Table of Contents
Understanding Integers
Integers extend the idea of whole counting numbers to include negative values and zero. In this chapter, we focus on what integers are, how they are arranged, and how they behave under basic operations. Ideas that depend on other number types (like fractions or decimals) will be left for later chapters.
What Integers Are
The set of integers is usually written as $\mathbb{Z}$ (from the German word “Zahlen,” meaning “numbers”).
An integer is any whole number that is:
- positive, like $1, 2, 3, \dots$
- negative, like $-1, -2, -3, \dots$
- zero, $0$
So we can list them as:
$$
\mathbb{Z} = \{ \dots, -3, -2, -1, 0, 1, 2, 3, \dots \}
$$
Key points:
- There are no fractional or decimal parts in integers.
- Between any two neighboring integers there is no other integer. For example, between $2$ and $3$ there is no integer.
- The integers go on forever in both directions: there is no largest positive integer and no smallest negative integer.
The Number Line for Integers
Integers are often shown on a horizontal line called the number line.
- Zero is placed in the center.
- Positive integers lie to the right of $0$.
- Negative integers lie to the left of $0$.
- Numbers increase as you move to the right and decrease as you move to the left.
If one integer is to the right of another on the number line, it is greater. If it is to the left, it is smaller.
For example:
- $3$ is to the right of $-2$, so $3 > -2$.
- $-5$ is to the left of $-1$, so $-5 < -1$.
Sign and Opposites
Every integer (except $0$) has a sign:
- A positive integer is usually written without a sign, like $5$. You can write $+5$, but we normally just write $5$.
- A negative integer is written with a minus sign, like $-5$.
Each nonzero integer has an opposite:
- The opposite of $5$ is $-5$.
- The opposite of $-5$ is $5$.
We say that $a$ and $-a$ are opposites. They are the same distance from $0$ on the number line, but on opposite sides.
Zero is its own opposite:
$$
-0 = 0
$$
Comparing Integers
To compare two integers, think about their positions on the number line.
- Any positive integer is greater than zero, and any negative integer is less than zero:
- If $n$ is positive, then $n > 0$.
- If $n$ is negative, then $n < 0$.
- Therefore, any positive integer is greater than any negative integer. For example, $1 > -100$.
- When both are positive:
- The one with the larger usual size is greater.
- For example, $7 > 3$.
- When both are negative:
- The one that is “less negative” (closer to zero) is greater.
- For example, $-2 > -5$ because $-2$ is to the right of $-5$ on the number line.
Inequality symbols you will see:
- $a < b$ means “$a$ is less than $b$.”
- $a > b$ means “$a$ is greater than $b$.”
- $a \le b$ means “$a$ is less than or equal to $b$.”
- $a \ge b$ means “$a$ is greater than or equal to $b$.”
Absolute Value of Integers
The absolute value of an integer is its distance from zero on the number line, ignoring the sign.
- The absolute value of $a$ is written $|a|$.
- $|a|$ is never negative.
Examples:
- $|5| = 5$
- $|-5| = 5$
- $|0| = 0$
In general:
$$
|a| =
\begin{cases}
a & \text{if } a \ge 0 \\
-a & \text{if } a < 0
\end{cases}
$$
Absolute value turns a number into its nonnegative size, but does not care about direction.
Adding Integers
Addition with integers extends the idea of counting to include moving in both directions on the number line.
Adding with the Same Sign
If both integers have the same sign:
- Add their absolute values.
- Keep the common sign.
Examples:
- $3 + 5 = 8$ (both positive, so answer is positive)
- $-3 + (-5) = -(3 + 5) = -8$ (both negative, answer is negative)
- $7 + 2 = 9$
- $-4 + (-1) = -5$
On the number line, adding a positive number moves you to the right; adding a negative number moves you to the left.
Adding with Different Signs
If the integers have different signs:
- Subtract the smaller absolute value from the larger absolute value.
- Give the result the sign of the number with the larger absolute value.
Examples:
- $7 + (-3)$: compare $|7|$ and $|{-3}|$. Since $7 > 3$,
$$
7 + (-3) = 7 - 3 = 4
$$ - $-7 + 3$: compare $|{-7}|$ and $|3|$. Since $7 > 3$,
$$
-7 + 3 = -(7 - 3) = -4
$$ - $-10 + 10$:
$$
|{-10}| = |10| = 10,\quad -10 + 10 = 0
$$
When the two numbers are opposites, their sum is $0$.
Subtracting Integers
Subtraction of integers can always be turned into addition by using opposites.
The key idea:
$$
a - b = a + (-b)
$$
That is, subtracting $b$ is the same as adding its opposite.
Examples:
- $7 - 3 = 7 + (-3) = 4$
- $7 - (-3) = 7 + 3 = 10$
- $-5 - 2 = -5 + (-2) = -7$
- $-5 - (-2) = -5 + 2 = -3$
On the number line:
- Subtracting a positive number moves you left.
- Subtracting a negative number moves you right (because it’s like adding a positive).
Multiplying Integers
Multiplication with integers follows definite sign rules.
Sign Rules for Multiplication
- Positive $\times$ positive = positive
- Negative $\times$ negative = positive
- Positive $\times$ negative = negative
- Negative $\times$ positive = negative
More symbolically:
- If $a$ and $b$ have the same sign and neither is zero, then $ab > 0$.
- If $a$ and $b$ have different signs, then $ab < 0$.
- If either $a$ or $b$ is $0$, then $ab = 0$.
Examples:
- $3 \times 4 = 12$
- $(-3) \times 4 = -12$
- $3 \times (-4) = -12$
- $(-3) \times (-4) = 12$
- $0 \times 5 = 0$
- $0 \times (-7) = 0$
Multiplying Several Integers
When multiplying more than two integers, the sign of the product depends on how many negative factors there are.
- If the number of negative factors is even, the product is positive.
- If the number of negative factors is odd, the product is negative.
Examples:
- $(-2) \times (-3) \times 4$: two negatives (even), so product is positive.
$$
(-2) \times (-3) = 6,\quad 6 \times 4 = 24
$$ - $(-2) \times 3 \times (-4) \times (-1)$: three negatives (odd), so product is negative.
First $(-2) \times 3 = -6$, then $-6 \times (-4) = 24$, then \times (-1) = -24$.
Dividing Integers
Division of integers uses the same sign rules as multiplication.
If $b \ne 0$, then $a \div b$ (or $a/b$) is the integer that, when multiplied by $b$, gives $a$, provided such an integer exists.
Sign Rules for Division
- Positive $\div$ positive = positive
- Negative $\div$ negative = positive
- Positive $\div$ negative = negative
- Negative $\div$ positive = negative
In brief:
- If $a$ and $b$ have the same sign (and $b \ne 0$), then $a/b$ is positive.
- If $a$ and $b$ have different signs, then $a/b$ is negative.
Examples:
- $12 \div 3 = 4$
- $(-12) \div 3 = -4$
- $12 \div (-3) = -4$
- $(-12) \div (-3) = 4$
Important:
- Division by zero is not allowed: $a \div 0$ is undefined for any integer $a$.
- Sometimes $a/b$ is not an integer (for example, $1/2$), in which case the result is not an integer. These situations will be handled in later chapters on rational numbers. Here we focus only on divisions that result in integers.
Basic Properties of Integer Operations
When we add or multiply integers, they behave in predictable ways. These patterns are important and will be used throughout mathematics.
Let $a, b, c$ be integers.
Closure
- If you add two integers, the result is an integer:
$$
a + b \in \mathbb{Z}
$$ - If you multiply two integers, the result is an integer:
$$
a \times b \in \mathbb{Z}
$$
(Closure generally does not hold for division: $a/b$ need not be an integer.)
Commutative Property
The order does not matter for addition or multiplication:
- Addition: $a + b = b + a$
- Multiplication: $ab = ba$
Associative Property
How you group numbers does not matter for addition or multiplication:
- Addition: $(a + b) + c = a + (b + c)$
- Multiplication: $(ab)c = a(bc)$
Distributive Property
Multiplication distributes over addition:
$$
a(b + c) = ab + ac
$$
This works for all integers $a, b, c$.
Identity Elements
- Additive identity: $0$ is the unique integer such that
$$
a + 0 = 0 + a = a
$$ - Multiplicative identity: $1$ is the unique integer such that
$$
a \times 1 = 1 \times a = a
$$
Additive Inverses
Every integer $a$ has an additive inverse $-a$ such that:
$$
a + (-a) = 0
$$
This is why opposites add to zero.
Integers in Context
Integers often describe situations with direction, gain, or loss. A few typical uses:
- Temperature: Degrees above $0^\circ$ as positive, below as negative (for example, $-10^\circ$).
- Money: Gains as positive, losses or debts as negative (for example, $-25$ dollars means owing $25$).
- Height or depth: Positions above sea level as positive, below as negative.
- Game scores: Points gained as positive, penalties as negative.
In each case, $0$ is a reference point (no gain or loss, sea level, freezing point, and so on), positive values are one “direction” from it, and negative values the opposite.
These interpretations help give meaning to the signs and operations you perform with integers, and connect the abstract number line to real situations you encounter.