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1/0 = 0 (2018)

hillelwayne.com

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Re: 1/0 = 0 (2018)

#21

Post by judofyr »

rsp1984 wrote: post #3

As long as lim(1/x)_x->0 = inf, 1/0 = 0 doesn't make a whole lot of sense, mathematically speaking. I might be wrong but I don't think it was addressed in the article either.

The arguments around limits are addressed towards the end (under "Update 8/12/2018"):

> > If 0/0 = 0 then lim_(x -> 0) sin(x) / x = sin(0) / 0 = 0, but by L’Hospitals’ Rule lim_(x -> 0) sin(x) / x = lim_(x -> 0) cos(x) / 1 = 1. So we have 0 = 1.

> This was a really clever one. The issue is that the counterargument assumes that if the limit exists and f(0) is defined, then lim_(x -> 0) f(x) = f(0). This isn’t always true: take a continuous function and add a point discontinuity. The limit of sin(x) / x is not sin(0) / 0, because sin(x) / x is discontinuous at 0. For the unextended division it’s because sin(0) / 0 is undefined, while for our extended division it’s a point discontinuity. Funnily enough if we instead picked x/0 = 1 then sin(x) / x would be continuous everywhere.

Similar examples can be constructed for any regular function which is discontinuous (e.g. Heaviside step function).

Re: 1/0 = 0 (2018)

#22

Post by blueblimp »

This sort of convenient semi-arbitrary extension of a partial function is ubiquitous in Lean 4 mathlib, the most active mathematics formalization project today. It turns out that the most convenient way to do informal math and formal math differ in this aspect.

Re: 1/0 = 0 (2018)

#23

Post by [deleted] »

moomin wrote: post #5

Honestly this hurts my head but Hillel is inevitably correct. You can define an explicitly undefined operation to do whatever you like. But what’s the point? There’s no new mathematics you can do with it, no existing behaviours you can extend like this. Normally, when you divide by a small number, you get a large number. Now for some reason it goes through zero. Why not five? Why not seven? Just because it’s formally…

[deleted]

Re: 1/0 = 0 (2018)

#24

Post by lupire »

v4vvdq wrote: post #10
rsp1984 wrote: post #3

As long as lim(1/x)_x->0 = inf, 1/0 = 0 doesn't make a whole lot of sense, mathematically speaking. I might be wrong but I don't think it was addressed in the article either.

I was also looking for this. And would like to add: lim(-1/x)_x -> 0 = -inf That is (in my opinion) the whole point why it is actually undefined. On one side of the y-axis it goes to infinity, on the other to minus infinity. I don't see a solution to this and therefore always have accepted that it is undefined.

No. 1/x^2 is undefined at 0 but has the same limit behavior, because limit behavior is not a function from "pairs of (functions from R to R, R)" to R

Infinity is not a real number.

Re: 1/0 = 0 (2018)

#25

Post by recursive »

WithinReason wrote: post #16
v4vvdq wrote: post #15

Earlier quoted context omitted.

Then take 10 and divide it by -10 = -1. 10 / -5 = -2. 10 / -0.5 = -20. So from the other side of the y-axis it behaves the exact opposite. It goes to minus infinity. So at x=0 we would have infinity and minus infinity at the same time. Imho that is why it is undefined.

on computers you can have negative zeros

Negative zero is equal to zero, so it's not really a distinct number, just another representation of the same value.

Re: 1/0 = 0 (2018)

#26

Post by tux3 »

codeflo wrote: post #12

This article invents a new binary operation, calls it "division" and uses the "/" operator to denote it. But the article repeats multiple times that this new operation isn't a multiplicative inverse, so it's not actually division. For example, (a/b)*b=a isn't true for this new operation.

Reusing symbols like +, *, or / to define operations that aren't the + or the / you're used to is pretty common in math. It's just notation.

At the end of the day, the / that we have in programming has the same problem as this article's /, almost all programming languages will return 5/2 = 2 when dividing integers, even though 2 * 2 is not 5! Division is not defined for all integers, but it's just convenient to extend it when programming.

So if some languages want to define 1/0 = 0, we really shouldn't be surprised that 0*0 is not 1, we already had the (a/b)*b != a problem all along!

Re: 1/0 = 0 (2018)

#27

Post by recursive »

lupire wrote: post #14
moomin wrote: post #5

Honestly this hurts my head but Hillel is inevitably correct. You can define an explicitly undefined operation to do whatever you like. But what’s the point? There’s no new mathematics you can do with it, no existing behaviours you can extend like this. Normally, when you divide by a small number, you get a large number. Now for some reason it goes through zero. Why not five? Why not seven? Just because it’s formally…

Because exceptions are expensive, and functions with holes are dumb. "Dumb" is purely a matter of aesthetic preference. Calling things "dumb" is dumb. > Normally, when you divide by a small number, you get a large number. Now for some reason it goes through zero. Zero is not a "small" number. Zero is the zero number. There is no number that is better result than 0 when dividing by 0; "Infinity" is not a real (or comp…

> Zero is not a "small" number. Zero is the zero number.

What do you mean by this? Zero is certainly a zero number, but it seems that it might also be a small number simultaneously.

Re: 1/0 = 0 (2018)

#28

Post by lupire »

jvanderbot wrote: post #7

Q on this post: Is the field rule "Every element Except Zero has ... " (the 9th rule) defined with respect to the additive identity "zero" or the magical other undefined "Zero" that is the number we're all familiar with? If so, how weirdly arbitrary that the additive zero is omitted for all multiplicative inverse definitions. (At least it seems to me). I always figured this was a consequence of our number systems, no…

I am not familiar with any "magical other undefined Zero". Please familiarize me :-)

What is your, uh, definition of this undefined* number you are familiar with?

Re: 1/0 = 0 (2018)

#29

Post by [deleted] »

codeflo wrote: post #12

This article invents a new binary operation, calls it "division" and uses the "/" operator to denote it. But the article repeats multiple times that this new operation isn't a multiplicative inverse, so it's not actually division. For example, (a/b)*b=a isn't true for this new operation.

[deleted]

Re: 1/0 = 0 (2018)

#30

Post by d_tr »

Whatever as long as the name does not imply that these are integers, because then it is just wrong. The same holds for overflowing results being clamped or resulting in smaller or negative values due to wraparound. These are not integers.

There is only one correct behavior for something named "int". Give the correct result or throw an error.

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