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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

a3 = 1·a·a·a
a2 = 1·a·a
a1 = 1·a
a0 = 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 00 ≝ 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


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