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 include no fractions or decimals. Two ideas carry almost every integer problem: on the number line the sign tells direction from $0$ while the size tells distance from $0$, and for multiplication and division same signs give a positive result while different signs give a negative one.

When To Use The Number-Line Method

Reach for the number line whenever you are adding or subtracting integers, because it turns signs into movement. Positive integers sit to the right of $0$, negative integers to the left, and numbers farther from $0$ have greater distance from zero. For example, $4$ is four units right of $0$ while $-4$ is four units left: same distance from zero, opposite direction. That is exactly why integers model gain and loss, temperature above or below zero, elevation, and horizontal position.

The Procedure For Each Operation

Addition and subtraction — read the sign as movement:

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

Multiplication and division — apply the sign rule, then handle the size:

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

One condition rides along: division by $0$ is undefined, and division by a nonzero integer does not always stay inside the integers. For example,

is a real number but not an integer.

The Whole Procedure On One Example

Evaluate $-2 + 7 - 4$ by tracking movement:

Start at $-2$. Adding $7$ moves $7$ units right:

Now subtract $4$, moving $4$ units left:

So

Read each sign as direction, then track the movement, and the whole expression resolves one step at a time.

Where Each Step Trips People Up, And How To Check

• Confusing integers with whole numbers. Whole numbers are $0, 1, 2, 3, \ldots$, but integers also include the negatives, so $-5$ is an integer but not a whole number.
• Forgetting that subtraction changes direction. In $3 - (-2)$ you subtract a negative, which adds a positive:

• Assuming division always lands on an integer. Integers are closed under addition, subtraction, and multiplication, but not under division, so two integers can divide to a non-integer. Self-check: after dividing, ask whether the quotient is really a whole value.

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, and once they feel natural on the number line, later topics such as absolute value, inequalities, and algebraic expressions read much more easily.

For a quick run-through, work $-6 + 9$, $4 - 11$, and $(-5)(-2)$ on your own number line, narrating the direction of each sign change before you write the answer.