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.
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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 $\,1$ |