Example 1. Adding across zero
Work out −6 + 10.
Solution
- Start at −6 on the number line.
- Adding 10 moves 10 places to the right.
- Six of those places reach zero, and four more go past it.
4
Chapter 02
Integers and negative numbers
Arithmetic starts above zero, where every number counts something you can hold. Pre-algebra carries the same arithmetic below zero, where the numbers count what is owed, how far below sea level, or how many degrees under freezing.
Nothing about the arithmetic changes. What changes is direction, and the rules below are the bookkeeping for it.
Integers and negative numbers
Integers are the whole numbers and their opposites. A negative number is a number below zero, and it obeys the same arithmetic as a positive one once you know which way the number line runs.
- The integers are −3, −2, −1, 0, 1, 2, 3 and so on in both directions. None of them has a fractional part.
- On a number line, negative numbers sit to the left of zero and positive numbers sit to the right.
- Adding a negative number moves left: 5 + (−3) = 2.
- Subtracting a negative number moves right: 5 − (−3) = 8.
- A negative times a positive is negative: (−3) × 4 = −12.
- A negative times a negative is positive: (−3) × (−4) = 12.
- The same two rules hold for division: −12 ÷ 4 = −3, and −12 ÷ −4 = 3.
Subtracting a negative
−7 − (−2)
Taking away −2 is the same as adding 2.
−7 + 2 = −5
Examples
Each one is set out the way a solution is written on paper: the question, the steps in order, then the answer.
Example 2. Two negatives in a product
Work out (−5) × (−6).
Solution
- Multiply the numbers as if both were positive: 5 × 6 = 30.
- Both signs are negative, so the answer is positive.
30
Where this leads
- The adding machine
- It prints a running total on tape, so a string of additions and subtractions below zero can be checked line by line.