# Recognizing Zero and One

Be on the lookout for the following special names for zero and one!

## Special Properties of $\,0\,$ and $\,1$

• For all real numbers $\,x\,,$   $\,x + (-x) = 0\,.$ A number added to its opposite always gives zero. The opposite of a number is also called the additive inverse.
• For all real numbers $\,x\,,$   $\,x\cdot 0 = 0\,.$ Any number multiplied by zero gives zero.
• For all nonzero real numbers $\,x\,,$   $\,\displaystyle\frac{x}{x} = x\cdot\frac{1}{x} = 1\,.$   A nonzero number divided by itself (or multiplied by its reciprocal) always gives one.

The number $\,\frac{1}{x}\,$ is called the reciprocal of $\,x\,$ or the multiplicative inverse of $\,x\,.$ Multiplying a number by its reciprocal gives the number $\,1\,.$ Every nonzero number has a reciprocal; zero does not have a reciprocal.

## Examples

Decide if the given number is $\,0\,,$ $\,1\,,$ or a different number:

 $\displaystyle\frac13 + (-\frac 13)$ Answer: $\,0\,$ $\displaystyle 0\cdot\frac 13$ Answer: $\,0\,$ $\displaystyle -2\cdot \frac{-1}2$ Answer: $\,1\,$ $\displaystyle\frac{1/7}{1/7}$ Answer: $\,1\,$ $\displaystyle 3\bigl(-\frac13\bigr)$ Answer: not $\,0\,,$ and not $\,1$

This number is:
0
1
not 0, and not 1