Vedic Maths Sutra 14 : Ekanyūnena Pūrveṇa
Sūtra 14 of 16
एकन्यूनेन पूर्वेण
Ekanyūnena Pūrveṇa
“By one less than the previous one”
Meaning & Essence
This Sūtra means “By one less than the previous one”.
It is the natural counterpart of Ekādhikena Pūrveṇa and is especially powerful for multiplication by numbers consisting entirely of 9s (9, 99, 999, 9999…).
Mathematical Principle
When multiplying a number \( N \) by a number consisting of \( k \) nines:
\[
N \times (10^k - 1) = (N - 1)\ ||\ (10^k - N)
\]
In practice:
- Left part = Multiplicand − 1
- Right part = the all-9s number − Left part
(or simply the complement of the original multiplicand to the next power of 10)
Worked Examples
Example 1 → \( 7 \times 9 \)
- Left part: \( 7 - 1 = 6 \)
- Right part: \( 9 - 6 = 3 \)
- Answer: 63
Answer: 63
Example 2 → \( 15 \times 99 \)
- Left part: \( 15 - 1 = 14 \)
- Right part: \( 99 - 14 = 85 \)
- Answer: 1485
Answer: 1485
Example 3 → \( 67 \times 99 \)
- Left part: \( 67 - 1 = 66 \)
- Right part: \( 99 - 66 = 33 \)
- Answer: 6633
Answer: 6633
Example 4 → \( 2463 \times 9999 \)
- Left part: \( 2463 - 1 = 2462 \)
- Right part: \( 9999 - 2462 = 7537 \)
- Answer: 24627537
Answer: 24627537
Interactive Demo – Multiply by all 9s
Enter a number and choose how many 9s to multiply by (1–6 recommended).
Please enter a valid number and number of 9s (1–8)
When to Use This Sūtra
• Multiplying any number by 9, 99, 999, 9999, etc.
• Especially fast when the multiplicand has the same number of digits as the string of 9s
• Mental calculation of products involving all-9s multipliers
• Useful complement to the Nikhilam and Ekādhikena methods