Math question: is there a middle number?

[Alka_Seltzer]

@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. 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.)

[ultrafilter]

@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.

[Alka_Seltzer]

@ultrafilter

No. This is true of every real number for any reasonable definition of size.

Can you explain why please?

[Thudlow_Boink]

@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).

[Alka_Seltzer]

@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).

[Thudlow_Boink]

@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.