<story><title>Why don&apos;t we define “imaginary” numbers for every “impossibility”? (2012)</title><url>https://math.stackexchange.com/questions/259584/why-dont-we-define-imaginary-numbers-for-every-impossibility</url></story><parent_chain><item><author>ndsipa_pomu</author><text>Well, we can define mathematical objects for every gap (impossibility), but most of them will turn out to be inconsistent with our existing mathematical objects, and thus not very useful or interesting. I&amp;#x27;d consider that mathematics is the study of consistency and what can be discovered using the simplest possible starting points (axioms).&lt;p&gt;The classic case would be if mathematicians wanted to assign a value to division by zero. It turns out that if you do allow that to take a value, then it becomes possible to &amp;quot;prove&amp;quot; that any number is equal to any other number. Quite simply, it makes maths less interesting to allow that, but instead having division by zero be undefined appears far more useful&amp;#x2F;interesting.</text></item></parent_chain><comment><author>nextaccountic</author><text>&amp;gt; The classic case would be if mathematicians wanted to assign a value to division by zero. It turns out that if you do allow that to take a value, then it becomes possible to &amp;quot;prove&amp;quot; that any number is equal to any other number. Quite simply, it makes maths less interesting to allow that, but instead having division by zero be undefined appears far more useful&amp;#x2F;interesting.&lt;p&gt;There are multiple extensions to the real numbers that allow division by zero. One is a real projective line, which has only one infinity so that 1 &amp;#x2F; 0 = -1 &amp;#x2F; 0 = infinity&lt;p&gt;&lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Real_projective_line&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Real_projective_line&lt;/a&gt;&lt;p&gt;Another is the extended real number line which has positive infinity and negative infinity, so 1 &amp;#x2F; 0 = +infinity and -1 &amp;#x2F; 0 = -infinity and they are different from each other&lt;p&gt;&lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Extended_real_number_line&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Extended_real_number_line&lt;/a&gt;&lt;p&gt;Those are all perfectly fine but they still can&amp;#x27;t define 0 &amp;#x2F; 0, which is a harder problem.</text></comment>
<story><title>Why don&apos;t we define “imaginary” numbers for every “impossibility”? (2012)</title><url>https://math.stackexchange.com/questions/259584/why-dont-we-define-imaginary-numbers-for-every-impossibility</url></story><parent_chain><item><author>ndsipa_pomu</author><text>Well, we can define mathematical objects for every gap (impossibility), but most of them will turn out to be inconsistent with our existing mathematical objects, and thus not very useful or interesting. I&amp;#x27;d consider that mathematics is the study of consistency and what can be discovered using the simplest possible starting points (axioms).&lt;p&gt;The classic case would be if mathematicians wanted to assign a value to division by zero. It turns out that if you do allow that to take a value, then it becomes possible to &amp;quot;prove&amp;quot; that any number is equal to any other number. Quite simply, it makes maths less interesting to allow that, but instead having division by zero be undefined appears far more useful&amp;#x2F;interesting.</text></item></parent_chain><comment><author>iamerroragent</author><text>Riemann Sphere:&lt;p&gt;&lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Riemann_sphere&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Riemann_sphere&lt;/a&gt;</text></comment>