# Math question: is there a middle number? **URL:** \ **Category:** Factual Questions\ **Created:** [January 11, 2013, 3:06am UTC](https://boards.straightdope.com/t/math-question-is-there-a-middle-number/646818 "2013-01-11T03:06:57Z")\ **Posts on this page:** 6\ **Page:** 3
**Author:** ![Alka\_Seltzer](https://avatars.discourse-cdn.com/v4/letter/a/c89c15/32.png) [@Alka\_Seltzer](https://boards.straightdope.com/u/Alka_Seltzer)\ **Post date:** [January 14, 2013, 4:21pm UTC](https://boards.straightdope.com/t/math-question-is-there-a-middle-number/646818/41 "2013-01-14T16:21:57Z")
> [@Chronos](#): > > It _does_ have the same number of positive and negative values, and in fact that’s easy to prove. But that’s not enough, since it also has the same number of values above and below 1, or above and below -57896210984.1789264, or any other number you choose. Yes, but zero is unique in the sense that the infinite set of numbers on each side of it has an equal size. (Not all infinities are [equally large](http://www.scientificamerican.com/article.cfm?id=strange-but-true-infinity-comes-in-different-sizes). For example, the set of real numbers is larger than the set of natural numbers, even though both are infinite. Infinity is really, really weird.) ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [January 14, 2013, 4:24pm UTC](https://boards.straightdope.com/t/math-question-is-there-a-middle-number/646818/42 "2013-01-14T16:24:23Z")
> [@Chronos](#): > > It _does_ have the same number of positive and negative values, and in fact that’s easy to prove. But that’s not enough, since it also has the same number of values above and below 1, or above and below -57896210984.1789264, or any other number you choose. This is true, but the set of positive numbers has a definition which actually doesn’t depend on the ordering of the reals, and so we can take it as something special. Let _P_ be a set of real numbers with the following two properties: [ol] [li]For every nonzero real _a_, either _a_ or its additive inverse is in _P_.[/li][li]If _a_ and _b_ are in _P_, then so are _a_ + _b_ and _ab_.[/li][/ol] Given this set, we can define _a_ \> _b_ to hold exactly when _a_ - _b_ is an element of _P_. In particular, _a_ \> 0 if and only if _a_ is in _P_. This is one argument for why 0 is a special point on the real line. Does that justify it as the middle? Not really, but if you’re going to insist that there is one, it suggests that 0 is a reasonable candidate. > [@Alka\_Seltzer](#): > > Yes, but zero is unique in the sense that the infinite set of numbers on each side of it has an equal size. No. This is true of every real number for any reasonable definition of size. ---
**Author:** ![Alka\_Seltzer](https://avatars.discourse-cdn.com/v4/letter/a/c89c15/32.png) [@Alka\_Seltzer](https://boards.straightdope.com/u/Alka_Seltzer)\ **Post date:** [January 14, 2013, 4:31pm UTC](https://boards.straightdope.com/t/math-question-is-there-a-middle-number/646818/43 "2013-01-14T16:31:36Z")
> [@ultrafilter](#): > > No. This is true of every real number for any reasonable definition of size. Can you explain why please? ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 14, 2013, 5:06pm UTC](https://boards.straightdope.com/t/math-question-is-there-a-middle-number/646818/44 "2013-01-14T17:06:30Z")
> [@Alka\_Seltzer](#): > > Can you explain why please? Take 17 for example: The set of numbers greater than 17 is equal in size (cardinality) to the set of numbers below 17, because they can be matched up in a one-to-one correspondance. Match each number above 17 to the number that is an equal distance below 17. So, 18 corresponds to 16, 19 corresponds to 15, 17.1 corresponds to 16.9, and in general, for any x \> 17, x is (x-17) above 17, so it corresponds to 17 - (x-17). ---
**Author:** ![Alka\_Seltzer](https://avatars.discourse-cdn.com/v4/letter/a/c89c15/32.png) [@Alka\_Seltzer](https://boards.straightdope.com/u/Alka_Seltzer)\ **Post date:** [January 14, 2013, 5:39pm UTC](https://boards.straightdope.com/t/math-question-is-there-a-middle-number/646818/45 "2013-01-14T17:39:37Z")
> [@Thudlow\_Boink](#): > > Take 17 for example: The set of numbers greater than 17 is equal in size (cardinality) to the set of numbers below 17, because they can be matched up in a one-to-one correspondance. I was taking a slightly different tack. For the set of positive real numbers, you can always find a matching number in the set of negative real numbers by inverting the sign. That’s a method that doesn’t work if you start at 17, while your algorithm does. Thinking it over, that’s probably just a flaw of the method I was using. If so, that leaves “the middle of infinity” as undefined or a useless concept (every real number). ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 14, 2013, 5:47pm UTC](https://boards.straightdope.com/t/math-question-is-there-a-middle-number/646818/46 "2013-01-14T17:47:33Z")
> [@Alka\_Seltzer](#): > > It’s impossible to measure the middle of anything precisely, only approximate it (admittedly, to a high degree of precision), or provide an _arbitrary_ definition, as you have done above. If you’re working strictly in the realm of mathematics, there’s a sense in which all definitions are arbitrary. [Previous page](https://boards.straightdope.com/t/math-question-is-there-a-middle-number/646818.md?page=2)