# 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: $(x+1)(x+2)$ |

In algebra and beyond, do NOT use the symbol ‘$\,\times\,$’ to denote multiplication, since it can be confused with the variable $\,x\,.$ (Exception: it is conventional to use an ‘$\,\times\,$’ for scientific notation.)

*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. |

The first way (the horizontal fraction) is the preferred
representation in algebra, and beyond.
In all these forms,
$\,x\,$ is called the *numerator* and
$\,y\,$ is called the *denominator*.

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

Note: 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.