# Integers

An *integer* is a number that does not have a fractional part. The set of integers is

$\mathbb{Z}=\{\cdots -4, -3, -2, -1, 0, 1, 2, 3, 4 \dots\}.$

The notation $\mathbb{Z}$ for the set of integers comes from the German word *Zahlen*, which means "numbers". Integers strictly larger than zero are **positive integers** and integers strictly less than zero are **negative integers**.

For example, $2$, $67$, $0$, and $-13$ are all integers (2 and 67 are positive integers and -13 is a negative integer). The values $\frac{4}{7}$, $10.7$, $\frac{34}{7}$, $\sqrt{2}$, and $\pi$ are not integers.

## Properties of Integers

The following are the properties of integers:

- The set of integers is closed under the operation of addition: if $a, b \in \mathbb{Z}$, then $a+b \in \mathbb{Z}$.
- The set of integers is closed under the operation of multiplication: if $a, b \in \mathbb{Z}$, then $ab\in \mathbb{Z}$.
- For any integer $a$, the additive inverse $-a$ is an integer.
- If $a$ and $b$ are integers such that $a \cdot b = 0$, then $a=0$ or $b=0$.
- The set of integers is infinite and has no smallest element and no largest element.

$(\in$ means "belongs to", as $a \in Z$ means $a$ is an element of the set $Z$ or $a$ belongs to the set $Z.)$

Note that the set of integers is not closed under the operation of division. As an example, $a=3$ and $b=4$ are integers, but $\frac{a}{b} = \frac{3}{4}$ is not an integer.

Which of the following are integers?

$\begin{array}{c}&\frac{4}{2}, &-8, &0. 2, &12-4, &\frac{10}{4} \end{array}$

Since $\frac{4}{2} = 2$, $12-4 = 8,$ and $2 < \frac{10}{4} < 3$, the integers in the list are $\frac{4}{2}, -8$, and $12-4.$ $_\square$

What is the smallest integer that is larger than $\frac{10}{3}?$

Since $3< \frac{10}{3} < 4,$ the smallest integer that is larger than $\frac{10}{3}$ is $4$. $_\square$

What is the largest integer that is smaller than $\frac{10}{3}?$

Since $3< \frac{10}{3} < 4,$ the largest integer that is smaller than $\frac{10}{3}$ is $3$. $_\square$

Using the properties of integers above, show that set of integers is closed under the operation of subtraction.

Consider any two integers $a$ and $b$. We would like to show $a-b$ is also an integer. By property $3,$ the additive inverse of $b$ is $-b$, which is an integer. Then

$a-b = a + (-b)$

is an integer by Property $1.$ Therefore, the set of integers is closed under the operation of subtraction. $_\square$