Numbers and sequences
Special numbers
Some numbers have special shapes and names. A square number comes from multiplying an integer by itself, and a cube number comes from multiplying an integer by itself and by itself again.
Square numbers and the ² notation
A square number is what you get when you multiply an integer by itself. You can show them as square patterns of dots: 1, 4, 9, 16, 25, ... We write 'squared' with a small 2. So 6² means 6 x 6 = 36. You can find the square number in any position, for example the ninth square number is 9 x 9 = 81.
Cube numbers and the ³ notation
A cube number is what you get when you multiply an integer by itself and by itself again. You can picture them as cubes of small blocks. We write 'cubed' with a small 3. So 2³ means 2 x 2 x 2 = 8. The cube numbers up to 5 are 1, 8, 27, 64 and 125.
Comparing squares and cubes
To compare something like 2³ and 3², work each one out fully before you decide. 2³ = 2 x 2 x 2 = 8 and 3² = 3 x 3 = 9, so 3² is larger, even though its base number is bigger only in the squaring.
Patterns with special numbers
Special numbers hide neat patterns. Some totals can be made from two square numbers, such as 9 + 36 = 45, or from two cube numbers, such as 125 + 27 = 152. Adding odd numbers in order also makes square numbers: 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16.
Worked examples
Find a square number in a given position
- The ninth square number means the 9th one in the list 1, 4, 9, ...
- Multiply the position by itself: 9 x 9.
Answer: The ninth square number is 81.
Work out a cube number
- 5³ means 5 cubed, which is 5 x 5 x 5.
- Do it in steps: 5 x 5 = 25, then 25 x 5.
Answer: 5³ = 125.
Compare a square and a cube (Worked example 2)
- Work out 2³ = 2 x 2 x 2 = 8.
- Work out 3² = 3 x 3 = 9, then compare 8 and 9.
Answer: 3² is larger because 9 > 8.
Find two square numbers with a given total
- List some square numbers: 1, 4, 9, 16, 25, 36.
- Look for two of them that add to 45; try 9 + 36.
Answer: 9 + 36 = 45.
Where you see this in real life
- Working out how many tiles fit in a square patio that is 6 tiles along each side.
- Counting the small cubes in a Rubik's-style cube that is 3 blocks along each edge.
- Arranging chairs in a square block for a class photo.
Key words
- Integer
- A whole number, with no fraction or decimal part.
- Square number
- A number you get by multiplying an integer by itself, such as 25 = 5 x 5.
- Cube number
- A number you get by multiplying an integer by itself and by itself again, such as 8 = 2 x 2 x 2.
- Squared
- Multiplied by itself; shown with a small 2, like 6².
- Cubed
- Multiplied by itself and by itself again; shown with a small 3, like 2³.