Integers — Rules for Operations & Number Line

Integers are the positive whole numbers, the negative whole numbers, and $0$: $\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$. They do not include fractions or decimals.

If you want the quick rule set, start with two ideas. On the number line, the sign tells direction from $0$ and the size tells distance from $0$. For multiplication and division, same signs give a positive result and different signs give a negative result.

What Integers Mean On a Number Line

The number line makes integers easier to read. Positive integers are to the right of $0$, negative integers are to the left, and numbers farther from $0$ have greater distance from zero.

For example, $4$ is four units to the right of $0$, while $-4$ is four units to the left. They have the same distance from zero but opposite directions.

That is why integers are useful for gain and loss, temperature above or below zero, elevation, and horizontal position.

Integer Rules for Addition, Subtraction, Multiplication, and Division

For addition and subtraction, think in terms of movement on the number line:

• Adding a positive integer moves right.
• Adding a negative integer moves left.
• Subtracting an integer means adding its opposite.

Example:

$$5 - 8 = 5 + (-8) = -3$$

For multiplication and division, use the sign rule:

• Same signs give a positive result.
• Different signs give a negative result.

$$(-3)(4) = -12$$ $$(-3)(-4) = 12$$ $$12 \div (-3) = -4$$

One condition matters here. Division by $0$ is undefined, and division by a nonzero integer does not always stay inside the integers. For example,

$$7 \div 2 = 3.5$$

The quotient is a real number, but it is not an integer.

Worked Example: Evaluate $-2 + 7 - 4$

Use the number line idea step by step:

$$-2 + 7 - 4$$

Start at $-2$. Adding $7$ means move $7$ units right:

$$-2 + 7 = 5$$

Now subtract $4$. That means move $4$ units left:

$$5 - 4 = 1$$

So

$$-2 + 7 - 4 = 1$$

This is the main pattern for integer addition and subtraction: read the sign as direction, then track the movement.

Common Mistakes with Integer Operations

Mixing Up Integers and Whole Numbers

Whole numbers include $0, 1, 2, 3, \ldots$, but integers also include the negative versions. So $-5$ is an integer, but it is not a whole number.

Forgetting That Subtraction Changes Direction

In $3 - (-2)$, you are subtracting a negative, which means adding a positive:

$$3 - (-2) = 3 + 2 = 5$$

Assuming Division Always Gives an Integer

Integers are closed under addition, subtraction, and multiplication, but not under division. That means dividing two integers can give a value that is not an integer.

When Integers Are Used

Integers appear in arithmetic, coordinate graphs, algebra, accounting, temperatures, elevations, and computer science. They are often the first number system where direction matters, not just size.

Once integers feel natural on the number line, later topics such as absolute value, inequalities, and algebraic expressions become much easier to read.

Try Your Own Version

Try these on your own number line: $-6 + 9$, $4 - 11$, and $(-5)(-2)$. If you want to check a longer expression after doing it by hand, try your own version in a solver and compare each sign change.