﻿ Basic Properties of Zero and One

# Basic Properties of Zero and One

Zero ( $0$ ) and one ( $1$ ) are very special numbers. This page summarizes their important 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.

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.