<story><title>Euclid&apos;s Proof that √2 is Irrational</title><url>https://www.mathsisfun.com/numbers/euclid-square-root-2-irrational.html</url></story><parent_chain><item><author>p-e-w</author><text>If I wasn&amp;#x27;t familiar with that concept already, then I would probably assume that math is no more rigorous than psychology after encountering this thread.&lt;p&gt;The exact disciplines are doing themselves a terrible disservice by muddying up established terminology like this (and &amp;quot;algebraic integers&amp;quot; are &lt;i&gt;far&lt;/i&gt; from the only such case).</text></item><item><author>kevinventullo</author><text>By “usual integer”, I mean what people usually refer to as an integer:&lt;p&gt;&lt;pre&gt;&lt;code&gt;  …, -2, -1, 0, 1, 2, …
&lt;/code&gt;&lt;/pre&gt;
As opposed to “algebraic integer”, which is a more general notion.</text></item><item><author>gerdesj</author><text>Maths is always a bit boggling.  You say that root two can be considered an integer despite being irrational.&lt;p&gt;What does &amp;quot;usual integer&amp;quot; mean?</text></item><item><author>kevinventullo</author><text>For those who are interested in connections to more advanced mathematics, there is a sense in which √2 is still an integer, even though it is irrational. Specifically there is the notion of “algebraic integers”, which are the set of all complex numbers expressible as the root of a &lt;i&gt;monic&lt;/i&gt; polynomial:&lt;p&gt;&lt;pre&gt;&lt;code&gt;  x^n + a_{n-1}x^(n-1) + … + a_1x + a_0.
&lt;/code&gt;&lt;/pre&gt;
Here each a_i is a usual integer in ℤ, and &lt;i&gt;monic&lt;/i&gt; refers to the leading coefficient being equal to 1.&lt;p&gt;It turns out the set of such roots is actually closed under multiplication, addition, and subtraction, and there is even an analogue of prime factorization if you squint. Moreover, the intersection of these “algebraic integers” and the rational numbers ℚ are exactly the usual integers ℤ. This is why you sometimes might hear an algebraic number theorist refer to ℤ as the set of “rational integers”.</text></item></parent_chain><comment><author>gjm11</author><text>I don&amp;#x27;t think that&amp;#x27;s fair.&lt;p&gt;Mathematicians pretty reliably say &amp;quot;algebraic integer&amp;quot; or &amp;quot;integer in [some specific class of numbers that has non-rational integers in it]&amp;quot; when they are talking about the broader notion, and if they&amp;#x27;re doing something where that broader notion is often relevant they will generally say something like &amp;quot;rational integer&amp;quot; when they mean the narrower notion. So in practice there is seldom any confusion.&lt;p&gt;And algebraic integers really _are_ like ordinary integers in important ways. Inventing a completely new term would not obviously be an improvement.&lt;p&gt;It&amp;#x27;s not like this sort of thing is unique to mathematics. Once upon a time a &amp;quot;language&amp;quot; was a thing human beings used to communicate with one another. Then along came &amp;quot;programming languages&amp;quot; which are not languages in that sense. And then things like &amp;quot;hypertext markup language&amp;quot; which isn&amp;#x27;t a language in the programming sense either.&lt;p&gt;(Arguably this is partly mathematicians&amp;#x27; fault since I think they were the first to use &amp;quot;language&amp;quot; to refer to purely formal constructs. But I think the use of &amp;quot;language&amp;quot; in computing arose mostly by analogy to human languages.)&lt;p&gt;And it happens plenty outside &amp;quot;the exact disciplines&amp;quot;. A republican is someone who favours a mode of government that doesn&amp;#x27;t have monarchs, but if you call someone a &amp;quot;Republican&amp;quot; in the US you mean something rather more specific and 
a few &amp;quot;Republicans&amp;quot; would actually quite like a system hard to distinguish from monarchy. A window is a transparent thing placed in a wall to let light in, but a window of opportunity is something quite different. A czar is the absolute ruler of Russia, but when someone says (rightly or wrongly) that Kamala Harris was &amp;quot;border czar&amp;quot; they don&amp;#x27;t mean &lt;i&gt;that&lt;/i&gt;. A star is a gigantic ball of stuff undergoing nuclear fusion and producing unimaginable amounts of energy, but even the most impressive rock stars don&amp;#x27;t do that, and some people called &amp;quot;rock stars&amp;quot; have never played or sung a note of rock music in their lives.</text></comment>
<story><title>Euclid&apos;s Proof that √2 is Irrational</title><url>https://www.mathsisfun.com/numbers/euclid-square-root-2-irrational.html</url></story><parent_chain><item><author>p-e-w</author><text>If I wasn&amp;#x27;t familiar with that concept already, then I would probably assume that math is no more rigorous than psychology after encountering this thread.&lt;p&gt;The exact disciplines are doing themselves a terrible disservice by muddying up established terminology like this (and &amp;quot;algebraic integers&amp;quot; are &lt;i&gt;far&lt;/i&gt; from the only such case).</text></item><item><author>kevinventullo</author><text>By “usual integer”, I mean what people usually refer to as an integer:&lt;p&gt;&lt;pre&gt;&lt;code&gt;  …, -2, -1, 0, 1, 2, …
&lt;/code&gt;&lt;/pre&gt;
As opposed to “algebraic integer”, which is a more general notion.</text></item><item><author>gerdesj</author><text>Maths is always a bit boggling.  You say that root two can be considered an integer despite being irrational.&lt;p&gt;What does &amp;quot;usual integer&amp;quot; mean?</text></item><item><author>kevinventullo</author><text>For those who are interested in connections to more advanced mathematics, there is a sense in which √2 is still an integer, even though it is irrational. Specifically there is the notion of “algebraic integers”, which are the set of all complex numbers expressible as the root of a &lt;i&gt;monic&lt;/i&gt; polynomial:&lt;p&gt;&lt;pre&gt;&lt;code&gt;  x^n + a_{n-1}x^(n-1) + … + a_1x + a_0.
&lt;/code&gt;&lt;/pre&gt;
Here each a_i is a usual integer in ℤ, and &lt;i&gt;monic&lt;/i&gt; refers to the leading coefficient being equal to 1.&lt;p&gt;It turns out the set of such roots is actually closed under multiplication, addition, and subtraction, and there is even an analogue of prime factorization if you squint. Moreover, the intersection of these “algebraic integers” and the rational numbers ℚ are exactly the usual integers ℤ. This is why you sometimes might hear an algebraic number theorist refer to ℤ as the set of “rational integers”.</text></item></parent_chain><comment><author>Someone</author><text>&amp;gt; The exact disciplines are doing themselves a terrible disservice by muddying up established terminology like this&lt;p&gt;So, what term do propose they use for this? “Hutyfreklop” or “gensym_167336871904” would probably be unique, but wouldn’t tell anything about the subject itself, “roots of polynomials with integer coefficients and a leading coefficient of one” would get cumbersome soon.</text></comment>