Chasing π

The idea

π, the ratio of a circle's circumference to its diameter, is irrational, so no single fraction can give it exactly; fractional values of π are only approximations.

A fraction for π?

π is the ratio of a circle's circumference to its diameter. For centuries, mathematicians looked for a fraction that equals it.

Is there a fraction that is exactly equal to π?

• Yes, such as 22/7
• No fraction is exactly π
• Not sure

Compare with π

Tap each fraction. Its digits turn green while they match π and red at the first digit that is wrong.

Only approximations

π, the ratio of a circle's circumference to its diameter, is irrational, so no single fraction can give it exactly. Fractional values of π are only approximations.

Āryabhaṭa (499 CE) gave the accurate value 3927/1250 = 3.1416, but called it an *asanna* (approximation). Lambert proved in 1761 that π is irrational.

Notes

π is irrational, so no single fraction gives it exactly. Fractional values of π are only approximations.

Check yourself

Āryabhaṭa called 3927/1250 an asanna. What does that word mean?

Which statement about π is true?

The fraction 3927/1250 gives 3.1416, and π is 3.14159265… After the decimal point, how many digits do they agree on?

Answer: 3

3.1416 and 3.1415…: they agree on 1, 4, 1 and differ at the fourth digit (6 against 5).

π is defined as the ratio of …

• The exact value of π. He said the opposite: it is only close to π.
• A circle. It means an approximation.
• An approximation — correct. Yes. He knew that a fraction could only come close to π.
• No fraction is exactly equal to π — correct. Yes. π is irrational.
• 22/7 is exactly π. 22/7 = 3.142857…, while π = 3.141592…
• π = 3.14 exactly. 3.14 stops, but π does not. 3.14 is only an approximation.
• a circle's area to its radius. π is the ratio of the circumference to the diameter.
• a circle's circumference to its diameter — correct. Yes. Even so, π is not a ratio of two integers.
• a square's diagonal to its side. That ratio is √2.