Question

Why is `1/0 = oo` ?

Answer 1

You can fit infinite zeroes into any number

Explanation:

When dividing, you are specifying how may parts of the denominator can fit into the numerator. Therefore, you can fit infinite `prop` 0 into any number.

Answer 2

`x/0` is normally undefined, not `oo`

Explanation:

`a/b` is the number which when multiplied by `b` gives `a`.

So if `c = a/b`, then `a = bc`.

Now if `b = 0`, then `bc = 0` for any `c`.

So if `a = 0`, `c` can take any value and if `a != 0`, there are no values of `c` which satisfy the equation.

So whether `a = 0` or `a != 0`, the quotient `a/0` is undefined.

So what is `oo`?

It's used in various ways, but often as a shorthand for 'unlimited'.

For example, we speak of the limit as `n -> oo`, meaning as `n` gets larger and larger without limit.

So we might write:

`lim_(n->oo) 1/n = 0`

Usually, we also have a negative infinity `-oo`, so we could write:

`lim_(n->-oo) 1/n = 0`

...referring to the limit as `n` gets more and more negative without limit.

We can imagine `+oo` and `-oo` as being at the extreme ends of the Real line - though the Real line doesn't really have ends.

The symbols `+oo` and `-oo` are also used to express the results of a limit process.

For example, we could write:

`lim_(x->oo) x^2 = +oo`

meaning that as `x` gets larger and larger without limit, so does `x^2`.

We also find:

`lim_(x->0) 1/x^2 = +oo`

meaning that as `x` approaches `0` closer and closer, `1/x^2` get more and more positive without limit.

There are also one sided limits. If we want to speak of the limit as `x` approaches `0` from the 'right' (i.e. `x > 0`) then we write:

`lim_(x->0+) 1/x = +oo`

or from the 'left' (i.e. `x < 0`):

`lim_(x->0-) 1/x = -oo`

So notice that the left and right limits are in stark disagreement about what value we might try to give `1/0`

When `1/0 = oo`

There are a couple of contexts in which it's meaningful to speak of `1/0 = oo`.

They are called the Projective Line `RR_oo` and Riemann Sphere `CC_oo`. These add a single point called '`oo`' to the Real line or to the Complex plane. You probably won't meet these properly for a few years yet.