Reference · module 1
Working with numbers
Everything this module explains, on one page.
Order of operations
Work out any expression in the right order, without a calculator.
- An expression is worked out in a fixed order: brackets first, then multiplication and division, then addition and subtraction. The order is a convention, and everyone who reads 2 + 3 × 4 must get the same number.
- So 2 + 3 × 4 is 2 + 12, which is 14 — not 5 × 4. Multiplication binds tighter than addition, and brackets are how you ask for the other order: (2 + 3) × 4 is 20.
- Operations of the same rank go left to right. 20 − 5 − 3 is 15 − 3, which is 12, and 24 ÷ 4 × 2 is 6 × 2, which is 12. Starting from the right gives a different, wrong answer.
The mistake you are about to make
2 + 3 × 4 = 20
2 + 3 × 4 = 14
Reading left to right is how we read words, so the sum seems to come first. Multiplication always goes before addition, unless brackets say otherwise.
Negative numbers
Add, subtract and multiply numbers below zero without guessing the sign.
- Negative numbers sit left of zero on the number line. −5 is smaller than −2, because it is further left, even though 5 is bigger than 2. A temperature of −10 is colder than −3.
- Adding a negative number moves you left: 3 + (−5) is −2. Subtracting a negative moves you right, because taking away a debt is a gain: 3 − (−5) is 3 + 5, which is 8.
- For multiplication and division the rule is short: same signs give a positive result, different signs give a negative one. So −4 × −3 is 12, and −4 × 3 is −12.
The mistake you are about to make
3 − (−5) = −2
3 − (−5) = 8
Two minus signs look like more subtraction, so the answer seems to shrink. Subtracting a negative is adding: the two minuses together make a plus.
Factors and primes
Tell at a glance what a number divides by, and spot prime numbers.
- Quick tests save long division. A number divides by 2 if its last digit is even, by 5 if it ends in 0 or 5, by 10 if it ends in 0, and by 3 if the sum of its digits divides by 3.
- A factor of a number divides it with nothing left over. The factors of 12 are 1, 2, 3, 4, 6 and 12. They come in pairs that multiply to the number: 1 × 12, 2 × 6, 3 × 4.
- A prime number has exactly two factors: 1 and itself. 2, 3, 5, 7, 11 and 13 are prime. 1 is not prime, because it has only one factor, and 2 is the only even prime.
The mistake you are about to make
1 is a prime number
2 is the smallest prime number
1 divides only by 1 and itself, so it looks like the purest prime. A prime needs exactly two different factors, and 1 has just one.