The square root of 2

Doubling a square

Around 800 BCE, Baudhayana asked how to build a square twice as big as a given square. Doubling the side fails, because it gives four times the area. His answer (Shulbasutra, verse 1.9) is to build the new square on the diagonal of the old one. Lines drawn east-west and north-south show why: the old square is made of 2 identical right triangles and the new one of 4. Repeating the step gives squares made of 2, 4 and 8 triangles, each doubling the last.

Halving works the same way in reverse. Fold the four corners of a square sheet to its centre: the creases join the midpoints of the sides and form a tilted square whose area is exactly half the sheet, because the four folded flaps cover it completely. A square with half the side would have only a quarter of the area.

The hypotenuse of a 1-1 right triangle

Cut a unit square PEAR along its diagonal ER and you get a right triangle whose two shorter sides are 1. The square REST on its hypotenuse is the doubled square, so its area is 2 × 1 = 2. If the hypotenuse has length c, then c² = 2, and c = √2, the number whose square is 2. For equal sides of any length a, the same argument gives c² = 2a², so c = a√2. Equal sides of 12 give c = √288, between 16 and 17; a hypotenuse of √72 gives sides of 6.

How big is √2?

Squares can trap √2 between two numbers. Since 1² = 1 and 2² = 4, √2 lies between 1 and 2; these are called a lower and an upper bound. Next, 1.4² = 1.96 and 1.5² = 2.25, so 1.4 < √2 < 1.5. Then 1.41² = 1.9881 and 1.42² = 2.0164, so 1.41 < √2 < 1.42. Then 1.414² = 1.999396 and 1.415² = 2.002225, so 1.414 < √2 < 1.415. Each step adds one more correct digit, and the process can go on as long as we like.

A decimal that never ends

Will the digits ever stop? Suppose √2 were a terminating decimal. Its last digit would be non-zero, and the square of such a number also ends in a non-zero digit after the decimal point, so it could never equal 2 or 2.000 exactly. The decimal of √2 must therefore go on for ever: √2 = 1.41421356…

Not a fraction either

Could √2 be a fraction m/n of counting numbers? Then 2 = m²/n², so 2n² = m². In the prime factorisation of a square, each prime appears an even number of times. So m² would contain the prime 2 an even number of times, while 2n² contains it an odd number of times, which is impossible. This proof was given by Euclid in his Elements, around 300 BCE. Numbers like √2 are studied more in higher classes. Continue with the theorem and Pythagorean triples, or see the Chapter 2 notes and NCERT solutions.