< **1** >

### One and the other

The number 1 has a special meaning in mathematics.

##### Example 1

The number 1 is also called factor 1, as a product keeps unchanged. You will understand that from the calculation

or written as

you can not conclude that 2 equals 3.

##### Example 2

The meaning of exponentiation is "Multiply the factor 1 so often with the base, as indicated by the exponent", so

a^{3}= 1·a·a·a

a^{2}= 1·a·a

a^{1}= 1·a

a^{0}= 1

Exponent zero indicates that the factor 1 is *not* to be multiplied with the base, and as a result only 1 is left. This notation produces 0^{0} ≝ 1 and that is just fine.

##### Example 3

There are more special cases like this, where you can't immediately determine whether it's allowed or not. Take the calculation

Is the answer *a = *7 correct, or should we say *'not possible'*. After all, the square root of a negative number is invalid. But what about