RECOGNIZING 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.

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

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 $\,1$
Master the ideas from this section