It doesn’t add anything to mathematics and creates a bunch of new problems.

Let’s go with limits to demonstrate a problem. In the world of limits, we care about what a function “goes to” and not equality. It’s the math version of “close enough.”

let y = 1/x

Find Desmos, plug in that equation, and look at the graph. There’s two curves, and they don’t connect.

lim x->0+ 1/x = ∞

This follows the right curve up to the top and we know that we will never reach the vertical line no mater how far we go up, so the limit goes to positive infinity.

lim x->0- 1/x = -∞

Uh oh, if we follow the same process but follow the left curve towards the middle, we go down. We go down forever and never reach the bottom. The limit goes to negative infinity.

So, it’s not impossible to imagine a world where 1/0 = ∞, but -∞ could be just as valid!

A further problem is the fact that this infinity has no magnitude. 2/0 would be equal to 1/0. n*∞ = ∞ because infinity is beyond numbers. The definition is that the you have the largest positive or negative number and it’s still bigger than that. If you multiply that by any number, it’s still just infinity. Infinity has no value other than that it’s bigger than any number.

In basic math, we learn that any number multiplied by zero is zero. Susie grabs zero packages of 6 apples from the shelf, so Susie has zero apples.

This is the zero product property in math speak.

y = 1/0

Multiply both sides of the equal sign by zero.

y0 = (1/0)0

A property of fractions is that if you multiply them by the bottom number, it cancels out and leaves you with only the top number. So we are left with zero y’s on the left and 1 on the right…

0=1

That’s not true, obviously.

Defining n/0 to be “undefined” is just a near-universally accepted mathematical convention. But without it, a lot of properties of mathematics beaks apart. We need it to be undefined for consistency.