Introduction to Integers

Up to this point, most of the numbers you have worked with have been zero or positive: counting objects, measuring lengths, and calculating totals. But many real-world situations involve values that go below zero. Temperatures drop below freezing, bank accounts go into overdraft, and elevations dip below sea level. To represent these situations mathematically, we need integers.

What Are Integers?

Integers are the set of whole numbers and their negatives:

$\{\ldots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \ldots\}$

This set includes:

• Positive integers: $1, 2, 3, 4, \ldots$ (numbers greater than zero)
• Zero: $0$ (neither positive nor negative)
• Negative integers: $-1, -2, -3, -4, \ldots$ (numbers less than zero)

Integers do not include fractions or decimals. For example, $\frac{1}{2}$ and $3.7$ are not integers, but $-5$ and $12$ are.

The Number Line

The number line is the essential tool for visualizing integers. Zero sits at the center, positive numbers extend to the right, and negative numbers extend to the left.

Key observations about the number line:

• Numbers increase as you move to the right.
• Numbers decrease as you move to the left.
• Every positive number has a corresponding negative number the same distance from zero on the opposite side. These are called opposites (e.g., $3$ and $-3$).

Real-World Contexts for Integers

Integers appear in many everyday situations. Recognizing these contexts helps you understand what negative numbers mean in practice.

Temperature

Temperatures frequently go below zero, especially if you live in a cold climate or work with freezers.

• A temperature of $-10°\text{F}$ means 10 degrees below zero.
• A temperature increase from $-5°\text{F}$ to $3°\text{F}$ is an 8-degree rise.

Money and Debt

In banking, a negative balance represents money you owe.

• A balance of -\150 means you are \150 overdrawn.
• Depositing $200 into an account at -\150 brings the balance to \50.

Elevation

Elevation is measured relative to sea level, which is defined as zero.

• The Dead Sea shore is approximately $-430$ meters (430 meters below sea level).
• Mount Everest’s peak is approximately $8{,}849$ meters above sea level.

Other Contexts

• Football: A loss of 7 yards can be written as $-7$ yards.
• Time zones: UTC$-5$ means 5 hours behind Coordinated Universal Time.
• Stock market: A stock dropping 3 points can be represented as $-3$.

Opposites of Integers

The opposite of an integer is the number that is the same distance from zero but on the other side of the number line.

$\text{The opposite of } a \text{ is } -a$

For example:

• The opposite of $7$ is $-7$.
• The opposite of $-4$ is $4$.
• The opposite of $0$ is $0$ (zero is its own opposite).

A number and its opposite always add up to zero:

$a + (-a) = 0$

Worked Examples

Example 1: Identifying Integers

Which of the following are integers? $\quad -3, \quad 4.5, \quad 0, \quad \frac{7}{2}, \quad 12, \quad -100$

Solution: The integers are $-3$, $0$, $12$, and $-100$. The values $4.5$ and $\frac{7}{2}$ are not integers because they are not whole numbers.

Example 2: Writing Situations as Integers

Express each situation as an integer:

• A submarine is 200 feet below sea level: $-200$
• A profit of $5,000: $+5{,}000$
• A temperature of 15 degrees below zero: $-15$
• A gain of 8 pounds: $+8$

Example 3: Finding Opposites

Find the opposite of each integer: $-9$, $14$, $0$.

• The opposite of $-9$ is $9$.
• The opposite of $14$ is $-14$.
• The opposite of $0$ is $0$.

Practice Problems

Problem 1: Which of the following are integers? $\quad -7, \quad 3.14, \quad 0, \quad \frac{5}{1}, \quad -\frac{2}{3}, \quad 45$

Problem 2: A diver descends 40 feet below the surface. Express this as an integer.

Problem 3: What is the opposite of $-23$?

Problem 4: The temperature at midnight was $-8°\text{F}$. By noon it had risen 20 degrees. What was the noon temperature?

Problem 5: List all the integers between $-4$ and $3$, not including $-4$ and $3$ themselves.

Key Takeaways

• Integers include all positive whole numbers, zero, and negative whole numbers. They do not include fractions or decimals.
• The number line shows integers in order, with negative numbers to the left of zero and positive numbers to the right.
• Negative integers represent values below zero: debt, below-sea-level elevations, temperatures below freezing, and losses.
• Every integer has an opposite that is the same distance from zero on the other side. A number and its opposite always sum to zero.
• Zero is an integer, is neither positive nor negative, and is its own opposite.

Return to Arithmetic for more foundational math topics.