Three approximations, each with a measured error
Show on a big screenNo fraction and no ending decimal equals π. So how close can we get?
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- 22/7 stands apart. 3.1416 and 355/113 look merged with π.
- Zoom in. 3.1416 separates. 355/113 still hides beside π.
- Zoom in again. 355/113 separates too. It is the closest of the three.
- Each zoom shows a smaller, real miss.
The rule
Very close. One fraction, 355/113, misses π by about 0.00000027.
Every fraction is an approximation of π. So is every decimal that ends. Each miss can be measured, not guessed.
An approximation is a close value, not the exact one. Its miss is how far it is from π.
Three people are named for this work. Archimedes showed that π lies between 223/71 and 22/7. 223/71 is a little too small. 22/7 is a little too big.
Aryabhata gave 3.1416. He called it asanna, which means "approaching". Zu Chongzhi gave 22/7 and also 355/113.
| Value | Its decimal | How far too big |
|---|---|---|
| 22/7 | 3.142857… | about 0.00126 |
| 3.1416 | 3.1416 | about 0.0000073 |
| 355/113 | 3.14159292… | about 0.00000027 |
All three are bigger than π. Each is closer than the one before.
Matching to 2 places means both round to the same 2-place number.
22/7 matches π to 2 decimal places (3.14). 3.1416 matches to 4. 355/113 matches to 6 (3.141593).
Try the lab to finish this lesson.