Vedic Math: 10 Ancient Indian Techniques That Still Work

In 1965, an Indian mathematician named Bharati Krishna Tirthaji published a book claiming to have rediscovered 16 mathematical sutras (formulas) from the ancient Vedic texts. Whether the historical provenance is precise or not, the techniques themselves are undeniably powerful — they turn complex arithmetic into patterns that can be solved in seconds.

Here are 10 Vedic math techniques that your student can learn and use today. Each one is a genuine shortcut, not a party trick.

1. Nikhilam — “All From 9, Last From 10”

This sutra gives you a lightning-fast way to multiply numbers close to a power of 10 (like 100, 1000, etc.).

Example: 97 × 94

1. Both numbers are near 100. Find their deficits: 100 − 97 = 3, 100 − 94 = 6.
2. Left part: Subtract either deficit from the other number: 97 − 6 = 91 (or 94 − 3 = 91).
3. Right part: Multiply the deficits: 3 × 6 = 18.
4. Answer: 9,118.

More examples:

• 96 × 93 → 96 − 7 = 89, 4 × 7 = 28 → 8,928
• 988 × 995 → 988 − 5 = 983, 12 × 5 = 060 → 983,060

For numbers above the base (like 103 × 107), the process is the same but you add instead of subtract: 103 + 7 = 110, 3 × 7 = 21 → 11,021.

2. Urdhva-Tiryak — “Vertically and Crosswise”

This is a general multiplication method that works for any two numbers. For two-digit numbers, it's faster than the standard algorithm.

Example: 23 × 14

1. Right column (vertical): 3 × 4 = 12. Write 2, carry 1.
2. Middle (crosswise): (2 × 4) + (3 × 1) = 8 + 3 = 11. Add the carry: 12. Write 2, carry 1.
3. Left column (vertical): 2 × 1 = 2. Add the carry: 3.
4. Read: 322.

The beauty is that this extends naturally to 3-digit, 4-digit, and larger numbers. You just add more crosswise steps.

3. Ekadhikena Purvena — “By One More Than the Previous One”

This sutra provides an instant way to square any number ending in 5.

Rule: Take the digit(s) before the 5, multiply by one more than itself, then append 25.

• 35² → 3 × 4 = 12, append 25 → 1,225
• 75² → 7 × 8 = 56, append 25 → 5,625
• 125² → 12 × 13 = 156, append 25 → 15,625
• 205² → 20 × 21 = 420, append 25 → 42,025

Why it works: Any number ending in 5 can be written as (10n + 5). Squaring gives 100n(n+1) + 25, which is exactly n × (n+1) followed by 25.

4. Paravartya Yojayet — “Transpose and Apply”

This technique simplifies division when the divisor is close to a power of 10. Instead of long division, you transpose the divisor's complement and use it as a multiplier.

Example: 1,352 ÷ 12

1. 12 is near 10. The complement is −2 (since 12 − 10 = 2, but we negate it).
2. Set up like synthetic division, using −2 as the multiplier.
3. Bring down 1. Multiply by −2: −2. Add to 3: get 1. Multiply by −2: −2. Add to 5: get 3. Multiply by −2: −6. Add to 2: get −4.
4. Result: quotient 112, remainder 8 (after adjusting the −4).

This is essentially the same as synthetic division, which itself is derived from this Vedic approach.

5. Shunyam Saamyasamuccaye — “If the Sum Is the Same, That Sum Is Zero”

This sutra helps solve certain equations instantly by spotting patterns.

Example: (x + 3)(x + 5) = (x + 3)(x + 7)

Both sides share the factor (x + 3). The equation is only true when that common factor equals zero: x + 3 = 0, so x = −3. No need to expand and simplify.

Another pattern: If the sum of the numerator and denominator is the same on both sides of an equation, that sum equals zero.

6. Anurupyena — “Proportionately”

Scale numbers to make calculations easier, then adjust proportionally.

• 48 × 5 → think of it as 24 × 10 = 240 (halved one, doubled the other)
• 35 × 16 → 70 × 8 = 560 (doubled one, halved the other)
• 125 × 32 → 250 × 16 → 500 × 8 = 4,000

The principle is simple: if you double one factor and halve the other, the product stays the same. Keep going until one factor becomes easy to work with.

7. Ekanyunena Purvena — “By One Less Than the Previous One”

This gives an instant formula for multiplying by a string of 9s.

Rule: The left part of the answer is one less than the multiplicand. The right part is the complement (9s minus the left part).

• 7 × 9 → left = 7 − 1 = 6, right = 9 − 6 = 3 → 63
• 7 × 99 → left = 7 − 1 = 6, right = 99 − 6 = 93 → 693
• 23 × 99 → left = 23 − 1 = 22, right = 99 − 22 = 77 → 2,277
• 45 × 999 → left = 44, right = 999 − 44 = 955 → 44,955

8. Sankalana-Vyavakalanabhyam — “By Addition and Subtraction”

This sutra solves simultaneous equations by simply adding or subtracting them.

Example:

• x + y = 10
• x − y = 4

Add both equations: 2x = 14, so x = 7. Subtract: 2y = 6, so y = 3.

While this seems obvious, the Vedic approach encourages students to always look for this direct path before reaching for more complex methods like substitution or matrices.

9. Puranapuranabhyam — “By Completion”

Complete an expression to a convenient form, then adjust.

Example: Solve x² + 6x = 16.

1. To “complete the square,” add (6/2)² = 9 to both sides.
2. x² + 6x + 9 = 25 → (x + 3)² = 25 → x + 3 = ±5.
3. x = 2 or x = −8.

This is the Vedic root of “completing the square” — a technique that's typically taught as a purely algebraic maneuver but was originally understood geometrically. You're literally completing a geometric square by adding a small corner piece.

10. Yavadunam — “Whatever the Extent of Its Deficiency”

This gives a fast way to square any number near a round base.

Example: 97²

1. Base is 100. Deficit = 3.
2. Left part: 97 − 3 = 94 (subtract the deficit from the number).
3. Right part: 3² = 09 (square the deficit, pad to 2 digits since base is 100).
4. Answer: 9,409.

More examples:

• 103² → 103 + 3 = 106, 3² = 09 → 10,609
• 992² → 992 − 8 = 984, 8² = 064 → 984,064
• 48² → base 50. 48 − 2 = 46 → 46 × 50/50... actually, use base 50: 25 − 2 = 23, 2² = 04 → 2,304

Should You Teach Vedic Math?

Vedic math isn't a replacement for understanding — it's a complement to it. These techniques work best when a student already understands the fundamentals of multiplication and algebra. Then, the Vedic shortcuts become genuine time-savers, not just memorized procedures.

The real value of Vedic math is that it teaches students to look for patterns. A student who learns Nikhilam starts asking: “Is this number close to a round number? Can I exploit that?” That kind of mathematical thinking — spotting structure and using it — is far more valuable than any individual trick.

This is Part 4 of our “Math Tricks Your Textbook Never Taught You” series. Next up: Why Asian Students Excel at Math: The Methods Behind the Results.