1.02 Add and subtract integers

We have seen that any way we might want to move back or forth along the number line can be expressed as a sum of integers. Subtraction is just a particular kind of addition.

However, the concept of subtraction is still useful; in many cases it can make more sense to take away an integer rather than add its opposite. So it is common to see expressions involving the addition of a negative integer rewritten as the subtraction of a positive integer. This process is called combining adjacent signs.

For a number line with the positive direction to the right, we find the following results.

• Adding a positive integer means we move to the right:3 + (+5) \quad = \quad 8 \quad = \quad 3 + 5
• Adding a negative integer means we move to the left:3 + (-5) \quad = \quad -2 \quad = \quad 3 - 5
• Subtracting a positive integer means we move to the left:3 - (+5) \quad = \quad -2 \quad = \quad 3 - 5
• Subtracting a negative integer means we move to the right:3 - (-5) \quad = \quad 8 \quad = \quad 3 + 5

In an expression like 3 + (-5), the + and - are adjacent signs which are combined into one subtraction operation to give 3 - 5. This is a common way to simplify expressions. However, notice that these adjacent signs have different meanings. The first tells us the operation (either addition or subtraction), while the second tells us the sign of the integer (either positive or negative).

With this in mind, an expression like 3 - (-5) would best be read as "3 subtract negative 5", or "3 take away negative 5", or even "3 minus negative 5". We would like to avoid reading it as "3 minus minus 5", since "minus" is an operation, not the sign of the integer -5.

Examples