π, a New Number പുതിയൊരു സംഖ്യ
Every circumference is a fixed number of times the diameter. That number – the circumference of the circle of diameter 1 – is so special that it has its own symbol, π. Here we meet it, use it, and follow 4000 years of people (including a Kerala mathematician) trying to pin it down.
- Circumference changes in the same proportion as diameter (page 5).
- Some lengths are not fractions: the diagonal of the unit square is √2, the height of the equilateral triangle of side 2 is √3 (chapter പുതിയ സംഖ്യകൾ).
1. The circle of diameter 1
Roll a circle of diameter 1 along a straight line for exactly one turn. The distance it covers is its circumference.
Like √2 and √3, this length cannot be written as a fraction – proved only in the 18th century. It is even stranger: unlike √2, √3 or √2 + √3, it cannot be written using roots of natural numbers or fractions at all. Mathematicians gave it a symbol: π, the Greek letter "pi" (പൈ). Since it is not a fraction we can only find fractions close to it. To four decimal places:
π ≈ 3.14162. Circumference = π × diameter
Diameter 1 cm ⇒ circumference π cm. Diameter 2 cm ⇒ 2π cm. Diameter 1½ cm ⇒ 32π cm.
വൃത്തത്തിന്റെ ചുറ്റളവ്, അതിന്റെ വ്യാസത്തിന്റെ π മടങ്ങാണ്.
circumference = π × diameter c = πdവൃത്തത്തിന്റെ ചുറ്റളവ്, അതിന്റെ ആരത്തിന്റെ 2π മടങ്ങാണ്.
circumference = 2π × radius c = 2πrWe usually draw circles with a given radius (ആരം), so this form is the one used most.
3. The wire problem, solved
Answers can be left in terms of π (5π cm, 10π m). Use a decimal only when a measurement is needed. For most practical work four decimals of π are plenty: 31.416 m is correct to the millimetre.
4. π through history ചരിത്രത്തിലൂടെ π
Click an approximation. The red mark shows where it sits on a magnified number line around the true π (green).
Archimedes' 96-sided polygons
Archimedes (Greece, 3rd century BC) used polygons with 96 sides inside and outside a circle of diameter 1 and proved 3 10/71 < π < 3 1/7, that is 3.1408 < π < 3.1428. His 3 1/7 = 22/7 was used for centuries.
π in Kerala π കേരളത്തിൽ
Madhavan, an astronomer and mathematician of Kerala, found a way to compute π to any accuracy using numbers alone, without geometry. One of his discoveries: the numbers 1, 1 − 13, 1 − 13 + 15, 1 − 13 + 15 − 17, … come closer and closer to π4. Gregory (Scotland) and Leibniz (Germany) found the same in the 17th century. This series is slow – about 4000 terms for Archimedes' accuracy – so Madhavan used another, faster method and got π ≈ 3.14159265359.
A bicycle wheel of diameter 70 cm travels π × 70 ≈ 220 cm in one turn, so it turns about 455 times in a kilometre. The odometer in a car or an autorickshaw counts wheel turns and multiplies by πd.
5. Check yourself
- π is the circumference of the circle of diameter 1; it is not a fraction (nor a root expression). π ≈ 3.1416.
- c = πd = 2πr. Diameter 5 cm ⇒ 5π ≈ 15.7 cm; radius 5 m ⇒ 10π ≈ 31.416 m.
- Archimedes: 3 10/71 < π < 3 1/7 using 96-gons. Madhavan (Kerala, 14th century): series and π ≈ 3.14159265359.