The story of π

The C/D ratio is called π ('pie'). Measure it at home: wrap a thin thread 20 times round a reel of diameter D. If the thread is L long, L/(20D) comes out close to π, between 3.1 and 3.2.

Trapping π with shapes. A regular hexagon inside a circle of radius 1 has perimeter 6, and the circle is longer, so π > 3. Mesopotamian mathematicians (c. 1900 BCE) used 3 1/8 = 3.125.

Archimedes (c. 250 BCE) squeezed π between polygons inside and outside the circle. Hexagons give 3 < π < 2√3 ≈ 3.46; 96-sided polygons give 3 10/71 < π < 3 1/7.

Better and better values:

• Ptolemy: 377/120 ≈ 3.14167;
• Zu Chongzhi (480 CE): 22/7, and 355/113 ≈ 3.1415929, unbeaten for over 800 years;
• Āryabhaṭa (499 CE): 62832/20000 = 3.1416, which he called āsanna, 'approximate';
• Brahmagupta (628 CE): √10 ≈ 3.1622, chosen because it was easy to work with.

Mādhava of Sangamagrāma found the first exact formula, an infinite series: π/4 = 1 – 1/3 + 1/5 – 1/7 + …. With series he got π to 11 decimal places.

π is irrational (Lambert, 1761): no fraction equals it. So π ≈ 22/7, but π ≠ 22/7. William Jones first used the letter π in 1706, from the Greek perimetros; Euler made it popular.