Numbers and sequences
Square & triangular numbers
Some numbers make a perfect square of dots. Others stack into a triangle. Both patterns hide a neat piece of arithmetic.
Square numbers
A SQUARE NUMBER is what you get when you multiply a whole number by itself. 1 × 1 = 1 2 × 2 = 4 3 × 3 = 9 4 × 4 = 16 So the square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, … They are called square numbers because that many dots make a perfect square: 9 dots make 3 rows of 3.
Writing them with an index
You can write a square number with a small raised 2: 1² = 1, 2² = 4, 3² = 9, 4² = 16 Read 3² as 'three squared'. It means 3 × 3, NOT 3 × 2 — a very common slip.
A hidden pattern in the odd numbers
Look at how each square grows from the one before: 1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 Adding the odd numbers in order gives the square numbers every time. Each new square adds one more row and one more column, plus the corner — which is always the next odd number.
Triangular numbers
A TRIANGULAR NUMBER makes a triangle of dots, with rows of 1, 2, 3, 4 and so on. 1 1 + 2 = 3 1 + 2 + 3 = 6 1 + 2 + 3 + 4 = 10 So the triangular numbers are 1, 3, 6, 10, 15, 21, 28, … To get the next one, add the next counting number. From 10, add 5 to get 15.
Worked examples
Drawing the next square number
- The pattern so far is 1, 4, 9, 16.
- Each term adds a row with two more dots than the last.
- The 5th square number is 5 × 5.
Answer: The 5th square number is 25
The 10th square number
- The 10th square number is 10 multiplied by itself.
- 10 × 10 = 100.
Answer: The 10th square number is 100
The next two triangular numbers
- The pattern starts 1, 3, 6, 10.
- To get 10 you added 4, so next add 5: 10 + 5 = 15.
- Then add 6: 15 + 6 = 21.
Answer: The next two are 15 and 21
Adding odd numbers
- Add the first four odd numbers: 1 + 3 + 5 + 7.
- That comes to 16.
- 16 is 4 × 4.
Answer: The first four odd numbers add to the 4th square number, 16
Is 30 a square number?
- Look for a whole number multiplied by itself.
- 5 × 5 = 25 and 6 × 6 = 36.
- 30 falls between them, so nothing gives exactly 30.
Answer: No — 30 is not a square number
Where you see this in real life
- A square patio of 49 tiles must be 7 tiles along each side.
- Bowling pins stand in a triangle of 1 + 2 + 3 + 4 = 10 — a triangular number.
- Seats arranged in a square block let you find the total by multiplying one side by itself.
Key words
- Square number
- The result of multiplying a whole number by itself.
- Triangular number
- A number that makes a triangle of dots, found by adding 1 + 2 + 3 and so on.
- Index
- The small raised number showing a power, as in 3².
- Squared
- Multiplied by itself.
- Term
- One number in a sequence.
- Consecutive
- Following one after another.