That number is not a fraction
Show on a big screenIs π just another fraction, like 22/7?
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- 1/4 as a decimal: 0.25. It ends.
- 1/3 as a decimal: 0.333… The remainder 1 keeps coming back.
- π as a decimal: 3.14159265… No block ever repeats.
- No fraction behaves like that. So no fraction equals π.
The rule
No. No fraction is equal to π.
A fraction's decimal always ends or repeats. π's decimal does neither.
Irrational — a number that is not a fraction of two whole numbers.
Long division on a fraction can only do two things. It can stop. Or it meets a remainder it met before. Then the digits repeat.
Take 22/7. Dividing by 7 leaves a remainder from 0 to 6. So within 7 steps the division stops or a remainder comes back.
In the lab, tap Next digit. The list of remainders grows. It stops when a remainder comes back. Pick π too. It has no division, so it has no remainders.
The digits of π go on for ever, with no repeating block. No fraction can do that. So π is . The proof needs later maths.
| Value | Its decimal | Ends or repeats? |
|---|---|---|
| 22/7 | 3.142857142857… | Repeats: the block 142857 |
| π | 3.14159265358… | Neither. No block ever comes back |
22/7 is close to π, but its decimal repeats. So 22/7 and π are two different numbers.
√2 is the positive number that squares to 2. √2 and 1.414 are two different numbers. 1.414 × 1.414 = 1.999396, not 2.
A fraction is only a close value for π. So is a decimal that ends. 22/7, 3.14 and 3.1416 are used because they are close enough.
Try the lab to finish this lesson.