   SEARCH HOME Math Central Quandaries & Queries  Question from Noni, a teacher: Q. 0÷0=1 Why it is wrong when 4÷4=1; 3÷3=1; 2÷2=1: 1÷1=1 ? Hi Noni,

Division is the inverse of multiplication so to ask "What is $35 \div 5?$" is equivalent to asking "What is $x$ if $5 \times x = 35?$".

"What is $4 \div 4?$" is equivalent to asking "What is $x$ if $4 \times x = 4?$".

The answer is clearly $1.$

"What is $3 \div 3?$" is equivalent to asking "What is $x$ if $3 \times x = 3?$".

The answer is clearly $1.$

"What is $2 \div 2?$" is equivalent to asking "What is $x$ if $2 \times x = 2?$".

The answer is clearly $1.$

"What is $1 \div 1?$" is equivalent to asking "What is $x$ if $1 \times x = 1?$".

The answer is clearly $1.$

"What is $0 \div 0?$" is equivalent to asking "What is $x$ if $0 \times x = 0?$".

The answer is here is that $x$ can be any number.

This creates a problem. Do you just define $0 \div 0 = 1$ and agree that in this case division is not the inverse of multiplication or do you find some other way out of this dilemma. Mathematicians have found it to be most useful to say that $0 \div 0$ is not a number and more generally that division by zero is not allowed.

I hope this helps,
Penny      Math Central is supported by the University of Regina and the Imperial Oil Foundation.