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.

The following are the properties of integers:

1. The set of integers is closed under the operation of addition: if $a, b \in \mathbb{Z}$, then $a+b \in \mathbb{Z}$.
2. The set of integers is closed under the operation of multiplication: if $a, b \in \mathbb{Z}$, then $ab\in \mathbb{Z}$.
3. For any integer $a$, the additive inverse $-a$ is an integer.
4. If $a$ and $b$ are integers such that $a \cdot b = 0$, then $a=0$ or $b=0$.
5. 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?

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

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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 $\frac{10}{3}?$

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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 $\frac{10}{3}?$

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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.

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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. $_\square$