Close, but not equal
Approximations of π are not equal to π
Is it exact?
Zu Chongzhi found two famous fractions for π: 227 and 355113. Many people write π = 227.
Is 227 exactly equal to π?
What this lesson covers
The idea
Since π is irrational there is no best fraction for it, as any fraction close to π can be bettered; so π ≈ 22/7 but π ≠ 22/7, and 355/113 is a much better approximation.
Is it exact?
Zu Chongzhi found two famous fractions for π: 22/7 and 355/113. Many people write π = 22/7.
Is 22/7 exactly equal to π?
- Yes, π is exactly 22/7
- No, it is close but not equal
- It depends on the size of the circle
Hunt for a fraction
Slide the denominator q. For each q the toy picks the best numerator p. Digits that agree with π turn green. Find a fraction that is closer to π than 22/7.
Always a closer fraction
Since π is irrational, there is no best fraction for π: any fraction close to π can be bettered by one that is closer. So π ≈ 22/7 but π ≠ 22/7. The fraction 355/113 is a much better approximation.
22/7 agrees with π only up to 3.14, which is good enough for most practical uses. 355/113 agrees to 6 decimal places, and no fraction with a denominator below 15,000 is as close. Even so, π ≠ 355/113: a closer fraction always exists, for example 103993/33102.
Notes
π is irrational, so no fraction equals π and there is no best fraction. We write π ≈ 22/7 and also π ≠ 22/7. The fraction 355/113 is a much better approximation.
Check yourself
Which statement is correct?
355/113 = 3.14159292… and π = 3.14159265… How many decimal places of 355/113 agree with π?
Answer: 6
Both start 3.141592. In the 7th place 355/113 has 9 but π has 6. So 6 places agree.
Use π ≈ 22/7. A circle has diameter 14 cm. About how long is its circumference?
Answer: 44 cm
C ≈ 22/7 × 14 = 22 × 2 = 44 cm.
Since π is irrational, what can we say about fractions that are close to π?
- π = 22/7. π is irrational, so it cannot be written as a fraction of two integers. 22/7 is only close to π.
- π ≈ 22/7, but π ≠ 22/7 — correct. Yes. They are close but not equal.
- π ≠ 22/7, so 22/7 is no use. 22/7 is a good approximation and is good enough for most practical uses.
- There is a best fraction, but nobody has found it yet. There is no best fraction at all: any fraction can be bettered.
- 355/113 is the best possible fraction. 355/113 is very close, but there is always a fraction even closer.
- For any such fraction there is another fraction even closer to π — correct. Yes. There is no best fraction for π.