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The World of Numbers

A spiral built one right triangle at a time

One triangle gave a length that squares to 2. What length does the next triangle give?

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Two legs of 1 at a right angle. The hypotenuse is root 2.
Step 1 of 3
  1. Two legs of 1 at a right angle. The hypotenuse is root 2.
  2. A new leg of 1 on that hypotenuse. The new hypotenuse is root 3.
  3. Another new leg of 1. The new hypotenuse is root 4, which is exactly 2.

How each stage is built

The next length squares to 3. Here is how each stage is built.

Start with two legs of 1 unit, at a right angle. The hypotenuse, the longest side, squares to 1 + 1 = 2. To square means to multiply a number by itself. Squares on the legs add to the square on the longest side.

Use that hypotenuse as one leg of a new right triangle. The other leg is a new 1. The new hypotenuse squares to 2 + 1² = 3.

Repeat this. Each new leg adds 1² = 1. So the square of the hypotenuse goes up by 1 each time.

Four stages of the spiral and the square of each new hypotenuse
StageSquare of the hypotenuseThe hypotenuse
12root 2
23root 3
34exactly 2
45root 5

Jump to lesson

This track

  1. 0 Start here
  2. 1 Every number so far is one integer over another
  3. 2 There is always another one in the gap
  4. 3 Which fractions stop, and what decides it
  5. 4 A block that repeats is the number written out
  6. 5 Turning a repeating decimal back into a fraction
  7. 6 Checkpoint: fractions and the decimals they make
  8. 7 A length you can draw and cannot write
  9. 8 The argument that no fraction can do it
  10. 9 A spiral built one right triangle at a time
  11. 10 Squeezing it between two fractions
  12. 11 Checkpoint: the numbers that are not fractions
  13. 12 One number can have two decimal names
  14. 13 A block that turns round: 142857
  15. 14 The decimal tells you if a number is rational
  16. 15 The big picture
  17. 16 Track quiz

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