Infinity isn't a number really, it's a concept, like the word many or the word few. If someone says they have many of something, you don't think is that odd or even you just know they have a lot of it. Infinity is kind of like that, it explains the idea of things going on forever, not an exact quantity of things like the number 10 or 11.
For any collection with a certain number of things (such as n potatoes), I can always name a collection with more things, such as n+1 potatoes.
"Infinity" is a word that we use for games like that.
PS: they seem to _know_ that endless*endless > endless but do not dare to admit it
x * x === x^2
The parent comment was alluding to the idea of set cardinality (https://en.wikipedia.org/wiki/Cardinality). Two sets have the same cardinality if you can establish a bijection (a one-to-one mapping) between elements of one set and elements of the other. The set of all natural numbers is said to have a "countably infinite" cardinality.
It turns out that for any countably infinite set S, the set S x S is also countably infinite (see Hilbert's hotel: https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...). For example, the set of 2-tuples of natural numbers (1, 1), (1, 2), (1, 3), ..., (2, 1), (2, 2), ... is the same size as the set of natural numbers. So in this sense, "endless*endless = endless". Whereas the set of infinitely-long tuples of natural numbers is "uncountably infinite;" it has a cardinality greater than that of the set of natural numbers. Thus, "you have to go exponential"; i.e. "endless^endless".
A circle is endless, and yet certainly isn't infinite.
> Infinity + Infinity
Infinity
> Infinity - Infinity
NaN
> Infinity == Infinity
true
> NaN == NaN
false
> NaN ** 0
1
> typeof(NaN)
'number'
> NaN + NaN
NaN
> 1/0
Infinity
>>> 1/0
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
ZeroDivisionError: division by zero
>>> float('inf') - float('inf')
nan
>>> float('inf') == float('inf')
True
>>> NAN=float('inf') - float('inf')
>>> NAN == NAN
False
I have my own little programming language - PYX [1] - and i don't allow this madness (even if it is a violation of the floating point spec ;-)
pyx
> mathconst.Infinity - mathconst.Infinity
Error: results in 'not a number' - that's not allowed here
#(1) mathconst.Infinity - mathconst.Infinity
|....................^
> 1/0
Error: Can't divide by zero
#(1) 1/0
|..^
>>> type(1.0)
<class 'float'>
>>> 1.0/0
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
ZeroDivisionError: float division by zero
>>> import numpy as np
>>>
>>> np.divide(1.0,0)
__main__:1: RuntimeWarning: divide by zero encountered in true_divide
inf>>> 1/0.0
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
>>> 1.0/0.0
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
It's been a while since I was doing this in a classroom, but I feel like those things you're thinking are nonuniform infinities could just as well be thought of as entities comprised of infinity and a (perhaps implied) coefficient. Divide out infinity to reveal the coefficient. (And the same for powers/logs, etc.)
In this model, infinities are indeed uniform (quite similar to a constant), though they are often augmented in any of infinitely many ways.
> NaN == NaN
false
There are natural numbers, like 0,1,2 and so on. Natural numbers can be odd or even. There is no such natural number as infinity. Therefore the question if 'infinity' is odd or even is meaningless. It does not even type-check.
In math people like well-formed questions, and generally don't like ill-formed questions.
He's given me quite a bit to think about in regard to NULL and the assumptions I make around the concept, which is fascinating in itself because his books are hot garbage.
This is not so simple, though. Ill formed questions can be interesting as a motivation to formalise them (ie make them well-formed) in generalising/abstracting concepts into new concepts. Eg how even/odd has been generalised to transfinite numbers.
If you try hard enough, you can find similar questions, that do type-check. You can talk with 6yo children about them if you want. Still, I stand with my answer. I would say this (also I think this is the best thing to say/I am capable of).
"There are many infinities! The smallest one is bigger than all the counting numbers, so you can't count up to it, but it's out there! We call it omega. You can make bigger infinities too, like omega + 1!"
Kids LOVE that, and it's good math too! (But gets tricky quickly, because addition of transfinite ordinals is not commutative, and standard transfinite ordinals don't allow subtraction)
It's easy to draw as a "number tree" too:
root
/ \
/ \
1,2,3,4... omega, omega+1,...
> If you ask a child what comes after infinity, "Infinity + 1" is pretty much the default answer. Any kid who knows multiplication knows "Infinity + Infinity" is the same as "Infinity Times Two". The answer of "Infinity TIMES Infinity" is also popular for kids to say when they know a number bigger than their friend (who just proclaimed infinity is the largest number).
Sensei: Grasshopper, lecture over today.