tag:blogger.com,1999:blog-14039116420811512482024-03-05T13:36:21.935+00:00A Drunkard's Walk Through MathematicsMy meandering walk through the interesting parts of mathematicsNathan Jameshttp://www.blogger.com/profile/17655487615203178002noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-1403911642081151248.post-15222751086271718122015-02-16T18:07:00.000+00:002015-02-16T22:00:15.930+00:00My Favourite MathematiciansContinuing the 'My favourite...' series, I will now describe a few of my favourite past and present mathematicians. Due to the extreme number of extremely good mathematicians, this list was especially hard to form but I think I have narrowed down my favourite five.<br />
<br />
<a href="http://upload.wikimedia.org/wikipedia/commons/9/9b/Carl_Friedrich_Gauss.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://upload.wikimedia.org/wikipedia/commons/9/9b/Carl_Friedrich_Gauss.jpg" height="200" width="155" /></a><b>Carl Friedrich Gauss</b><br />
A truly incomparable mathematician who had a major influence in various areas of maths even at a very young age. I cannot name even a fraction of his various contributions to mathematics but they include:<br />
<br />
<ul>
<li>The construction of a heptadecagon (17 sided shape) with just a straight edge and compass, as shown in <a href="https://www.youtube.com/watch?v=87uo2TPrsl8&feature=youtu.be" target="_blank">Numberphile's recent video</a>.</li>
<li>The Prime Number Conjecture</li>
<li>The fundamental theorem of algebra </li>
<li>Advances in Modular Arithmetic (a post on this should soon follow)</li>
<li>and over <a href="http://en.wikipedia.org/wiki/List_of_things_named_after_Carl_Friedrich_Gauss" target="_blank">100 different theorems, ideas, methods and lemmas</a> named after him</li>
</ul>
<div>
Suffice it to say, he has had an incredible influence in almost all areas of maths since the 18th century.</div>
<br />
<br />
<b>Euclid</b><br />
<a href="http://upload.wikimedia.org/wikipedia/commons/9/9e/Euklid.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://upload.wikimedia.org/wikipedia/commons/9/9e/Euklid.jpg" height="200" width="115" /></a>Euclid, the classic Greek mathematician, was a person who revolutionised maths, introducing and developing whole fields of mathematics, especially geometry (he is known as the '<i>Father of Geometry</i>') and axiomatic theory.<br />
His most prominent work, the Elements, set the way for maths in the following millenia and until the 21st century was the second most sold work worldwide, following only the Bible.<br />
<br />
One of his most important ideas he put into practice was the use of axioms, specifically the 5 Postulates from which all his proofs came:<br />
1. A straight line segment can be drawn joining any two points.<br />
2. Any straight line segment can be extended indefinitely in a straight line.<br />
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.<br />
4. All right angles are congruent.<br />
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.<br />
<br />
You'll note that the 5th is rather more complex than the other four, but that is also a story for another post...<br />
<br />
<b>Leonhard Euler</b><br />
<a href="http://upload.wikimedia.org/wikipedia/commons/6/60/Leonhard_Euler_2.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://upload.wikimedia.org/wikipedia/commons/6/60/Leonhard_Euler_2.jpg" height="200" width="160" /></a>I don't know whether I can put this better than Laplace when he said:<br />
"<i>Read Euler, read Euler, he is the master of us all.</i>"<br />
Euler made great discoveries, like all in this list, in various areas of maths but there is one which I associate most strongly with Euler: Euler's Identity.<br />
Euler's identity is what some, if not most, mathematicians call the greatest work of maths as it links seemingly distinct areas of maths in a wonderfully simple and concise equation:<br />
<img alt="e^{i \pi} + 1 = 0" class="mwe-math-fallback-image-inline tex" src="http://upload.wikimedia.org/math/f/8/9/f897005615c391e14cd50112cda44665.png" style="background-color: white; border: none; color: #252525; display: inline-block; font-family: sans-serif; font-size: 14px; line-height: 22.3999996185303px; vertical-align: middle;" /><br />
where:<br />
<br />
<ul>
<li>'e' is the base of the natural logarithm, fundamental in calculus and discovered by Gauss to be linked strongly with the primes</li>
<li>'π' is the ratio between the circumference and diameter of a circle, fundamental to Euclidean and non-Euclidean geometry (named after Euclid of course)</li>
<li>1 is the multiplicitive identity, i.e. anything multiplied by 1 is itself</li>
<li>0 is the additive identity, i.e. anything add 0 is itself</li>
</ul>
<br />
This simple equation links all of these together and is in my opinion the greatest result in mathematics ever.<br />
<br />
<b>Pierre de Fermat</b><br />
<a href="http://upload.wikimedia.org/wikipedia/commons/f/f3/Pierre_de_Fermat.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://upload.wikimedia.org/wikipedia/commons/f/f3/Pierre_de_Fermat.jpg" height="200" width="149" /></a>Unlike the others in this list, Fermat was an amateur mathematician but, as E.T. Bell put it, the 'Prince of Amateurs'.<br />
He was most famous probably for not what he did, but what he didn't do. He was famed for claiming he had solved big problems without releasing his proofs, most famously in the case of 'Fermat's Last Theorem':<br />
<i>"It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain."</i><br />
<br />
The jury is still out as to whether he actually solved it or not, but given that it was only solved 358 years later using areas of mathematics as yet undiscovered, it seems unlikely.<br />
However that does not detract from his brilliance as a mathematician.<br />
<br />
<b>Srinivasa Ramanujan</b><br />
<a href="http://upload.wikimedia.org/wikipedia/commons/c/c1/Srinivasa_Ramanujan_-_OPC_-_1.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://upload.wikimedia.org/wikipedia/commons/c/c1/Srinivasa_Ramanujan_-_OPC_-_1.jpg" height="200" width="145" /></a><br />
Ramanujan, the most modern mathematician in this list, was an exceptional person. Untrained, self-taught and a bit mad (who isn't?), he was brought over to England by G. H. Hardy, author of A Mathematician's Apology, after the exchange of a number of letters which revealed Ramanujan to be a truly incredible mathematician.<br />
In his comparatively short mathematical life, he came up with around 3900 results notably making progress towards areas of maths including infinite series, continued fractions,...<br />
As Erdős recollected, <i>"Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself a score of 25, J.E. Littlewood 30, David Hilbert 80 and Ramanujan 100."</i><br />
<br />
But what he is now most famous for outside of mathematical circles is the taxicab story, which highlights the brilliance of this man:<br />
<i>"I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No", he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."</i><br />
<br />
<b>A few others:</b><br />
<ul>
<li>Blaise Pascal</li>
<li>Pythagoras</li>
<li>Archimedes</li>
<li>Isaac Newton</li>
<li>Gottfried Wilhelm Leibniz</li>
<li>René Descartes</li>
<li>David Hilbert</li>
<li>Évariste Galois</li>
<li>Kurt Gödel</li>
<li>Andrew Wiles</li>
</ul>
Nathan Jameshttp://www.blogger.com/profile/17655487615203178002noreply@blogger.com0tag:blogger.com,1999:blog-1403911642081151248.post-50136531394118780482015-02-10T20:30:00.000+00:002015-02-10T20:30:25.278+00:00My Favourite Maths Books<div class="MsoNormal">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwobCUok-_DDeSaWoCIeBBLfCf4ZHLSf83751v5Z81C5RsNP3dDyWXiFqGpoQh-Y90xOi10kR3tpD_AGaoCyCJB9wXlIg2f_H1uh66TPCMFiYAi9NtIhl983ci4GazV_cyF0JGW7QaAXg/s1600/B1+-+Fermat's%2BLast%2BTheorem.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwobCUok-_DDeSaWoCIeBBLfCf4ZHLSf83751v5Z81C5RsNP3dDyWXiFqGpoQh-Y90xOi10kR3tpD_AGaoCyCJB9wXlIg2f_H1uh66TPCMFiYAi9NtIhl983ci4GazV_cyF0JGW7QaAXg/s1600/B1+-+Fermat's%2BLast%2BTheorem.jpg" height="200" width="132" /></a><span style="background-color: black; color: white;">I have a (relatively) large selection of maths books which I
really enjoy reading and which were what I would say sparked my love for maths. I
would love to share with you my top 5 of these.</span></div>
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<span style="background-color: black;"><span style="color: white;"><br /></span></span></div>
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<span style="background-color: black;"><span style="color: white;"><span style="text-indent: -18pt;"><br /></span></span></span>
<span style="background-color: black;"><span style="color: white;"><span style="text-indent: -18pt;">1) <b>Fermat’s Last Theorem by Simon Singh</b></span></span></span></div>
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<span style="background-color: black;"><span style="color: white;"><o:p></o:p></span></span></div>
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<span style="background-color: black;"><span style="color: white;">This is the book that really kick-started my love for maths.
It provides a great history of maths from the Greeks to modern days whilst
keeping the running theme of Fermat’s Last Theorem. It was the first book to
introduce me to the ideas of proof, I read it when I was 11 or 12, and it really
left an imprint in my mind about what maths is all about.</span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZUxXZOrS4L9VVmEyY20e5aox3nWW1Pf2Qt_EEBp-tmDqgzdqewfuOVTtclz4q8iDAJ51wrHcOSK5xbZUUxxnutKwJ9fVZFw8SW_X4tgRLDNfNJkW-qPYYshmupcURrwsG8SNcmJDxHmQ/s1600/B2+-+Adventures+in+Numberland.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZUxXZOrS4L9VVmEyY20e5aox3nWW1Pf2Qt_EEBp-tmDqgzdqewfuOVTtclz4q8iDAJ51wrHcOSK5xbZUUxxnutKwJ9fVZFw8SW_X4tgRLDNfNJkW-qPYYshmupcURrwsG8SNcmJDxHmQ/s1600/B2+-+Adventures+in+Numberland.jpg" height="200" width="132" /></a><span style="background-color: black;"><span style="color: white;">It also describes the actual problem extremely well, and
shows its difficulty through the vast number of mathematicians attempting it
over the 358 years it was unsolved. It is definitely my favourite maths book
and will always have a place in my heart. </span></span></div>
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<span style="background-color: black; text-indent: -18pt;"><b><span style="color: white;"><br /></span></b></span>
<span style="background-color: black; text-indent: -18pt;"><b><span style="color: white;">2) Alex’s Adventures in Numberland by Alex Bellos</span></b></span></div>
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<span style="background-color: black;"><span style="color: white;"><o:p></o:p></span></span></div>
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<span style="color: white;"><span style="background-color: black;">Another book which has shaped my mathematical journey is
this one. It contains such a great variety of information, from ancient
counting systems to casino </span><span style="background-color: black;">probabilities, and shows the applications of maths in
everyday life really well. It was definitely a fascinating read and a book I’d
recommend to both maths-addicts and people without a maths background alike.</span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOMMv0R9BVkpqam4VNyf9JXUwrYItpRsylM8QqgP1vn0Ece9iOHI9oevc0ZWghm-o6JUTfnASofJgP-EUuW-WDG_Caf2RHwaZs8TpB1QLrWpmwOR_cIaTfkxD9ZM1Gd9Fm6oR3yATyXiU/s1600/B3+-+Music+of+Primes.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOMMv0R9BVkpqam4VNyf9JXUwrYItpRsylM8QqgP1vn0Ece9iOHI9oevc0ZWghm-o6JUTfnASofJgP-EUuW-WDG_Caf2RHwaZs8TpB1QLrWpmwOR_cIaTfkxD9ZM1Gd9Fm6oR3yATyXiU/s1600/B3+-+Music+of+Primes.jpg" height="200" title="" width="131" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOMMv0R9BVkpqam4VNyf9JXUwrYItpRsylM8QqgP1vn0Ece9iOHI9oevc0ZWghm-o6JUTfnASofJgP-EUuW-WDG_Caf2RHwaZs8TpB1QLrWpmwOR_cIaTfkxD9ZM1Gd9Fm6oR3yATyXiU/s1600/B3+-+Music+of+Primes.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><span style="background-color: black; color: white;"></span></a><br />
<b style="text-indent: -18pt;"><span style="background-color: black; color: white;"><br /></span></b>
<b style="text-indent: -18pt;"><span style="background-color: black; color: white;">3) The Music of the Primes by Marcus du Sautoy</span></b></div>
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<span style="color: white;"><span style="background-color: black;">This books also concentrates on a particular problem</span><span style="background-color: black;">, in
this case the Riemann Hypothesis, and I think it describes it extremely well in
a similar way to Fermat’s Last Theorem in that it gives a deep background of
both prime numbers in general and the core question giving a historical context
throughout.<o:p></o:p></span></span></div>
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<span style="color: white;"><span style="background-color: black;">Unlike the previous two, it also goes into some quite
complex maths which was very interesting and explai</span><span style="background-color: black;"><span style="font-family: inherit;">ns it very well.</span></span></span><br />
<span style="background-color: black; font-family: inherit; line-height: 1.295; white-space: pre-wrap;"><span style="color: white;"><br /></span></span>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmEM732oMq6knxepygpwvsqDVa6yj_gUWZjOJhyphenhyphenLcfwfImdX7BC9RVAHZRJdxenTm9Ps36D7oOU8rNoOzsd-pmW3T4S_a35HAgR0VDGg1Vcy5kWT5zf-9KnyF0qIrhrvxM6Bfgy72ScwQ/s1600/B4+-+1089.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><span style="color: white;"><img alt="" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmEM732oMq6knxepygpwvsqDVa6yj_gUWZjOJhyphenhyphenLcfwfImdX7BC9RVAHZRJdxenTm9Ps36D7oOU8rNoOzsd-pmW3T4S_a35HAgR0VDGg1Vcy5kWT5zf-9KnyF0qIrhrvxM6Bfgy72ScwQ/s1600/B4+-+1089.jpg" height="200" title="" width="130" /></span></a><b><span style="color: white;"><span style="background-color: black; font-family: inherit;"></span></span></b><br />
<b><b><span style="background-color: black; font-family: inherit;"><span style="color: white;"><br /></span></span></b></b>
<b><span style="color: white;"><span style="background-color: black;">4) </span><span style="background-color: black; font-family: inherit; line-height: 1.295; white-space: pre-wrap;">1089 and All That by David Acheson</span></span></b><br />
<span style="background-color: black; font-family: inherit; line-height: 1.295; white-space: pre-wrap;"><span style="color: white;">1089 and All That, despite being smaller than the others in the list, contains an excellent summing up of what maths is about. Its topics range from the fundamental ideas beh</span></span><span style="background-color: black; color: white; font-family: inherit; line-height: 1.295; white-space: pre-wrap;">ind mathematics, “Wonderful Theorems, Beautiful Proofs and Great Application”, to a brilliant finale deriving Euler’s Identity. Definitely a great book for providing insight into all areas of maths and deservedly has a place in my Top 5.</span><br />
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<span style="background-color: black; color: white; font-family: inherit; line-height: 1.295; white-space: pre-wrap;"><b><br /></b></span></div>
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<span style="background-color: black; color: white; font-family: inherit; line-height: 1.295; white-space: pre-wrap;"><b>5) The Drunkard’s Walk by Leonard Mlodinow</b></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkUn-dGtk1pQGaT2P-z1IJzVv3AitiIwrt3Wdt8EXJPp1z__tKEM6Yldmw-JjOtl9wSVVo77-isaQQRcTYv0E6iijEn1OnYAagZkRJirtpM9x2qglJp36Er13tITL3AV0Di1wF46ebQHM/s1600/B5+-+Drunkard's%2BWalk.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkUn-dGtk1pQGaT2P-z1IJzVv3AitiIwrt3Wdt8EXJPp1z__tKEM6Yldmw-JjOtl9wSVVo77-isaQQRcTYv0E6iijEn1OnYAagZkRJirtpM9x2qglJp36Er13tITL3AV0Di1wF46ebQHM/s1600/B5+-+Drunkard's%2BWalk.jpg" height="200" width="130" /></a><span style="background-color: black; color: white; font-family: inherit; line-height: 1.295; white-space: pre-wrap;">This book, after which this blog is named, is an excellent description of a more specific topic than the others, randomness and uncertainty. I, for one, really found it interest as it was an area I knew little about before and it, like most of the others, showed how well maths can relate to the real world whilst also being beautiful in itself. Suffice it to say, it is an excellent book which I would recommend to anyone interested in the subject</span></div>
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<span style="background-color: black; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span style="color: white; font-family: inherit;"><br /></span></span></div>
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<span style="background-color: black; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span style="color: white; font-family: inherit;">A few others:</span></span></div>
<ul style="margin-bottom: 0pt; margin-top: 0pt;">
<li dir="ltr" style="font-style: normal; font-variant: normal; font-weight: normal; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.295; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: black; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span style="color: white; font-family: inherit;">A Mathematician’s Apology by G. H. Hardy – An insight into the life of one of history’s greatest mathematicians and a defence of maths for the sake of maths</span></span></div>
</li>
<li dir="ltr" style="font-style: normal; font-variant: normal; font-weight: normal; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.295; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: black; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span style="color: white; font-family: inherit;">The Joy of X by Steven Strogatz – A history of algebra and more maths besides</span></span></div>
</li>
<li dir="ltr" style="font-style: normal; font-variant: normal; font-weight: normal; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.295; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: black; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span style="color: white; font-family: inherit;">Number Freak by Derrick Niederman – A run-through of various facts about all the numbers from 0 to 200</span></span></div>
</li>
<li dir="ltr" style="font-style: normal; font-variant: normal; font-weight: normal; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.295; margin-bottom: 0pt; margin-top: 0pt;">
<span style="background-color: black; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span style="color: white; font-family: inherit;">From 0 to Infinity in 26 Centauries by Chris Waring – A history of mathematics from the Greeks to the Renaissance to the future</span></span></div>
</li>
<li dir="ltr" style="font-style: normal; font-variant: normal; font-weight: normal; list-style-type: disc; text-decoration: none; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.295; margin-bottom: 8pt; margin-top: 0pt;">
<span style="background-color: black; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span style="color: white; font-family: inherit;">Perfect Rigour by Masha Gesson – A biography of Perelman, who solved one of the seven Millennium Problems, the Poincaré Conjecture, and turned down the million dollars</span></span></div>
</li>
</ul>
<span style="color: white; font-family: inherit;"><span id="docs-internal-guid-f9688652-7515-44c1-ed5a-a0375e22ee4d" style="background-color: black;"></span></span><br />
<div dir="ltr" style="line-height: 1.295; margin-bottom: 8pt; margin-top: 0pt;">
<span style="background-color: black; font-style: normal; font-variant: normal; font-weight: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span style="color: white; font-family: inherit;">I’d love to hear what maths books you like and have inspired you in your mathematical journey…</span></span></div>
</div>
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<span style="background-color: black;"><span style="color: white;">Continuing my previous post, we will now look at happy and
palindromic numbers...<br />
<br />
<b><u>Happy Numbers<br />
</u></b>For any number, if you add up the squares of each digit and repeat this
you will eventually either reach 1 or start an endlessly repeating
loop. Numbers which result in a value of 1 are called Happy Numbers.<br />
For example, 19 is a happy number because 1<sup>2</sup> + 9<sup>2</sup> =
82, 8<sup>2</sup> + 2<sup>2</sup> = 68, 6<sup>2</sup> + 8<sup>2</sup> =
100, 1<sup>2</sup> + 0<sup>2</sup> + 0<sup>2</sup> = 1. As you
can see, this ends in 1 and 19 is therefore a happy number.<br />
20, however, is not a happy number as 2<sup>2 </sup>+ 0<sup>2</sup>= 4, 4<sup>2 </sup>=
16, 1<sup>2 </sup>+ 6<sup>2</sup>=37, 3<sup>2</sup>+7<sup>2</sup>=58, 5<sup>2 </sup>+8<sup>2</sup>=89,
8<sup>2</sup>+9<sup>2</sup>=145, 1<sup>2</sup>+4<sup>2</sup>+5<sup>2</sup>=42,
4<sup>2</sup>+2<sup>2</sup>=20 which is back where we began. In
fact, every non-happy number ends up in this sequence; 20, 4, 16, 37, 58, 89,
145, 42, 20.<br />
The first 10 happy numbers are 1, 7, 10, 13, 19, 23, 28, 31, 32 and
44. Happy numbers which are also prime are, inventively, called Happy
Primes; examples of which are 7 and 19. <o:p></o:p></span></span></div>
<br />
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<span style="background-color: black;"><span style="color: white;"><br />
<b><u>Palindromic Numbers<br />
</u></b>Palindromic numbers, in the same way as palindromic words or phrases,
are numbers which are the same when reversed, e.g. 131.<br />
In themselves they may not be that interesting, there are infinitely many and
quite regular, but they can be combined with other sets of numbers to give
interesting results. For example Palindromic Primes (palprimes) include
313, 797 and Belphegor's Prime: 1000000000000066600000000000001.<br />
<br />
By their nature, all Mersenne and Fermat primes are palindromic when written in
binary as they are either a run of '1's or a run of '0's with '1's at both
ends. For example 31, 11111, is a Mersenne prime and 17, 10001, is a Fermat
prime.<br />
Combining these gives Palindromic Happy Prime Nunbers, the first one being 10<sup>150006</sup> +
7426247×10<sup>75000</sup> + 1 which has over 150 thousand digits.</span></span><o:p></o:p></div>
Nathan Jameshttp://www.blogger.com/profile/17655487615203178002noreply@blogger.com2tag:blogger.com,1999:blog-1403911642081151248.post-47067368637305270332014-06-26T20:17:00.001+01:002015-02-08T14:54:07.699+00:00What are vampire/perfect/happy/palindromic numbers? - Part 1<div style="line-height: 115%; margin-bottom: 10.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 0cm;">
<span style="background-color: black; color: white;"><span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;">There are many different types of numbers and
classifications of numbers, some of which you will have heard of such as the
obvious integers, negatives, evens and the like. However, there are also some rarer,
more interesting types a few examples of which I will give now.</span><span style="font-family: "Georgia","serif";"><o:p></o:p></span></span></div>
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<b><u><span style="background-color: black; font-family: Georgia, serif;"><span style="color: white;">Vampire Numbers<o:p></o:p></span></span></u></b></div>
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<span style="background-color: black; font-family: Georgia, serif; font-size: 9pt; line-height: 107%;"><span style="color: white;">Vampire Numbers are whole numbers which can be factorised
into two numbers which together have the same digits as the original number.
The two factors must have half the number of digits of the original number, and
the number itself must therefore have an even number of digits (the factors
must also not both have trailing zeroes). It is easier to explain these with an
example, so I will give one. 1395 is a vampire number because it is 15
multiplied by 93 and 1395 has the same digits as 15 and 93.Similarly 1260 is a
vampire number as it is 21 times 60 and 1435 is 35 multiplied by 41.<o:p></o:p></span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjoUoRRIBiiZlMkbJN1-cM7qOXXx62_olrlZr0eIQdhDjtIpuLk2TEDSxy8lTpOEg34uFRPAqk4vPKvF5V9WFw_BQliLkQH1SJ5nBGNEHFjv4hb2HkL_oTj264AmYwN-Vji_AUEFfeMJiU/s1600/Vampire1.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjoUoRRIBiiZlMkbJN1-cM7qOXXx62_olrlZr0eIQdhDjtIpuLk2TEDSxy8lTpOEg34uFRPAqk4vPKvF5V9WFw_BQliLkQH1SJ5nBGNEHFjv4hb2HkL_oTj264AmYwN-Vji_AUEFfeMJiU/s1600/Vampire1.png" height="141" width="200" /></a></div>
<span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 107%;"><span style="background-color: black; color: white;">There are also different subsets of vampire numbers such as
numbers which have two or more possible pairs of ‘fangs’ (as the factors are
known). The first one of these is 125460 which is 204 × 615 and 246 × 510, the
first number with three possible pairs of factors is 13078260, which equals
1620 × 8073, 1863 × 7020 and 2070 × 6318. I could go on. Also there are
Pseudovampire numbers whose fangs are not equal to half the number of digits of
the original number. There are lots of these, well infinitely many, such as 126
which equals 6×21. Finally, there are Prime vampire numbers whose fangs are its
(only) prime factors as in 117067 w</span><span style="background-color: black;"><span style="color: white;">hich is 167*701.</span><o:p></o:p></span></span></div>
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<span style="background-color: black; font-family: "Georgia","serif"; font-size: 9pt; line-height: 107%;"><br /></span></div>
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<b><u><span style="background-attachment: initial; background-clip: initial; background-color: black; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: Georgia, serif;"><span style="background-color: black; color: white;">Perfect Numbers<o:p></o:p></span></span></u></b></div>
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<span style="color: white;"><span style="font-family: "Georgia","serif"; font-size: 9pt; line-height: 115%;"><span style="background-color: black;">Perfect numbers are numbers wh</span><span style="background-color: black;">ose divisors (not including
itself) sum to the original number. The first example of this is the number 6,
whose factors are 1, 2 and 3, which clearly add up to six, and the second
perfect number is 28, which is 1 + 2 + 4 + 7 + 14. What is interesting about
Perfect Numbers is that they are indefinitely linked with Mersenne Primes,
which are primes which are 1 less than a power of 2 such as 3 which is 1 less than
4, 7 (1 less than 8) and 31 (1 less than 32). For every Mersenne Prime, m,
m(m+1)/2 is a Perfect Number. For example, the first Mersenne Prime is 3 and
3(3+1)/2 = 12/2 = 6, the first Perfect Number. Leonhard Euler, one of my
favourite mathematicians, proved that all perfect numbers are like this. As a
concept, they are very old. The Greeks knew the first four (6, 28, 496, 8128),
and the fifth was first recorded in the mid 15</span></span><sup style="background-color: black;"><span style="font-family: "Georgia","serif"; font-size: 5.5pt; line-height: 115%;">th</span></sup><span style="background-color: black; font-family: "Georgia","serif"; font-size: 9pt; line-height: 115%;">
century.<o:p></o:p></span></span></div>
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<b><span style="background-color: black; font-family: Georgia, serif; font-size: 10pt; line-height: 115%;"><span style="color: white;">Amicable
and Sociable Numbers<o:p></o:p></span></span></b></div>
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<span style="background-color: black; color: white;"><span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;">Perfect numbers are linked with Amicable and Sociable
numbers, which both involving adding up the factors. Amicable numbers come in
pairs, such as 220 and 284, where the sum of the factors of the first number is
the second number and vice versa. Other pairs include 1184 and 1210 as well as
2620 and 2924. Sociable numbers are groups of numbers that form a chains so
that the sum of the factors of A equals B, the sum of B equals C and the sum of
C equals A. One such group is of four numbers; 1,264,460 -> 1,547,860 ->
1,727,636 -> 1,305,184 which goes back to 1,264,460. One delightful use of
amicable numbers is as a nerdy, romantic gesture as shown by the people at
Maths Gear - </span><span style="font-family: "Georgia","serif";"><a href="http://mathsgear.co.uk/products/amicable-numbers-pair-of-keyrings-nerd-romance"><span style="font-size: 9.0pt; line-height: 115%;">http://mathsgear.co.uk/products/amicable-numbers-pair-of-keyrings-nerd-romance</span></a></span><span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;">.<o:p></o:p></span></span></div>
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Nathan Jameshttp://www.blogger.com/profile/17655487615203178002noreply@blogger.com0tag:blogger.com,1999:blog-1403911642081151248.post-88367836797782238522014-06-25T10:18:00.001+01:002014-06-26T20:18:45.106+01:00Why does 5 round up? – The maths of rounding and approximating<div style="margin-bottom: .0001pt; margin: 0cm;">
<span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;">I was taught from a young age,
as I’m sure you were too, that 5 and numbers greater than 5 round up, whilst
numbers smaller than 5 round down. But why? <o:p></o:p></span></div>
<div style="margin-bottom: .0001pt; margin: 0cm;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4drWTDCa5H95_3_Lk-VLEtva0AWALTox8eyRa1S5OX_9uJoYfYnQTDS0nsqlMRy0991XR7b_LltIuh_3Z2gPgZmwwX1p5RRk8FLlbWS3od2_eiaSX6EAsNrdK8vsZXLn5CkM7d6dpJKA/s1600/Picture1.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4drWTDCa5H95_3_Lk-VLEtva0AWALTox8eyRa1S5OX_9uJoYfYnQTDS0nsqlMRy0991XR7b_LltIuh_3Z2gPgZmwwX1p5RRk8FLlbWS3od2_eiaSX6EAsNrdK8vsZXLn5CkM7d6dpJKA/s1600/Picture1.png" height="320" width="269" /></a></div>
<div style="margin-bottom: .0001pt; margin: 0cm;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiob-VZMVjBWyM3KAWWUptDsdoMSiY5tpKhjKvczZhafvE8eBXw67ZJoDNZwgjU54bTIhAbIUhTdP0gWxyuO-fjiyjGwj3CYbTOmIUqbd2HfgBKUNCDiSR7M5GFo58Xuqdfz0-b9LvVOGk/s1600/Picture1.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><span style="background-color: white; clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><u></u></span></a><span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;">The essence of rounding numbers
is to get an accurate, long value such as 6.72 and make it an easy, simple
value like 7. In order for the final number to be as close as possible to the
original, whilst still being simple, you must find the nearest ‘nice’ (which
depends on the context) number. The important word here is nearest.<o:p></o:p></span></div>
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<br /></div>
<div style="margin-bottom: .0001pt; margin: 0cm;">
<span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;">This means that if you were
rounding 6.72 you would chose 7 as that is closer to 6.72 than 6. In the same
way, 6.42 would round down to 6, as 6.42 is close to 6 than it is to 7. It
follows that 6.49 would round down while 6.51 would round up, 6.499 down and
6.501 up, 6.4999 down and 6.5001 up. The division is clearly going to be at
6.5, but that is equidistant from both 6 and 7 so which way does 6.5 go?<o:p></o:p></span></div>
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<br /></div>
<div style="margin-bottom: .0001pt; margin: 0cm;">
<span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;">The answer lies purely in
convention, from a purely mathematical perspective it should do
neither, but then again rounding isn’t a purely mathematical concept.
There are many different methods of sorting this, from the odd/even rule
to the 5 always rounds up to the view that nothing in real life would give a
value of exactly five. It also depends on context, however, for example if the
maximum voltage for a circuit is 4.5V you should round that down to 4V rather
than face the danger of going above the exact value. On the other hand, if you
need a minimum fuse of 3.5A you should round that up to 4A in order for it not
to fuse when a normal current goes through it (I admit my knowledge of
electronics is limited). All in all, the general convention is that 5
rounds up, albeit with little mathematical basis.<o:p></o:p></span></div>
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<br /></div>
<div style="margin-bottom: .0001pt; margin: 0cm;">
<span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;">Rounding also leads to other
interesting fallacies and mistakes. Recently someone asked me whether 4.46
would round up or down. Immediately I said 4, with my opinion that 4.46 is
closer to 4 than 5. However, their argument was that 4.46 would round up
to one decimal place to give 4.5 and that would then round up (as we have
decided that convention dictates 5 rounds up) to 5. This shows the issue with
rounding a rounded number, which can cause numerous differences such as in
this case the large difference between 4 (a lovely, even, perfect square) and 5
(a nasty, odd, prime number). In these situations, it is important to go
back to the trick of, is it closer to 4 or 5?<o:p></o:p></span></div>
<div style="margin-bottom: .0001pt; margin: 0cm;">
<span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;">Another fallacy caused by
rounding is the practice of performing calculations on a rounded number which
is perfectly shown in this classic joke:<o:p></o:p></span></div>
<div align="center" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 43.2pt; margin-right: 43.2pt; margin-top: 0cm; text-align: center;">
<i><span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;">Museum
goer: How old is this dinosaur?</span></i><span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div align="center" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 43.2pt; margin-right: 43.2pt; margin-top: 0cm; text-align: center;">
<i><span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;">Tour
guide: 70 million years and 2 weeks</span></i><span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div align="center" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 43.2pt; margin-right: 43.2pt; margin-top: 0cm; text-align: center;">
<i><span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;">Museum
goer (shocked): How do you know?</span></i><span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div align="center" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 43.2pt; margin-right: 43.2pt; margin-top: 0cm; text-align: center;">
<i><span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;">Tour
guide: When I first started working here the manager told me the dinosaur was
70 million years old. And I’ve been working here for 2 weeks.<o:p></o:p></span></i></div>
<div align="center" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 43.2pt; margin-right: 43.2pt; margin-top: 0cm; text-align: center;">
<br /></div>
<div style="margin-bottom: .0001pt; margin: 0cm;">
<a href="https://www.blogger.com/null" name="h.gjdgxs"></a><span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;">This
shows the issues with false precision, which in other words is taking a rounded
number literally.<o:p></o:p></span></div>
<br />
<div style="margin-bottom: .0001pt; margin: 0cm;">
<span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;">In conclusion, although
rounding is essential in the real world, it is easy to make mistakes and
important to question the essence behind the mathematics of rounding.<o:p></o:p></span><br />
<span style="background-color: #fcfcfc; font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; line-height: 16px;"><span style="display: none;">claimtoken-53ac6d9e8a2a9</span></span><br />
<span style="background-color: #fcfcfc; font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; line-height: 16px;"><span style="display: none;">claimtoken-53ac1b21eb3dc</span></span></div>
Nathan Jameshttp://www.blogger.com/profile/17655487615203178002noreply@blogger.com0tag:blogger.com,1999:blog-1403911642081151248.post-36137637299065660232014-06-25T10:18:00.000+01:002014-06-26T20:20:38.087+01:00Why do I love maths? – A maths student’s apology<div class="MsoNormal">
<span style="background-color: black; font-family: Georgia, serif; font-size: 9pt; line-height: 115%;"><span style="color: white;">To start off my entry into the ‘blogosphere’, I am going to
explain, in the style of G.H Hardy’s <i>A Mathematician’s
Apology</i>, why I love maths and why you should too.<o:p></o:p></span></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="background-color: black; font-family: Georgia, serif; font-size: 9pt; line-height: 115%;"><span style="color: white;">Firstly, I love maths because it ranges from the simplest
ideas, such as 1+1=2, to such incredibly complex thoughts as Graham’s Number
and the Riemann Hypothesis. This means that anyone can do maths; it is not
limited to a select few as neurobiology is, and you don’t need to be in the top
100 people to understand it as is the case with quantum physics. Also, all of the complex, difficult and high-level
pieces of mathematics are all dependant on the simplest of axioms and can be
most proven from them. This amazes me, how so many proofs, conjectures,
theorems, theories and hypotheses can all stem from the numbers 1, 2, 3 etc.<o:p></o:p></span></span></div>
<blockquote class="tr_bq">
<span style="background-color: black; background-position: initial initial; background-repeat: initial initial; font-family: Georgia, serif; font-size: 9pt; line-height: 115%;"><span style="color: white;">“Mathematics allows for no hypocrisy and no
vagueness.” Stendhal</span></span></blockquote>
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<br /></div>
<div class="MsoNormal">
<span style="background-color: black; font-family: Georgia, serif; font-size: 9pt; line-height: 115%;"><span style="color: white;">Maths is also useful yet remains self-rewarding. Although
maths nowadays is used for so much of our modern life, from binary in
technology to cyphers for internet banking, the thrill of maths often comes
from the doing of the maths. Maths for the sake of maths. This gives maths an
artistic quality, and both areas are similar in many ways. In the same way some
may find a Mondrian painting beautiful, I find Pascal’s triangle beautiful and
in the same way multiple artists can see one painting in many different ways,
many different results can come from one mathematical concept.<o:p></o:p></span></span></div>
<blockquote class="tr_bq">
<span style="background-color: black; background-position: initial initial; background-repeat: initial initial; font-family: Georgia, serif; font-size: 9pt; line-height: 115%;"><span style="color: white;">“A mathematician, like a painter or a poet,
is a maker of patterns” G.H Hardy</span></span><br />
<br /></blockquote>
<div class="MsoNormal" style="background-position: initial initial; background-repeat: initial initial; margin-bottom: 1.2pt; margin-left: 1.2pt;">
<span style="background-color: black; font-family: Georgia, serif; font-size: 9pt; line-height: 115%;"><span style="color: white;">In addition, maths is permanent; it was as
true one thousand years ago as it is today and will be in one thousand years
time. As Hardy wrote in A Mathematician’s Apology:<o:p></o:p></span></span></div>
<blockquote class="tr_bq" style="background-position: initial initial; background-repeat: initial initial; margin-bottom: 1.2pt; margin-left: 1.2pt; text-align: center;">
<span class="QuoteChar"><span style="background-color: black; font-family: Georgia, serif; font-size: 9pt; line-height: 115%;"><span style="color: white;">“Archimedes will be remembered when Aeschylus is
forgotten, because languages die and mathematical ideas do not. ‘Immortality’
may be a silly word, but probably a mathematician has the best chance of
whatever it may mean.”</span></span></span></blockquote>
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<span style="background-color: black; font-family: Georgia, serif; font-size: 9pt; line-height: 115%;"><span style="color: white;">In the same way, maths can be generalised
meaning there will never be a right-angled triangle that Pythagoras’s Theorem
does not apply to and there will never be another even prime after two. As well
as this, the answer to a maths problem is the answer, there are no different
interpretations or other possible viewpoints. If you have the mark scheme to a
maths exam, you will be able to get full marks unlike an English or History
exam in which it is about your opinion, your interpretation or the way you put
forward your ideas. This, to me, gives mathematics a sense of permanence, which
is becoming more and more rare in this ever-changing world.<o:p></o:p></span></span></div>
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<span style="background-color: black; font-family: Georgia, serif; font-size: 9pt; line-height: 115%;"><span style="color: white;">Maths is also fundamental to many subjects
and real life. As Galileo wrote:<o:p></o:p></span></span></div>
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<span style="background-color: black; font-family: Georgia, serif; font-size: 9pt; line-height: 115%;"><span style="color: white;">“Mathematics is the language with which God has written the
universe.”</span></span></blockquote>
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<span style="background-color: black; font-family: Georgia, serif; font-size: 9pt; line-height: 115%;"><span style="color: white;">Every science requires mathematics for the calculations and
ideas it holds, all engineers require mathematics, artists and architects use
mathematics perhaps without knowing it, all computers and computer scientists
rely on mathematics… Almost everything needs mathematics and in that way maths
is fundamental to our society and everyday life. But I don’t think that means
we should do maths, as I said before maths is self-rewarding not done for its
results but for the process of finding them.<o:p></o:p></span></span></div>
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<span style="background-color: black; background-position: initial initial; background-repeat: initial initial; font-family: Georgia, serif; font-size: 9pt; line-height: 115%;"><span style="color: white;">“All science requires mathematics.” Roger
Bacon</span></span></blockquote>
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<span style="background-color: black; font-family: Georgia, serif; font-size: 9pt; line-height: 115%;"><span style="color: white;">All-in-all, mathematics is beautiful, aesthetically
pleasing, simple yet complex, accessible, fun, permanent, artistic, useful yet
self-rewarding, fundamental and interesting. That is why I love it.</span></span></div>
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<span style="background-color: black; font-family: Georgia, serif; font-size: 9pt; line-height: 115%; text-align: center;"><span style="color: white;">“Pure
mathematics is, in its way, the poetry of logical ideas.” Albert Einstein</span></span></blockquote>
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<span style="background-color: black; font-family: Georgia, serif; font-size: 9pt; line-height: 115%;"><span style="color: white;">I would love to hear your opinions on why you do or don’t
love maths so please leave a comment below.<i><o:p></o:p></i></span></span></div>
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<span style="font-family: "Georgia","serif"; font-size: 9.0pt; line-height: 115%;"><span style="background-color: black;"><span style="color: white;">Nathan.</span></span><o:p></o:p></span></div>
Nathan Jameshttp://www.blogger.com/profile/17655487615203178002noreply@blogger.com0tag:blogger.com,1999:blog-1403911642081151248.post-61234068796823384302014-06-25T10:00:00.000+01:002014-06-25T10:19:25.655+01:00About<div class="MsoNormal">
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<b><u><span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 115%;">About
me</span></u></b></h3>
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<span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 115%;">Hello! I’m Nathan James and this is my blog about the fun
and interesting side of mathematics. I’m a student from Surrey, I love maths
and my aim is to make others love it to0. I also love music (playing and
listening to), technology and photography.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 115%;"><br /></span></div>
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<b><u><span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 115%;">About
the blog</span></u></b></h3>
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<span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 115%;">Because of this, I have decided to set up this blog with the
aim of exposing the hidden wonders of mathematics and I hope you will enjoy it.
I will also create a series of guides to various aspects of mathematics, </span><span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 115%;">A walk through…</span><span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 115%;"> As the
name suggests, I intend to have a drunkard’s walk through mathematics, with no
particular aim just exploring the more interesting side of mathematics.</span></div>
<span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 115%;">
Although the predominant topic of this blog will be mathematics I also intend
to write, from time to time, about my other interests such as photography (you
can see my photos on my </span><a href="http://nathanjamesphotography.weebly.com/"><span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 115%;">website</span></a><span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 115%;"> and my
</span><a href="http://www.flickr.com/photos/custardcream98/"><span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 115%;">flickr
account</span></a><span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 115%;">) and technology.<o:p></o:p></span><br />
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<b><u><span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 115%;">About
the name</span></u></b></h3>
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<span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 115%;">The name refers to a mathematical concept of The Drunkard’s
Walk, which is a random walk over two dimensions. As Wikipedia explains it:</span></div>
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<span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 107%;">“Imagine now a drunkard walking randomly in an idealized
city. The city is effectively infinite and arranged in a square grid, and at
every intersection, the drunkard chooses one of the four possible routes
(including the one he came from) with equal probability. Will the drunkard ever
get back to his home from the bar?”<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 115%; mso-ansi-language: EN-GB; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;">By the way, yes he will.</span></div>
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<span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 115%;">I hope you enjoy it!<o:p></o:p></span></div>
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<b><span style="font-family: "Georgia","serif"; font-size: 10.0pt; line-height: 115%;">Nathan<o:p></o:p></span></b></div>
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Nathan Jameshttp://www.blogger.com/profile/17655487615203178002noreply@blogger.com0