Ekanyunena Purvena

Meaning: "By one less than the previous one". This sutra is a powerful and elegant method for specific types of multiplication.

Primary Application: Multiplication by Nines

This sutra is extremely useful and efficient for multiplying any number by a series of nines (e.g., x 9, x 99, x 999, etc.).

The Method: The answer is obtained in two parts:

1. First Part (LHS - Left Hand Side): Subtract 1 from the multiplicand (the number being multiplied by nines). This is the "Ekanyunena" part – "one less than" the original number.
   LHS = Multiplicand - 1.
2. Second Part (RHS - Right Hand Side): Subtract the LHS (obtained in Step 1) from the multiplier (the number consisting of nines).
   Alternatively, and often easier, apply the "Nikhilam Navatashcaramam Dashatah" (All from 9, the last from 10) sutra to the digits of the LHS, ensuring the number of digits in the RHS matches the number of nines in the multiplier. If the LHS has fewer digits than the number of nines, pad it with leading zeros before applying Nikhilam or subtraction from the nines.

Case 1: Number of digits in multiplicand = Number of nines in multiplier (e.g., 46 x 99)

• Multiplicand = 46, Multiplier = 99.
• LHS = 46 - 1 = 45.
• RHS = 99 - 45 = 54. (Or, Nikhilam on 45: (9-4)(9-5) -> 54).
• Answer: 4554.

Case 2: Number of digits in multiplicand < Number of nines in multiplier (e.g., 7 x 99)

• Multiplicand = 7, Multiplier = 99.
• LHS = 7 - 1 = 6. (Treat as 06 for RHS calculation if needed for clarity).
• RHS = 99 - 06 = 93. (Or, Nikhilam on 06 for two digits: (9-0)(9-6) -> 93).
• Answer: 693. (The LHS is just 6, not 06 in the final answer unless it's part of a larger number).

Case 3: Number of digits in multiplicand > Number of nines in multiplier (e.g., 345 x 99) - This requires a slight modification or extension of the basic method, often involving splitting the multiplicand or using general multiplication if simpler.

One way for 345 x 99 (345 x (100-1)): 34500 - 345 = 34155.

Using Ekanyunena variation:
Split 345 based on number of 9s (two 9s): 3 | 45.
LHS part 1: (Digit(s) to left of split + 1) => 3+1 = 4.
345 - 4 = 341 (This forms the initial part of the LHS of the final answer).
For RHS: Nikhilam of the split part (45) from 100 => 100 - 45 = 55.
Answer: 34155. (This variation is more complex and shows limitations of direct application for this case).

Simpler general rule for Case 3 (like 345 x 99):
LHS: (345 - 1) = 344. This is NOT the full LHS yet.
Append as many zeros as there are nines: 34500.
Subtract original number: 34500 - 345 = 34155. (This is essentially (N x 100) - N). Ekanyunena is more direct when digits match or multiplicand is smaller.

Advantages:

• Extremely fast for multiplications by 9, 99, 999, etc., especially when digit counts are favorable.
• Reduces complex multiplication to simple subtraction.