Types of numbers and place value

An integer is a whole number: positive, negative or zero, such as −4, 0 and 17.

• A factor of a number divides into it exactly: the factors of 12 are 1, 2, 3, 4, 6 and 12.
• A multiple is in the number's times table: the multiples of 12 are 12, 24, 36, …
• A prime number has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, …

To find all the factors of a number, write its factor pairs, trying 1, 2, 3, … in turn, until the pairs repeat.

Example: find the factors of 36.

1. 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6
2. Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36

• Square numbers come from multiplying a number by itself: 1, 4, 9, 16, 25, … (5² = 25).
• Cube numbers come from n × n × n: 1, 8, 27, 64, 125, … (4³ = 64).
• A square root undoes squaring: √49 = 7 (and −7 is also a square root of 49).
• A cube root undoes cubing: ∛125 = 5.
• The reciprocal of a number is 1 ÷ the number. For a fraction, turn it upside down: 2/3 becomes 3/2.

Learn the squares up to 15² = 225 and the cubes up to 5³ = 125, and 10³ = 1000.

Example: work out √144 + ∛64, then find the reciprocal of 0.2.

1. √144 = 12 and ∛64 = 4
2. √144 + ∛64 = 12 + 4 = 16
3. Reciprocal of 0.2 = 1 ÷ 0.2 = 5

Place value is what a digit is worth because of its column. The columns, from left to right, are millions, hundred thousands, ten thousands, thousands, hundreds, tens and ones, then, after the decimal point, tenths, hundredths and thousandths.

In 4 506 312.087, the 4 is worth 4 000 000, the 5 is worth 500 000 and the 7 is worth 7 thousandths, 0.007.

Multiplying by 10, 100 or 1000 moves every digit 1, 2 or 3 columns to the left; dividing moves them to the right.

Example: what is the 8 worth in 4 506 312.087? Then work out 3.75 × 100.

1. The 8 is in the second column after the point: the hundredths.
2. It is worth 0.08.
3. Each digit moves two places left: 3.75 × 100 = 375

To put numbers in order:

1. Write fractions and percentages as decimals.
2. Give every decimal the same number of decimal places, adding zeros.
3. Compare from the left, digit by digit.

On a number line, numbers get bigger to the right, so −7 is smaller than −2.

Example: write 0.6, 5/8, 58% and 0.65 in order, smallest first.

1. 0.6 = 0.600
2. 5/8 = 5 ÷ 8 = 0.625
3. 58% = 0.580
4. 0.65 = 0.650
5. Order: 58%, 0.6, 5/8, 0.65

Harder example: order 3, −5, 0, −1.5 and −8, smallest first.

1. The negatives come first; the one furthest from 0 is smallest.
2. Order: −8, −5, −1.5, 0, 3