<story><title>Can 1/3 and 1/3 = 2/6? It seemed so</title><url>http://www.marilynburnsmathblog.com/can-1-3-1-3-2-6-it-seemed-so/</url></story><parent_chain><item><author>twic</author><text>&amp;gt; x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn&amp;#x27;t mean \frac{1}{f(x)}.&lt;p&gt;That notation for inverse functions is truly appalling. I don&amp;#x27;t know how the first mathematician to think of that didn&amp;#x27;t immediately discard it as nonsensical and misleading.</text></item><item><author>wcarey</author><text>This is a lovely (edit: having been in similar shoes, also terrifying-in-the-moment) example of a broader problem in teaching mathematics: the language we use to describe mathematical reasoning is a natural language, like English or Latin, and therefore full of the sorts of bizarre irregularities you&amp;#x27;d find in a natural language. Mathematics is also a language about rigorously and precisely defined objects. The conceptual shear between those two things is murder for lots of students.&lt;p&gt;Just like Tacitus omits his verbs (!), when we describe fractions we often omit the implicit definition of the whole. Turns out that&amp;#x27;s a problem for many students.&lt;p&gt;It&amp;#x27;s a bit like trying to learn a context-dependent programming grammar with an inconsistent API, but worse, because it&amp;#x27;s your first &amp;quot;mathematical&amp;quot; language so you&amp;#x27;re also trying to learn what the abstract objects the language manipulates &lt;i&gt;are&lt;/i&gt;.&lt;p&gt;Some other lovely examples:&lt;p&gt;3(5) means three times five. 3(x) means three times x. 35 means three times ten plus five. 3x means three times x. x(3) means that x is the name of a function taking, in this instance, 3 as its input.&lt;p&gt;x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn&amp;#x27;t mean \frac{1}{f(x)}.&lt;p&gt;\sin{30}. Radians or degrees? Probably the writer means degrees, but there&amp;#x27;s no way to tell.&lt;p&gt;There are many more.</text></item></parent_chain><comment><author>rocqua</author><text>It&amp;#x27;s not because &amp;#x27;function powers&amp;#x27; make sense, yhey are just iterated function application. That&amp;#x27;s how they work for the natural numbers and when you extend that to the integers, you immediately get f^-1 for the inverse.&lt;p&gt;Notation in higher level maths is almost always very ambiguous. Because many concepts are analogues of each other and to reflect that notation is just taken from the analogue.
Within a single domain, (like high-school arithmetic) you will usually not have this ambiguous problem. But once you move past that, it is something to get used to.</text></comment>
<story><title>Can 1/3 and 1/3 = 2/6? It seemed so</title><url>http://www.marilynburnsmathblog.com/can-1-3-1-3-2-6-it-seemed-so/</url></story><parent_chain><item><author>twic</author><text>&amp;gt; x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn&amp;#x27;t mean \frac{1}{f(x)}.&lt;p&gt;That notation for inverse functions is truly appalling. I don&amp;#x27;t know how the first mathematician to think of that didn&amp;#x27;t immediately discard it as nonsensical and misleading.</text></item><item><author>wcarey</author><text>This is a lovely (edit: having been in similar shoes, also terrifying-in-the-moment) example of a broader problem in teaching mathematics: the language we use to describe mathematical reasoning is a natural language, like English or Latin, and therefore full of the sorts of bizarre irregularities you&amp;#x27;d find in a natural language. Mathematics is also a language about rigorously and precisely defined objects. The conceptual shear between those two things is murder for lots of students.&lt;p&gt;Just like Tacitus omits his verbs (!), when we describe fractions we often omit the implicit definition of the whole. Turns out that&amp;#x27;s a problem for many students.&lt;p&gt;It&amp;#x27;s a bit like trying to learn a context-dependent programming grammar with an inconsistent API, but worse, because it&amp;#x27;s your first &amp;quot;mathematical&amp;quot; language so you&amp;#x27;re also trying to learn what the abstract objects the language manipulates &lt;i&gt;are&lt;/i&gt;.&lt;p&gt;Some other lovely examples:&lt;p&gt;3(5) means three times five. 3(x) means three times x. 35 means three times ten plus five. 3x means three times x. x(3) means that x is the name of a function taking, in this instance, 3 as its input.&lt;p&gt;x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn&amp;#x27;t mean \frac{1}{f(x)}.&lt;p&gt;\sin{30}. Radians or degrees? Probably the writer means degrees, but there&amp;#x27;s no way to tell.&lt;p&gt;There are many more.</text></item></parent_chain><comment><author>StavrosK</author><text>&amp;gt; I don&amp;#x27;t know how the first mathematician to think of that didn&amp;#x27;t immediately discard it as nonsensical and misleading.&lt;p&gt;Maybe they did.</text></comment>