Integers and Directed Numbers

Understand positive and negative whole numbers, how to place them on a number line, and how to add and subtract them.

What Are Integers?

An integer is any whole number — positive, negative, or zero. There are no fractions or decimals.

Examples of integers: $\ldots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \ldots$

Positive integers are greater than zero: $1, 2, 3, \ldots$

Negative integers are less than zero: $-1, -2, -3, \ldots$

Zero ( $0$ ) is an integer that is neither positive nor negative.

We sometimes call negative and positive numbers directed numbers because they have a direction — either above or below zero. A useful real-life example is temperature: $-5°C$ is 5 degrees below zero, and $+3°C$ is 3 degrees above zero.

Note

Integers appear everywhere in real life — bank accounts (overdrafts use negative numbers), floors in a building (basements are negative), and sea level (depths below sea level are negative).

Adding and Subtracting Integers

Adding a positive number moves you to the right on a number line.

Adding a negative number is the same as subtracting: $5 + (-3) = 5 - 3 = 2$

Subtracting a negative number is the same as adding: $5 - (-3) = 5 + 3 = 8$

A helpful rule: two negatives make a positive.

Example 1: $-4 + 7$

Start at $-4$ , move 7 places right: $-4 + 7 = \mathbf{3}$

Example 2: $-3 - 5$

Start at $-3$ , move 5 places left: $-3 - 5 = \mathbf{-8}$

Example 3: $-2 - (-6)$

Two negatives → add: $-2 + 6 = \mathbf{4}$

Tip

Draw a number line and physically count steps. Moving right = adding, moving left = subtracting. This always gives the correct answer.

Ordering Integers

To put integers in order, remember that further left on a number line = smaller.

Order from smallest to largest: $3, -5, 0, -1, 7$

Answer: $-5, -1, 0, 3, 7$

Comparing: $-8 < -2$ because $-8$ is further left (more negative).

Warning

Do not confuse "bigger" with "further from zero". $-10$ is not bigger than $-2$ just because it is further from zero — it is actually smaller.

Real world: building floors and basements

A hotel has floors numbered $-2$ (underground car park), $-1$ (basement), $0$ (ground), $1$ to $15$ . Moving from floor $-2$ to floor $7$ is $7 - (-2) = 9$ floors. Directed numbers give lifts and building management systems a precise way to track position.

Real world: bank account overdraft

A student's account shows a balance of $-35\text{ EUR}$ (overdrawn by €35). They deposit €50. New balance $= -35 + 50 = +15\text{ EUR}$ . Banks use integer arithmetic constantly — every debit and credit is an addition of a directed number.

Real world: temperature change in science

An experiment cools a liquid from $+8°C$ to $-13°C$ . Temperature change $= -13 - 8 = -21°C$ (a drop of 21 degrees). Scientists express temperature changes as directed numbers to indicate both magnitude and direction of change.

Tip Subtracting a negative is the same as adding a positive:

a - (-b) = a + b

. Think of removing a debt as gaining money.

Practice Problems

Calculate: $-6 + 10$

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Start at $-6$, move 10 right: $-6 + 10 = 4$.

Calculate: $3 - (-5)$

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Two negatives make a positive: $3 - (-5) = 3 + 5 = 8$.