Building √n, and π

"How do we draw a line of irrational length?" With a right triangle.

Finding √2 on the number line:

• mark OA = 1;
• at A draw a perpendicular and mark B with AB = 1;
• then OB = √(1² + 1²) = √2;
• with the compass point at O and radius OB, draw an arc to cut the number line at P. Now OP = √2.

Keep going. At B draw a unit perpendicular to OB, ending at C: OC = √(2 + 1) = √3. Then √4 = 2, then √5… A unit step at right angles always turns √n into √(n + 1). The triangles wind round into the square root spiral of Fig. 3.14.

π is irrational too. π is the ratio of a circle's circumference to its diameter. Āryabhaṭa (499 CE) gave 3927/1250 = 3.1416, but called it āsanna, an approximation, as if he knew no exact fraction exists. Lambert proved in 1761 that π is irrational.

Mādhava of Sangamagrama, who began the Kerala School in the 14th century, found an exact formula as an infinite sum: π = 4 × (1 – 1/3 + 1/5 – 1/7 + …). The more terms you add, the closer you get to π.

So the number line holds the rationals and the irrationals together.