Why is 1/0 = oo ?

2 Answers
Oct 12, 2015

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.

Oct 13, 2015

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.