My 6 and 7 yo's call infinity the "endless number". Well, at least it is a NaN number :)
PS: they seem to _know_ that endless*endless > endless but do not dare to admit it
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".
> 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
A circle is endless, and yet certainly isn't infinite.