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Comment by mm321 | Hacker News Reader

there is a difference between infinity and not a number, NaN isn't event equal to itself, in nodejs you get:

  > Infinity + Infinity
  Infinity
  > Infinity - Infinity
  NaN
  > Infinity == Infinity
  true
  > NaN == NaN
  false

The fun thing: NaN to the power of zero (because technically the type of NaN is a ... number)

  > NaN ** 0   
  1
  > typeof(NaN)
  'number'

By the same logic, NaN is also even because

  > NaN + NaN
  NaN

but i think it's better not explain that to your six year old...

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Huh. Most of those make sense to me, but infinity == infinity being true definitely feels like risky business. Algebraic limits is full of even some pretty trivial scenarios where infinity divided by a lesser-infinity turns out to be a real number— those cases where the two infinities are definitely not equal to each other.

Also: in javascript you can divide by zero

  > 1/0
  Infinity

But python throws a divide by zero error:

  >>> 1/0
  Traceback (most recent call last):
    File "<stdin>", line 1, in <module>
  ZeroDivisionError: division by zero

Still you do have infinity and nan in python - because these are part of the floating point spec.

  >>> float('inf') - float('inf')
  nan
  >>> float('inf') == float('inf')
  True
  >>> NAN=float('inf') - float('inf')
  >>> NAN == NAN
  False

However that's not mathematics, it's computers (these are even stranger...)

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
     |..^

[1] PYX - https://github.com/MoserMichael/jscriptparse - it's supposed to be an educational programming language, where I am trying to have detailed error messages, my side project.

Dividing by 0 or -0 is a valid floating-point operation because there's an infinity in the number system, and JS uses double precision floating point for all numbers. Python has an integer type and a double type, and division by 0 is disallowed for integers, but okay for doubles.

your explanation makes sense, however python doesn't allow division of floating point number by 0 either:

  >>> type(1.0)
  <class 'float'>
  >>> 1.0/0
  Traceback (most recent call last):
    File "<stdin>", line 1, in <module>
  ZeroDivisionError: float division by zero

However numpy lets you do it - it is only a warning

  >>> import numpy as np

  >>>
  >>> np.divide(1.0,0)
  __main__:1: RuntimeWarning: divide by zero encountered in    true_divide
  inf

Wow, Python really does hold your hand.

>>> 1/0.0

Traceback (most recent call last):

  File "<stdin>", line 1, in <module>

ZeroDivisionError: float division by zero

>>> 1.0/0.0

Traceback (most recent call last):

  File "<stdin>", line 1, in <module>

ZeroDivisionError: float division by zero

> those cases where the two infinities are definitely not equal to each other

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

Reminds me of SQL, where NULL isn't equal to NULL.

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