<story><title>How Did Anyone Do Math in Roman Numerals?</title><url>http://www.washingtoncitypaper.com/columns/straight-dope/article/20854121/how-did-anyone-do-math-in-roman-numerals</url></story><parent_chain><item><author>WalterBright</author><text>It&amp;#x27;s interesting how the notation used can encourage or retard progress. For example, Leibniz&amp;#x27;s calculus notation was vastly superior to Newton&amp;#x27;s, and calculus theory advanced much more quickly where Leibniz&amp;#x27;s notation was used.&lt;p&gt;&lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Leibniz%27s_notation&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Leibniz%27s_notation&lt;/a&gt;</text></item></parent_chain><comment><author>nnq</author><text>About this particular case: why hasn&amp;#x27;t Euler&amp;#x27;s notation (&lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Notation_for_differentiation#Euler.27s_notation&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Notation_for_differentiation#E...&lt;/a&gt;) become &lt;i&gt;the clear winner?&lt;/i&gt;&lt;p&gt;I mean, its the only one that is both &lt;i&gt;not confusing for beginners&lt;/i&gt; (because it doesn&amp;#x27;t trick them into the &amp;quot;cool, let&amp;#x27;s `simplify` the dx at the denominator with the next one&amp;quot; mindset) and it also translates easily to code (or other 1D encoding), like you can write &amp;quot;second derivative of f with respect to x&amp;quot; as D[f, x, 2] and &amp;quot;integral of a with respect to t&amp;quot; as &amp;quot;D[a, t, -1]&amp;quot;.&lt;p&gt;My point is mainly that &lt;i&gt;there is nothing rational at all in how humans choose notations...&lt;/i&gt; without some obscure historic events, we&amp;#x27;d probably still be using some derivative of roman numerals or maybe even sexagesimals! (&lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Sexagesimal&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Sexagesimal&lt;/a&gt;)&lt;p&gt;(I imagine the reason is because &amp;quot;highly performing&amp;quot; individuals have their own &amp;quot;internal&amp;quot; language to think in, so general language is just a for communication, so a political decision... unfortunately for &lt;i&gt;education :(&lt;/i&gt;)</text></comment>
<story><title>How Did Anyone Do Math in Roman Numerals?</title><url>http://www.washingtoncitypaper.com/columns/straight-dope/article/20854121/how-did-anyone-do-math-in-roman-numerals</url></story><parent_chain><item><author>WalterBright</author><text>It&amp;#x27;s interesting how the notation used can encourage or retard progress. For example, Leibniz&amp;#x27;s calculus notation was vastly superior to Newton&amp;#x27;s, and calculus theory advanced much more quickly where Leibniz&amp;#x27;s notation was used.&lt;p&gt;&lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Leibniz%27s_notation&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Leibniz%27s_notation&lt;/a&gt;</text></item></parent_chain><comment><author>jaymzcampbell</author><text>Another excellent example of the power of good notation to aid understanding and increase efficiency is Dirac, or Bra-Ket notation[1] used in Quantum Mechanics.&lt;p&gt;This allows you to do all sort of calculations with wave functions without constantly grinding to a halt bogged down by integrals and conjugates all over the place.&lt;p&gt;The ladder operators (aka raising &amp;amp; lowering), really put this notation to good use[2]. It would be tremendously tedious to manipulate such expressions repeatedly without the simplifying properties of Bra-Kets. Once you are done manipulating them you then use the rules of the bra-kets to transform it back into a plain old integral, evaluate, and you&amp;#x27;re done.&lt;p&gt;I used to think they were very mysterious until I realised they were (essentially) notational convenience.&lt;p&gt;[1]: &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Bra%E2%80%93ket_notation&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Bra%E2%80%93ket_notation&lt;/a&gt;&lt;p&gt;[2]: &lt;a href=&quot;http:&amp;#x2F;&amp;#x2F;www.tcm.phy.cam.ac.uk&amp;#x2F;~bds10&amp;#x2F;aqp&amp;#x2F;lec3_compressed.pdf&quot; rel=&quot;nofollow&quot;&gt;http:&amp;#x2F;&amp;#x2F;www.tcm.phy.cam.ac.uk&amp;#x2F;~bds10&amp;#x2F;aqp&amp;#x2F;lec3_compressed.pdf&lt;/a&gt;</text></comment>