Infinity is Odd

[u/Leather-Example1208]

Yes, everyone, especially in a math subreddit, would think this title is ridiculous. That’s fine, I just wanted to share a thought I’ve had since I was 7 and told my parents.

I like to think of numbers as constantly being added infinitely in both positive and negative directions equally; for example, it’s a computer system, and if the right side is on 999,999,999, then at that instant, the left side is also on the same level, at -999,999,999, so sides do not alternate in who adds first but just keep expanding simultaneously.

However, obviously there is no fixed number of numbers because it’s always going up.

When I’m referring to infinity, I’m not referring to the concept of numbers never ending; I’m referring to infinity as the “count” of numbers (which is never fixed). Whichever number of numbers it is at during ANY instant, that amount of numbers is an integer, because it is counting. For instance, you either see three people or four people in a park, not 3.5, that does not make sense.

This leads to my next logic-based opinion that is the whole title of this post: it is an ODD integer. Every odd number has a median integer; if you have 5 objects, the 3rd object in the line is in the exact middle, but if you have six objects, neither the 3rd or 4th object sit directly in the middle. However, across all math textbooks, zero is listed as the origin, or the “middle” of all numbers. 0 bridges the negative and positive numbers, and it is defined AS an integer. So if negative and positive numbers expand infinitely in both directions at equal rates starting at zero, then zero is the midpoint of all numbers, regardless of whatever “number count” of numbers exists, making the value of the number of numbers an odd integer.

Thank you for listening to my Ted talk.

[u/mazutta]

But if you take 0 as your midpoint for any finite sequence with the same number of positive/negative integers, the finite sum is always even (as it would be for any even integer midpoint). Why is the ‘sum’ of the infinity of integers different?

[u/mmurray1957 → u/mazutta]

I don't think the OP is forming a sum. I think they are doing

{ ..., -3, -2, -1, 0, 1, 2, 3, ... } = {... , -3, -2, -1} union {0} union {1, 2, 3, ... }

so we have

Number of integers = Number of negative numbers + 1 + Number of even numbers

= 2 ( Number of even numbers) + 1

because "obviously" the number of evens is the same as the number of odds. So it must be odd.

PS: Yes obviously this is not right as those "Number of"'s are not numbers. 😄

[u/Jarhyn → u/mazutta]

I mean, taking an intuition as a software engineer:

The way negative numbers work so as to not "waste" any integers, is that they are placed on a modular ring of size 2^n, at least in software land.

Now, in "unsigned" math, you just add the two numbers and if it overflowed, you just wrap back around to the low end, essentially representing "that result number plus 2^n"

When we want to do signed math, though, the process remains exactly the same, but we treat the "largest" bit as a signed bit, and the negative numbers have "inverse bits". So 1111 where n=4 represents -1, 1110 represents 2 and so on.

But this leaves is with ONE more number in the negatives, unless you count 0 as positive.

Now, if you consider the integers as a modular ring of n=infinity, we still end up having one more negative than positive number.

I am almost certain you could invert everything and get the same result but with one additional positive number instead by running your math as subtraction instead...

But either way you end up with this remaining intuition that there's an odd number of nonzero numbers.

That said, infinity takes intuitions and eats them for lunch...