# Division by zero

1. Jan 26, 2007

### Ahmed Ismail

Goodmorning all
I want to know why 0 divide anything = zero, and why division by zero is undefined?

regards

2. Jan 26, 2007

### HallsofIvy

Saying a/b= c means that a= bc. If a/0= c then a= 0c. But 0 times any number is 0- the equation a= 0c for a not equal to 0 is impossible so a/0= c is also impossible. If a= 0, theres a different problem. 0/0= c becomes 0= 0c which true no matter what c is! We still are unable to say that 0/c is any particular number.

Many math textbooks use the term "undefined" for a/0 when a is not zero and "undetermined" for 0/0 because of that difference.

3. Jan 26, 2007

### arildno

1. There is no such SEPARATE thing called "division" as you've learnt it(contrary to school teaching).
What you DO have, is the following:
Given any non-zero real number "a", there exists another number, that we have given the name "(1/a)", that has the property when multiplied with "a" yields the number 1, that is: a*(1/a)=1.
We can prove that for any particular number "a", the number "(1/a)" is UNIQUE, so the identity a*(1/a)=1 can be said to DEFINE (1/a) (just like we DEFINE "2" to be the number so that 1+1=2, really)

Whenever we multiply (1/a) with some other number, say b, that is we form the product b*(1/a), we find this notation so cumbersome so that in short hand, we introduce the notation b/a=b*(1/a).

Now, we can prove that whenever we multiply a number with 0, then we get 0, so for any "a", we have 0/a=0*(1/a)=0, providing the answer to your first question.

The answer to your second question is also contained here, because if 0*anything=0, then, since 1 isn't 0, then there CAN'T exist a number (1/0) having the property 0*(1/0)=1 (remember I required that a had to be non-zero earlier?)