Why can you divide 0 by 3 but not 3 by 0?

Answer 1

Division can be thought of as the opposite of multiplication:

0 ÷ 3 is the same as saying "what number when multiplied by 3 results in 0"; answer: 0.

3 ÷ 0 is the same as saying "what number when multiplied by 0 results in 3"; no number when multiplied by 0 results in 3 (as 0 times anything is 0), thus it can't be done.

Alternatively, division tells you how many times you need to, or can, subtract the divisor from the dividend to get to zero.

If you start with a dividend of zero and a non-zero divisor, you don't need to, nor can you, subtract the divisor to get to zero.

If you start with a non-zero dividend, and a zero divisor, no matter how many times you subtract the divisor you will never get to zero - the dividend stays the same.

With a zero dividend and a zero divisor, you have reached zero when you start, BUT you can subtract the divisor and the dividend will then become (stay) zero; thus zero divided by zero is any number you want - in calculus there are rules which specify the value to use in different circumstances.