ditto for cube root, nth root, etc.
In: 5
A better way to think about operators like plus, minus, multiply, divide, etc, is to picture what it does to a number line.
For instance, if you add 5 to some number, imagine a number line where you pick some value x on it. Now what does adding 5 do? It takes every value and sends it right 5 units, in effect “shifting” the entire number line 5 units over. Subtraction does the exact same thing, except it shifts the number line left for every value.
what does multiplying do to a number line? It stretches it. So if you point to some value x and multiply by three, what happens? Well, think about sticking a pin in 0 because that doesn’t move, and then pulling left and right and stretching it until 1 goes to 3. What happened to the value you were pointing at? It went from x to 3x.
Dividing is the same thing, except instead of stretching, you’re contracting the number line.
Now think about what happens if you multiply by zero, what does that do to the number line? It collapses the entire thing into a single point at 0. So what does dividing by 0 do? Well, it takes 0 and unpacks it to every value on the number line.
This is why dividing by zero is not “infinity” but, rather, undefined. It doesn’t actually send 0 to infinity, it sends it to every value on the number line simultaneously.
Why is this undefined, though? Couldn’t we just say it’s “everything”? Yes we could, actually. Except there’s a definition in mathematics for a function, and that definition says that a function is a beastie that takes some input value and maps it to a *unique* output value. Remember, when you graph a function, the one thing you cannot have is multiple y values associated with a single x value, if you draw a vertical line at some x value, it can only hit one corresponding y value.
All of math is built around this assumption that this definition of functions is respected. So you can definitely decide that you want to do math where you define some new thing instead of functions that doesn’t follow that definition, but then you aren’t allowed to use any other tools that rely on the definition of function (which is most of them).
Brief digression: I encourage you to think of *every* function not as an x-y graph, but instead as some complicated way of stretching/shrinking/shifting/etc a number line. The reason is that you quickly start to think in a very mathy way. For example, when I said multiply you stick a pin in zero, knowing that value doesn’t change is very interesting. It’s a characteristic of that particular operation.
Now if you think of some other function, like some arbitrary polynomial, as just a single complicated operation where you’re non-uniformly squishing/stretching/etc this number, sending every value to some other value, it likely also has some characteristic behaviors. And if you look at similar functions, you’ll see that those probably have the same general characteristics when you switch the numbers around, except it just moves where the pins are. So after you become familiar with this way of thinking about things, you can start to look at how number lines are moved around by some arbitrary function and recognize certain things about that function similar to other things you already know.
The problem is this
If 5*0 = 0, then 0/0 = 5.
But then you do the same with another number
If 8*0 = 0, 0/0 = 8
So you get 5 = 8. Which obviously isn’t true. So to avoid that we have to say that anything divided by 0 is undefined.
Another reason is that anything divided by itself is 1. But anything multiplied by 0 is 0. So we have something that should be equal to both 1 and 0 at the same time, so it’s undefined
You can… in the trivial ring.
In other division rings, unfortunately, it’s not possible, because there is no multiplication inverse of 0. I’ll mark the multiplication inverse of a number n as n^(T)
0 0^(T) = 1 (definition of multiplication inverse)
0 = 1 (definition of 0)
That statement is only compatible with a trivial ring. To show that, we pick any element of a ring R. Then
0r = 1r (multiplying both sides with r)
0 = r (definition of 0 and 1)
This shows that all elements of that ring are equal to zero. This means that the ring is just {0}
There is a way to work around that, by working with a wheel instead of a division ring, Wheel is similar to a ring, but with a slightly different axiom: [https://en.wikipedia.org/wiki/Wheel_theory](https://en.wikipedia.org/wiki/Wheel_theory)
A common way to obtain a wheel is by adjoining a bottom element to a projective line. The bottom would be the solution of 0/0
If we have to pick a number for 0/0, then it should be 1. Because for any other real number x/x = 1. So if you use lim x->0 (x/x), if you approach from both side, you will get 1.
It’s also why 0^0 is usually consider as 1.
Several people already explained how 0/0 can be literally any number, but I just want to chime in that for any number besides 0, x/0 is called “undefined” because an answer does not exist. On the other hand, 0/0 is called “indeterminate,” because there are infinite answers.
When we talk about dividing, what we are really doing is multiplying by the inverse. By that I mean that 7/9 is shorthand for 7*(1/9). So when you ask why we cannot divide by 0, you are really asking why 1/0 isn’t defined.
The reason 1/0 isn’t defined is because imagine 1/0=x. Then multiple both sides by 0 to get 1=x*0. However anything times 0 is 0, so we get 1=0, which is clearly wrong. Therefore 1/0 must be undefined.
Think of the plot of [y=1/x](https://upload.wikimedia.org/wikipedia/commons/4/43/Hyperbola_one_over_x.svg), compared to [y=x^2](https://study.com/cimages/multimages/16/2f02bcfd-854d-486a-ab39-30961b3337c4_yx2.jpg)
Think of multiplication as “taking from a box of apples” and division as “putting into a box of apples”.
Using those definitions, “0 * [any number]” is the same as “taking 0 apples from the box [any number] times”. At the end, how many apples do you have? Zero because you never actually took anything from the box.
For division, “[any number] divided by zero” is the same as “how many times can you put 0 apples into the box if you start with [any number]”. It’s obviously a pointless exercise so they call it “undefined”.
We want things to have a single definitive correct answer, so I can say:
x + 2 = 4
and the only right answer for “x” is “2”. “4 – 2 = 2”, that’s it, that’s the answer. We know because we can plug it back in as “x” and get the right result.
But while 0*0=0, we also have 0*1=0 and 0*2=0 and 0*1234567890=0.
So there’s no way to have a single right answer for 0/0, because when we get to “x * 0 = 0”,
any value can be plugged in for “x” and we get the right result.
By your logic, since 0 * 10 = 0, we could say that 0/0 = 10.
That’s the problem with 0/0 – it’s undefined, because you could make it equal whatever you want.
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