BASIC PROPERTIES OF ZERO AND ONE

Zero ( $0$ ) and
one ( $1$ ) are very special numbers.

This page summarizes their important properties.

Jump right to the properties!

Some of these properties require multiplication and division, so a quick review is in order:

*Ways to denote multiplication*:

A number
$\,x\,$ multiplied by a number
$\,y\,$ can be denoted in several ways:

$x \cdot y$ | using a centered dot | Juxtaposition (see below) is simpler and preferred, for variables. The centered dot is useful for constants: e.g., $2 \cdot 3 = 6\,$. |

$xy$ | using juxtaposition (placing things side-by-side) |
It is conventional to write a constant before a variable. For example, write $\,3x\,$, NOT $\,x3\,$. |

$(x)(y)$ | using parentheses | Juxtaposition (see above) is simpler and preferred, for variables. Parentheses are needed in situations like this: $(x+1)(x+2)$ |

*Ways to denote division*:

A number
$\,x\,$ divided by a number
$\,y\,$ can be denoted in several ways:

$\displaystyle\frac{x}{y}$ | using a horizontal fraction bar | There are implied parentheses in the numerator and denominator in this form. For example: $\displaystyle\frac{x+1}{x+2}$ means $(x+1)/(x+2)$ |

$x/y$ | using a forward slash | Be careful! Normal order of operations is at work here. For example: $x+1/x+2$ means $x + \frac 1x + 2\,$, NOT $\displaystyle\frac{x+1}{x+2}$ |

$x \div y$ | using the division symbol ‘$\div$’ | This style is rarely used in algebra, and beyond. |

In all these forms, $\,x\,$ is called the

*Addition Property of Zero*:

Adding zero to a number does not change it:

For all real numbers
$\,x\,$,
$x + 0 = 0 + x = x\,$.

*Multiplication Property of Zero*:

Multiplying a number by zero always gives zero:

For all real numbers
$\,x\,$,
$x \cdot 0 = 0 \cdot x = 0\,$.

*Multiplication Property of One*:

Multiplying a number by one does not change it:

For all real numbers
$\,x\,$,
$x \cdot 1 = 1 \cdot x = x\,$.

Recall that
$2^5$ (read as ‘two to the fifth power’ or simply ‘two to the fifth’)

is a shorthand for
$\,2 \cdot 2 \cdot 2 \cdot 2\cdot 2\,$ (five factors of two).

*Powers of One*:

The number one, raised to any power, equals one:

For all real numbers
$\,n\,$,
$\,1^n = 1\,$.

(Even though you may not know about negative and fractional powers yet, don't worry!

Just start getting used to the fact that the number one, raised to *any* power, is always one.)

*Powers of Zero*:

The number zero, raised to any allowable power, equals zero:

For
$\,n = 1,2,3,\ldots\,$,
$\,0^n = 0\,$.

In particular, zero to the zero power
($\,0^0\,$) is not defined.

*Zero as a numerator* :

Zero, divided by any nonzero number, is zero:

For all real numbers
$\,x\ne 0\,$,
$\,\frac{0}{x} = 0\,$.

Note:
$\frac{0}{0}$ is not defined.

*Division by zero is not allowed* :

Any division problem with zero as the denominator is not defined.

For example,
$\,\frac{0}{0}\,$,
$\,\frac{2}{0}\,$, and
$\,\frac{5.7}{0}\,$
are not defined.

*Names for the number one*:

Any nonzero number divided by itself equals one:

For all real numbers
$\,x\ne 0\,$,
$\,\frac xx = 1\,$.

*Names for the number zero*:

The numbers
$\,3\,$ and
$\,-3\,$ are *opposites*;

they are the same distance from zero, but on opposite sides of zero.

Any number added to its opposite is zero:

For all real numbers
$\,x\,$,
$\,x + (-x) = (-x) + x = 0\,$.

Note: the ‘opposite of
$\,x\,$’ is also called the ‘additive inverse of
$\,x\,$’:

it is the number which, when added to
$\,x\,$, gives zero.

Master the ideas from this section

by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:

Mixed Basic Add/Subtract/Multiply/Divide Practice

by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:

Mixed Basic Add/Subtract/Multiply/Divide Practice

On this exercise, you will not key in your answer.

However, you can check to see if your answer is correct.

However, you can check to see if your answer is correct.