Vedic mathematics: mental arithmetic and its limits

The name refers to a collection of calculation techniques popularised by Bharati Krishna Tirtha and the book “Vedic Mathematics”, published in 1965. The validity of an algorithm should not be confused with proof of its ancient origin. These are calculation methods, not an alternative probability theory or evidence that sixteen rules solve every mathematical problem.

98 is 2 below 100. Use (100−2)² = 10,000−400+4 = 9,604. In the shortcut, subtract another 2 from 98 to obtain 96, then append the two-digit value of 2²: 04. Two digits are required because the base is 100. Writing 964 would be incorrect.

For 97 × 96, the deficits from 100 are 3 and 4. The left part is 97−4 = 93, and the right part is 3 × 4 = 12, giving 9,312. This follows from (100−a)(100−b) = 100(100−a−b)+ab. If ab has more than two digits, a carry is required. Simply joining two strings is not always sufficient.

For a positive integer of the form 10a+5, (10a+5)² = 100a(a+1)+25. Thus 35² = 100 × 3 × 4+25 = 1,225, and 75² = 100 × 7 × 8+25 = 5,625. Ordinary multiplication verifies the identity. It requires neither faith in a method’s name nor special properties of the digits.

They may speed up arithmetic but provide no information about a future draw. Changing how multiplication is performed does not change the number of possible selections. Historical frequency, averages and probabilities need the same definitions whether calculated by a person, calculator or program.

Write the calculation and its algebraic justification, then check the answer with a different method. Include an example requiring a carry so that the shortcut’s limitations are understood. Use Pewniaki calculators to check arithmetic, rather than to validate unsupported stories about its origins.