Question #d0dfe

2 Answers
Mar 25, 2015

When you divide, say, 3/0 you are trying to find a result such as:
3/0=a
But this number a should be a number that multiplied by 0 gives 3!
3/0=a so rearranging 3=0*a
But this is not possible!
So, it is not possible to divide by 0.

On the other hand have a look at what happens if you get "near" to zero but not zero.
Try 0.01, 0.0001, 0.000001 and see what happens!

Mar 25, 2015

You can't do it.
(Any attempt to define division by zero will "break arithmetic" somewhere.)

Reason 1:

a/b = c exactly when b*c=a

But if b=0, we have

a/0 = c exactly when 0*c=a

0*c=a has no solution for a!=0 because 0*c=0 for all c.

(For example: 5/0=c would require 0*c=5 which cannot happen.)

Reason 2:

I am an algebraist, I define division to be multiplication by a reciprocal.

A reciprocal of a is a multiplicative inverse. That is, it is a solution to a*x="multiplicative identity"

For any number, x, we can show that 0*x=0 So 0 has no multiplicative inverse (no reciprocal).

0x+x=0x+1x=(0+1)x=1x=x
0x+x=x implies that 0x=0 (Subtract x from both sides.)

(General case)
In any ring whose additive identity is denoted 0,
we have 0 x=0 and x0=0 for all x.
So the only ring in which 0 has a reciprocal is the trivial ring: {0}.
(The trivial ring has one thing in it. That thing is the additive and multiplicative identities. In non-trivial rings, it is not possible for both identities to be the same.)