Yavadunam

Meaning: "By the deficiency" or "Whatever the extent of its deficiency". This is a specific case of the Nikhilam sutra, primarily applied to squaring numbers.

Primary Application: Squaring Numbers Near a Base

This sutra is exceptionally useful for squaring numbers that are close to a power of 10 (like 10, 100, 1000, etc.).

How it works (for a number N near a base B):

1. Find the Deficiency (or Surplus): Calculate the difference between the number (N) and the chosen base (B). Let this be 'd'.
   If N < B, then d = B - N (deficiency). The working value is -d.
   If N > B, then d = N - B (surplus). The working value is +d.
2. Calculate the First Part (LHS) of the Answer: Subtract the deficiency from the number (or add the surplus to the number). This is equivalent to Number ± deficiency/surplus from base.
   LHS = N - d (if N < B, so N - (B-N) which is wrong, it's N + (-d)) or more simply: N + (N-B). So, for 98 (base 100, deficiency 2), LHS = 98 - 2 = 96. For 103 (base 100, surplus 3), LHS = 103 + 3 = 106.
3. Calculate the Second Part (RHS) of the Answer: Square the deficiency (or surplus).
   RHS = d².
4. Adjust and Combine: The number of digits in the RHS should be equal to the number of zeros in the base. If d² has fewer digits, pad with leading zeros. If it has more digits, carry over the excess to the LHS.

Example 1 (Number below base): Square 98

• Base = 100. Number = 98.
• Deficiency (d) = 100 - 98 = 2.
• LHS = Number - Deficiency = 98 - 2 = 96.
• RHS = Deficiency² = 2² = 4. Since base 100 has two zeros, RHS should be 04.
• Answer: 9604.

Example 2 (Number above base): Square 103

• Base = 100. Number = 103.
• Surplus (d) = 103 - 100 = 3.
• LHS = Number + Surplus = 103 + 3 = 106.
• RHS = Surplus² = 3² = 9. Since base 100 has two zeros, RHS should be 09.
• Answer: 10609.

Why it works: For a number N = B-d, N² = (B-d)² = B² - 2Bd + d². This can be written as B(B-2d) + d². The sutra simplifies this to (N-d) | d². For N = B+d, N² = (B+d)² = B² + 2Bd + d² = B(B+2d) + d². The sutra simplifies this to (N+d) | d².