# Integers

The integers are the set of whole numbers and their opposites. Fractions and decimals are not included in the set of integers.

For example, $2,5,0,-12,244,-15$ and $8$ are all integers.

The numbers such as $8.5,\frac{2}{3}$ and $4\frac{1}{3}$ are not integers.

(Note that a number can be an integer even if it is written as a decimal or a fraction: for example, $-3.00$ and $\frac{8}{2}$ are both integers, because they are equal to $-3$ and $4$ , respectively.)

The set of integers is usually represented by the symbol $ℤ$ .

$ℤ=\left\{...,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,...\right\}$

We can plot the integers as equally spaced points on a number line , as shown in the figure. The arrows at the left and right sides show that the integers continue forever in both directions. The whole numbers greater than $0$ are called positive integers . Their opposites, which are less than $0$ , are called negative integers . Zero is neither positive nor negative.

If two numbers are opposite, they are the same distance away from zero. For example, $4$ and $-4$ are opposites, and each is $4$ units from zero.

The sum, difference, or product of two integers is an integer. For example,

Sum: $3+4=7$

Difference: $3-5=-2$

Product: $\left(-2\right)\left(3\right)=-6$

The quotient of two integers is not always an integer.

For example, $8÷\left(-2\right)=-4$ is an integer because it divides evenly.

However, $-2÷8=\frac{-2}{8}=-\frac{1}{4}$ is not an integer. When a quotient of integers does not divide evenly, the result is a fraction .

Example:

Which of the following numbers is an integer?

$6.5,\sqrt{5},\frac{2}{3},-24$

$6.5$ is greater than the integer $6$ and less than the integer $7$ . The $.5$ at the end of the number indicates a fractional part. So, this is not an integer.

The number $\sqrt{5}$ has a square root sign; its value is greater than the integer $2$ but less than the integer $3$ . Since $5$ is not a perfect square like $4$ or $9$ , $\sqrt{5}$ is not an integer.

The number $\frac{2}{3}$ is a fraction greater than $0$ but less than $1$ , so this is not an integer.

The number $-24$ is in the set $\left\{...,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,...\right\}$ .

So, $-24$ is an integer.