Irrational Numbers

[Kublai Khan]

Hey mathamascummers.. A quick question.

The square root of -1 is an irrational impossible number. So mathematicians represent that concept with the letter i.

How come we can't do the same with "divide by zero"?

[Something_Smart]

That's what infinity is

Only problem is, it's not as well defined, because say 1/0 = 2/0 = ∞ and if you allowed normal algebra with infinity you could prove 1 = 2 which is ridiculous. (As opposed to imaginary numbers where you actually can do that stuff and get away with it.)

[Flubbernugget]

Look at 1/x in a graphing calculator and see what happens at x=0. What would it mean to actually be able to plot that point?

[talah]

[Annadog40]

Look at 1/x in a graphing calculator and see what happens at x=0. What would it mean to actually be able to plot that point?

On an imaginary plain?

[Flubbernugget]

Just think about it on the real line.

The same underlying idea exists on the complex plane, but it's a lot harder to visualize.

[Scigatt]

There are rules of 'number-like' things (i.e. field axioms) where addition, subtraction, multiplication, and division behave like we expect.

If a particular structure follows these rules, then we can do useful and interesting things with it, like construct vector spaces over it. Thus, the field axioms are important in themselves. Structures that uphold the axioms are called fields. Examples of fields are the real and rational numbers.

The field axioms outright forbid division by zero, so that's a non-starter.

The 'square root of -1'

i

is a root of the polynomial P = x² + 1. This can be taken as a polynomial over the reals

R

, since its coefficients are in

R

.

The field axioms thus guarantee that if P has no roots in

R

, then it possible to extend

R

to a field

C

that contains both

R

and

i

. This is what we do to get complex numbers from the reals.

[popsofctown]

Division is defined as an undoing of multiplication.

If you have 15, and I'm like, you used a 5 to get there, tell me the other number you used to get there, you'll say, 3. And I'll say yeah, I buy that.

If you have the "the ratio of a circle's diameter to its circumference", and you used a 1 to get there, and you tell me 3.14 and I say nah man, I'm checking it, that comes out too low, and then you tell me 3.1416 and I say nah man, I'm checking it, it comes out too high, and then you say, there's this number in between those two, it's really finicky and hard to write exactly, I'll call it pi. I can buy that.

If I have 36 and I'm like, you used a 0 to get there, what number did you multiply by to get to 36, and you say, 20, and I say, that's not high enough, and then you say 248975817328471, and then I say, that's not high enough, and then you say, it's a number let's call it &, I'm going to say, no, I'm starting to think this isn't a thing, stop trying to make fetch happen.

[talah]

I'm guessing I'm way out of my depth here but from what I understand, "Infinity" isn't a constant and there are ways to define scales of infinity and ways to quantify at least ratios of different infinities.

I don't see a reason that we can't discriminate between types of zero unless the definition of zero is strictly defined as a "something", whereas zero by that definition is necessarily a "nothing". So that doesn't quite make sense.

Anyway... here's another Mathologer video which seems tangentially relevant.