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Introduction to integer subtraction

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Introduction to Integer Subtraction

This lesson introduces integer subtraction for students who are new to working with positive and negative numbers. It explains what subtraction means for integers, connects it to the add-the-opposite rule, and shows how to use a number line to visualize the process before moving on to more advanced integer operations.

What Does It Mean to Subtract Integers?

Integers are the set of whole numbers and their opposites: numbers like \(-4, -3, -2, -1, 0, 1, 2, 3, 4\) and so on, with no fractions or decimals. Subtracting integers means finding the difference between two of these numbers, but unlike subtracting whole numbers you already know from arithmetic, integer subtraction can involve negative numbers on either side of the minus sign, and the answer itself can be negative. Before diving into integer subtraction, it helps to be comfortable with comparing and ordering numbers, since knowing which integer is larger or smaller makes it much easier to predict whether an answer will be positive or negative.

The single most useful idea in integer subtraction is this: subtracting a number is the same as adding its opposite. In symbols,

\(a - b = a + (-b)\)

This works no matter what sign \(a\) and \(b\) have. For example, \(5 - 8\) becomes \(5 + (-8)\), which gives \(-3\). And \(6 - (-2)\) becomes \(6 + 2\), which gives \(8\). Notice that subtracting a negative number turns into addition. This one rule lets you turn every subtraction problem into an addition problem, which you likely already practiced with adding integers. Once you rewrite the subtraction as an addition, all the same addition rules apply.

A number line is a great way to see why the add-the-opposite rule works. To subtract a positive number, move to the left. To subtract a negative number, move to the right (since it is the same as adding). The diagram below shows \(5 - 8\): start at \(5\) and move \(8\) units to the left, landing on \(-3\).

−10 −5 0 5 10 −3 move 8 units left start: 5 5 − 8 = −3
Subtracting a positive integer moves you to the left on the number line.

Example 1: \(4 - 9\)

Rewrite as addition: \(4 + (-9) = -5\).

Example 2: \(-3 - 5\)

Rewrite as addition: \(-3 + (-5) = -8\).

Example 3: \(7 - (-2)\)

Rewrite as addition: \(7 + 2 = 9\). Subtracting a negative acts like addition, so the value gets larger.

Example 4: \(-6 - (-10)\)

Rewrite as addition: \(-6 + 10 = 4\).

Many students lose track of the negative signs when two of them appear next to each other, such as in \(-6 - (-10)\). Slow down and rewrite the two signs as a single operation first: two minus signs together become a plus sign. It also helps to double-check your answer's sign using the number line: if you moved left, your result should generally be smaller than where you started, and if you moved right, it should be larger. This introduction covers the basic idea, and the lesson on subtracting integers builds on it with more practice problems and additional strategies for harder cases.

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