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1052&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">A honeybee chooses to land on the zero option.</span>
<span class="attribution"><span class="source">Scarlett Howard/RMIT University</span></span>
</figcaption>
</figure>
<p>Surprisingly, bees trained to the “less than” rule preferred to visit the empty set rather than any other higher value number. This means bees understood an empty set was lower in number than a set containing actual elements.</p>
<p>In further experiments, other bees were able to place zero at the low end of the numerical continuum and demonstrated numerical distance effects. Numerical distance effects are demonstrated when accuracy increases as the difference between two numbers increases. The study showed that while bees could differentiate between zero and one, they performed better when the numbers were further apart, such as in the case of zero vs. six.</p>
<p>The next step for research on the processing of zero is to understand how small and seemingly simple brains (like those of bees) represent zero in a neurological sense. </p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/plants-use-advertising-like-strategies-to-attract-bees-with-colour-and-scent-92673">Plants use advertising-like strategies to attract bees with colour and scent</a>
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</em>
</p>
<hr>
<p>Andreas Nieder, an expert in numerical competency in animals from the University of Tübingen in Germany <a href="http://science.sciencemag.org/content/360/6393/1069">writes</a>: </p>
<blockquote>
<p>The advanced numerical skills of bees and other animals raise the question of how their brains transform “nothing” into an abstract concept of zero. </p>
</blockquote>
<p>This new research on bees has created many new questions in the field and also makes it clear that brain size and complexity does not fully determine intelligence and, in particular, numerical ability.</p><img src="https://counter.theconversation.com/content/97316/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Scarlett Howard received funding from The Company of Biologists (JEB Travelling Fellowship) and has an Australian Government Research Training Program Scholarship.</span></em></p><p class="fine-print"><em><span>Adrian Dyer receives funding from The Australian Research Council.</span></em></p><p class="fine-print"><em><span>Aurore Avarguès-Weber a reçu des financements de la fondation Fyssen. </span></em></p>The Romans may not have had a symbol for zero, but bees understand what it means beyond just the simple assumption “there’s nothing there”.Scarlett Howard, PhD candidate, RMIT UniversityAdrian Dyer, Associate Professor, RMIT UniversityAurore Avarguès-Weber, Researcher , Université de Toulouse III – Paul SabatierLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/843322017-09-21T11:02:15Z2017-09-21T11:02:15ZFive ways ancient India changed the world – with maths<figure><img src="https://images.theconversation.com/files/186896/original/file-20170920-16437-hxdak9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Bakhshali manuscript.</span> <span class="attribution"><span class="source">Bodleian Libraries, University of Oxford</span></span></figcaption></figure><p>It should come as no surprise that the first recorded use of the number zero, <a href="https://www.theguardian.com/science/2017/sep/14/much-ado-about-nothing-ancient-indian-text-contains-earliest-zero-symbol">recently discovered</a> to be made as early as the 3rd or 4th century, happened in India. Mathematics on the Indian subcontinent has a rich history <a href="http://www.tifr.res.in/%7Earchaeo/FOP/FoP%20papers/ancmathsources_Dani.pdf">going back over 3,000 years</a> and thrived for centuries before similar advances were made in Europe, with its influence meanwhile spreading to China and the Middle East.</p>
<p>As well as giving us the concept of zero, Indian mathematicians made seminal contributions to the study of <a href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html">trigonometry, algebra, arithmetic and negative numbers among other areas</a>. Perhaps most significantly, the decimal system that we still employ worldwide today was first seen in India.</p>
<h2>The number system</h2>
<p>As far back as 1200 BC, mathematical knowledge was being written down as part of a large body of knowledge known as <a href="https://www.ancient.eu/The_Vedas/">the Vedas</a>. In these texts, numbers were commonly expressed as <a href="http://www.thehindu.com/sci-tech/science/understanding-ancient-indian-mathematics/article2747006.ece">combinations of powers of ten</a>. For example, 365 might be expressed as three hundreds (3x10²), six tens (6x10¹) and five units (5x10⁰), though each power of ten was represented with a name rather than a set of symbols. It is <a href="http://www.tifr.res.in/%7Earchaeo/FOP/FoP%20papers/ancmathsources_Dani.pdf">reasonable to believe</a> that this representation using powers of ten played a crucial role in the development of the decimal-place value system in India.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=126&fit=crop&dpr=1 600w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=126&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=126&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=158&fit=crop&dpr=1 754w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=158&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=158&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Brahmi numerals.</span>
<span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/Brahmi_numerals#/media/File:Indian_numerals_100AD.svg">Wikimedia</a></span>
</figcaption>
</figure>
<p>From the <a href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html">third century BC</a>, we also have written evidence of the <a href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_numerals.html">Brahmi numerals</a>, the precursors to the modern, Indian or Hindu-Arabic numeral system that most of the world uses today. Once zero was introduced, almost all of the mathematical mechanics would be in place to enable ancient Indians to study higher mathematics. </p>
<h2>The concept of zero</h2>
<p>Zero itself has a much longer history. The <a href="http://www.bodleian.ox.ac.uk/news/2017/sep-14">recently dated first recorded zeros</a>, in what is known as the Bakhshali manuscript, were simple placeholders – a tool to distinguish 100 from 10. Similar marks had already been seen in the <a href="https://www.scientificamerican.com/article/history-of-zero/">Babylonian and Mayan cultures in the early centuries AD</a> and arguably in <a href="https://www.scientificamerican.com/article/history-of-zero/">Sumerian mathematics as early as 3000-2000 BC</a>.</p>
<p>But only in India did the placeholder symbol for nothing progress to become a <a href="https://theconversation.com/nothing-matters-how-the-invention-of-zero-helped-create-modern-mathematics-84232">number in its own right</a>. The advent of the concept of zero allowed numbers to be written efficiently and reliably. In turn, this allowed for effective record-keeping that meant important financial calculations could be checked retroactively, ensuring the honest actions of all involved. Zero was a significant step on the route to the <a href="https://blog.sciencemuseum.org.uk/illuminating-india-starring-oldest-recorded-origins-zero-bakhshali-manuscript/">democratisation of mathematics</a>.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.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">No abacus needed.</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
</figcaption>
</figure>
<p>These accessible mechanical tools for working with mathematical concepts, in combination with a strong and open scholastic and scientific culture, meant that, by around 600AD, all the ingredients were in place for an explosion of mathematical discoveries in India. In comparison, these sorts of tools were not popularised in the West until the early 13th century, though <a href="http://www.springer.com/gb/book/9780387407371">Fibonnacci’s book liber abaci</a>. </p>
<h2>Solutions of quadratic equations</h2>
<p>In the seventh century, the first written evidence of the rules for working with zero were formalised in the <a href="https://archive.org/details/Brahmasphutasiddhanta_Vol_1">Brahmasputha Siddhanta</a>. In his seminal text, the astronomer <a href="http://www.storyofmathematics.com/indian_brahmagupta.html">Brahmagupta</a> introduced rules for solving quadratic equations (so beloved of secondary school mathematics students) and for computing square roots.</p>
<h2>Rules for negative numbers</h2>
<p>Brahmagupta also demonstrated rules for working with negative numbers. He referred to <a href="https://nrich.maths.org/5961">positive numbers as fortunes and negative numbers as debts</a>. He wrote down rules that have been interpreted by translators as: “A fortune subtracted from zero is a debt,” and “a debt subtracted from zero is a fortune”.</p>
<p>This latter statement is the same as the rule we learn in school, that if you subtract a negative number, it is the same as adding a positive number. Brahmagupta also knew that “The product of a debt and a fortune is a debt” – a positive number multiplied by a negative is a negative. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.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">Negative cows.</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
</figcaption>
</figure>
<p>For the large part, European mathematicians were reluctant to accept negative numbers as meaningful. Many took the view that <a href="https://books.google.co.uk/books?id=STKX4qadFTkC&pg=PA56&redir_esc=y#v=onepage&q&f=false">negative numbers were absurd</a>. They reasoned that numbers were developed for counting and questioned what you could count with negative numbers. Indian and Chinese mathematicians recognised early on that one answer to this question was debts.</p>
<p>For example, in a primitive farming context, if one farmer owes another farmer 7 cows, then effectively the first farmer has -7 cows. If the first farmer goes out to buy some animals to repay his debt, he has to buy 7 cows and give them to the second farmer in order to bring his cow tally back to 0. From then on, every cow he buys goes to his positive total. </p>
<h2>Basis for calculus</h2>
<p>This reluctance to adopt negative numbers, and indeed zero, held European mathematics back for many years. Gottfried Wilhelm Leibniz was one of the first Europeans to use zero and the negatives in a systematic way in his <a href="https://books.google.co.uk/books?id=CXG6CgAAQBAJ&pg=PA165&lpg=PA165&dq=Leibniz+zero+negatives+calculus&source=bl&ots=NsKOzdZL7Y&sig=dE2KJvCXPFovF4uyFdgHMJOAQr8&hl=en&sa=X&ved=0ahUKEwjdxKv8_LPWAhXhAcAKHR0XBcUQ6AEIMjAC#v=onepage&q=Leibniz%20zero%20negatives%20calculus&f=false">development of calculus</a> in the late 17th century. Calculus is used to measure rates of changes and is important in almost every branch of science, notably underpinning many key discoveries in modern physics.</p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=759&fit=crop&dpr=1 600w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=759&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=759&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=954&fit=crop&dpr=1 754w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=954&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=954&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Leibniz: Beaten to it by 500 years.</span>
</figcaption>
</figure>
<p>But <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Bhaskara_II.html">Indian mathematician Bhāskara</a> had already discovered many of Leibniz’s ideas <a href="https://ijrier.com/published-papers/volume-1/issue-8/origin-of-concept-of-calculus-in-india.pdf">over 500 years earlier</a>. Bhāskara, also made major contributions to algebra, arithmetic, geometry and trigonometry. He provided many results, for example on the solutions of certain “Doiphantine” equations, that <a href="https://www.amazon.co.uk/Mathematical-Achievements-Pre-modern-Mathematicians-Elsevier/dp/0123979137#reader_0123979137">would not be rediscovered in Europe for centuries</a>.</p>
<p><a href="https://link.springer.com/referenceworkentry/10.1007%2F978-1-4020-4425-0_8683">The Kerala school of astronomy and mathematics</a>, founded by <a href="https://en.wikipedia.org/wiki/Madhava_of_Sangamagrama">Madhava of Sangamagrama</a> in the 1300s, was responsible for many firsts in mathematics, including the use of mathematical induction and some early calculus-related results. Although no systematic rules for calculus were developed by the Kerala school, its proponents first conceived of many of the results that would <a href="http://www.jstor.org/stable/1558972?origin=crossref&seq=1#page_scan_tab_contents">later be repeated in Europe</a> including Taylor series expansions, infinitessimals and differentiation. </p>
<p>The leap, made in India, that transformed zero from a simple placeholder to a number in its own right indicates the mathematically enlightened culture that was flourishing on the subcontinent at a time when Europe was stuck in the dark ages. Although its reputation <a href="http://www.cbc.ca/news/technology/calculus-created-in-india-250-years-before-newton-study-1.632433">suffers from the Eurocentric bias</a>, the subcontinent has a strong mathematical heritage, which it continues into the 21st century by <a href="http://m.ranker.com/list/famous-mathematicians-from-india/reference?page=1">providing key players at the forefront of every branch of mathematics</a>.</p><img src="https://counter.theconversation.com/content/84332/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Christian Yates 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>High school students can blame ancient India for quadratic equations and calculus.Christian Yates, Senior Lecturer in Mathematical Biology, University of BathLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/842322017-09-20T14:14:28Z2017-09-20T14:14:28ZNothing matters: how the invention of zero helped create modern mathematics<figure><img src="https://images.theconversation.com/files/186837/original/file-20170920-16403-yazsqf.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>A small dot on an old piece of birch bark marks one of the biggest events in the history of mathematics. The bark is actually part of an ancient Indian mathematical document known as the Bakhshali manuscript. And the dot is the first known recorded use of the number zero. What’s more, researchers from the University of Oxford <a href="https://www.theguardian.com/science/2017/sep/14/much-ado-about-nothing-ancient-indian-text-contains-earliest-zero-symbol">recently discovered</a> the document is 500 years older than was previously estimated, dating to the third or fourth century – a breakthrough discovery.</p>
<p>Today, it’s difficult to imagine how you could have mathematics without zero. In a positional number system, such as the decimal system we use now, the location of a digit is really important. Indeed, the real difference between 100 and 1,000,000 is where the digit 1 is located, with the symbol 0 serving as a punctuation mark.</p>
<p>Yet for thousands of years we did without it. The Sumerians of 5,000BC employed a <a href="http://www.storyofmathematics.com/sumerian.html">positional system</a> but without a 0. In some <a href="https://www.scientificamerican.com/article/what-is-the-origin-of-zer/">rudimentary form</a>, a symbol or a space was used to distinguish between, for example, 204 and 20000004. But that symbol was never used at the end of a number, so the difference between 5 and 500 had to be determined by context.</p>
<p>What’s more, 0 at the end of a number makes multiplying and dividing by 10 easy, as it does with adding numbers like 9 and 1 together. The invention of zero immensely simplified computations, freeing mathematicians to develop vital mathematical disciplines such as algebra and calculus, and eventually the basis for computers. </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/pV_gXGTuWxY?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
</figure>
<p>Zero’s late arrival was partly a reflection of the negative views some cultures held for the concept of nothing. Western philosophy is plagued with grave misconceptions about nothingness and the mystical powers of language. The fifth century BC <a href="https://plato.stanford.edu/entries/parmenides/">Greek thinker Parmenides</a> proclaimed that nothing cannot exist, since to speak of something is to speak of something that exists. This Parmenidean approach kept prominent <a href="http://www.nothing.com/Heath.html">historical figures</a> busy for a long while. </p>
<p>After the advent of Christianity, religious leaders in Europe argued that since God is in everything that exists, anything that represents nothing must be satanic. In an attempt to save humanity from the devil, they <a href="http://yaleglobal.yale.edu/history-zero">promptly banished</a> zero from existence, though merchants continued secretly to use it.</p>
<p>By contrast, in Buddhism the concept of nothingness is not only devoid of any demonic possessions but is actually a central idea worthy of much study en route to nirvana. <a href="http://www.huffingtonpost.com/lewis-richmond/emptiness-most-misunderstood-word-in-buddhism_b_2769189.html">With such a mindset</a>, having a mathematical representation for nothing was, well, nothing to fret over. In fact, the English word “zero” is <a href="http://www.etymonline.com/index.php?term=zero">originally derived</a> from the Hindi “sunyata”, which means nothingness and is a central concept in Buddhism.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=299&fit=crop&dpr=1 600w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=299&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=299&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=376&fit=crop&dpr=1 754w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=376&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=376&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The Bakhshali manuscript.</span>
<span class="attribution"><span class="source">Bodleian Libraries</span></span>
</figcaption>
</figure>
<p>So after zero finally emerged in ancient India, it took almost 1,000 years to set root in Europe, much longer than in China or the Middle East. In 1200 AD, the Italian mathematician Fibonacci, who brought the decimal system to Europe, <a href="http://www.springer.com/gb/book/9780387407371">wrote that</a>: </p>
<blockquote>
<p>The method of the Indians surpasses any known method to compute. It’s a marvellous method. They do their computations using nine figures and the symbol zero.</p>
</blockquote>
<p>This superior method of computation, clearly reminiscent of our modern one, freed mathematicians from tediously simple calculations, and enabled them to tackle more complicated problems and study the general properties of numbers. For example, it led to the work of the seventh century Indian <a href="http://www.storyofmathematics.com/indian_brahmagupta.html">mathematician and astronomer Brahmagupta</a>, considered to be the beginning of modern algebra.</p>
<h2>Algorithms and calculus</h2>
<p>The Indian method is so powerful because it means you can draw up simple rules for doing calculations. Just imagine trying to explain long addition without a symbol for zero. There would be too many exceptions to any rule. The ninth century Persian mathematician Al-Khwarizmi was the first to meticulously note and exploit these arithmetic instructions, which <a href="https://books.google.co.uk/books?id=zTQrDwAAQBAJ&pg=PA47&lpg=PA47&dq=al+khwarizmi+abacus&source=bl&ots=PakFxbCVwk&sig=FWjwHlnppHAU9i_zgAficOcw4ug&hl=en&sa=X&ved=0ahUKEwii-46257PWAhUhBcAKHaWtCRcQ6AEIajAP#v=onepage&q=al%20khwarizmi%20abacus&f=false">would eventually</a> make the abacus obsolete.</p>
<p>Such mechanical sets of instructions illustrated that portions of mathematics could be automated. And this would eventually lead to the development of modern computers. In fact, the word “algorithm” to describe a set of simple instructions <a href="https://en.oxforddictionaries.com/definition/algorithm">is derived</a> from the name “Al-Khwarizmi”.</p>
<p>The invention of zero also created a new, more accurate way to describe fractions. Adding zeros at the end of a number increases its magnitude, with the help of a decimal point, adding zeros at the beginning decreases its magnitude. Placing infinitely many digits to the right of the decimal point corresponds to <a href="https://www.youtube.com/watch?v=JmyLeESQWGw&list=PLYoCqdGsxmn9HOU84Ln2PhPKpxRfaEO9h&index=17">infinite precision</a>. That kind of precision was exactly what 17th century thinkers Isaac Newton and Gottfried Leibniz needed to develop calculus, the study of continuous change.</p>
<p>And so algebra, algorithms, and calculus, three pillars of modern mathematics, are all the result of a notation for nothing. Mathematics is a science of invisible entities that we can only understand by writing them down. India, by adding zero to the positional number system, unleashed the true power of numbers, advancing mathematics from infancy to adolescence, and from rudimentary toward its current sophistication.</p><img src="https://counter.theconversation.com/content/84232/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>Turning zero from a punctuation mark into a number paved the way for everything from algebra to algorithms.Ittay Weiss, Teaching Fellow, Department of Mathematics, University of PortsmouthLicensed as Creative Commons – attribution, no derivatives.