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निखिलं नवतश्चरमं दशतः

Nikhilaṃ Navataścaramaṃ Daśataḥ

"All from 9 and the last from 10"

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What this sutra solves

Use this when both numbers sit just below or above a round base like 100 or 1000.

A warehouse ships 97 pallets, each carrying 96 cartons, and you want the carton count without a calculator.

becomes97×9697 \times 96
⚡

Deviations −3 and −4 from 100: 93 | 12 → 9312.

A print run of 103 booklets at 104 pages each — total pages?

becomes103×104103 \times 104
⚡

Surplus +3 and +4 above 100: 107 | 12 → 10712.

Live Demo
Working Base
10
near 10
9
near 10
8
1

Choose the base

Both numbers are below base 10. Our working base is 10.

1 / 6

⚡ Speed Advantage

Vedic
3 steps
Traditional
12 steps

4× faster with Vedic Mathematics

Best for

  • • Multiplying 2–4 digit numbers near powers of 10

Use when

  • • Both numbers within 10–15% of a power of 10 (10, 100, 1000)

Avoid when

  • • Numbers far from any base (use Urdhva-Tiryakbyham)

Intuition

Find how far each number is from the nearest power of 10. Those distances do the multiplication for you.

Story Mode

The Complement Trick

Every number near 100 tells you two things: what it is, and how far it falls short. 94 is 6 short; 97 is 3 short. These two 'distances' interact in a beautiful way — their product fills the right side, while one number borrows the other's distance on the left. The ancient name 'all from 9 and last from 10' is the recipe for finding those distances from 99…9 (all 9s) and 100.

Vedic vs conventional

Conventional

97×9497\times 94 via long multiplication (12 partial products).

Vedic

deficiencies 3 and 6, left=97−6=9197-6=91, right=3×6=18→91183\times 6=18 \to 9118 (3 steps).

3 steps vs 12+ steps for multiplying numbers near a base.

Applications

Both numbers below base

Both numbers are less than the base (100, 1000 etc.).

97×9497 \times 9498×9798 \times 97994×997994 \times 997

Both numbers above base

Both numbers exceed the base — surpluses are added instead of subtracted.

103×107103 \times 107104×112104 \times 1121003×10071003 \times 1007

Mixed — one above, one below

One number is above base, one is below — result of cross-add is negative, requiring complement.

97×10497 \times 104103×98103 \times 98

Common Mistakes to Avoid

✕

Wrong number of digits in the right part

Wrong approach

97×9497\times 94: deficiency product 3×6=183\times 6=18, but writing '18' for base 100 is fine. For 994×997994\times 997: deficiency product 6×3=186\times 3=18, must write '018' (3 digits for base 1000).

Correct approach

✓
Right part must have exactly as many digits as zeros in the base. Pad with leading zeros.

Why this happens

💡 Students forget to pad the right part to match the base's digit count.

Why It Works

Let B be the working base; define deficiencies x and y:

a=B−x,b=B−y(x,y≥0)a = B - x, \quad b = B - y \quad (x, y \geq 0)

Expand the product:

a×b=(B−x)(B−y)=B2−B(x+y)+xya \times b = (B - x)(B - y) = B^2 - B(x + y) + xy

Factor out B from the first two terms:

=B(B−x−y)+xy=B(a−y)+xy= B\bigl(B - x - y\bigr) + xy = B(a - y) + xy

Identify left and right parts:

Left part:  a−y  (=b−x)Right part:  x⋅y\text{Left part:}\; a - y \;(= b - x) \qquad \text{Right part:}\; x \cdot y

∴ Subtract either deficiency from the opposite number to get the left part. Multiply the deficiencies for the right part. Right part must have as many digits as there are zeros in B.

For numbers near base B: let a=B−xa = B-x, b=B−yb = B-y (deficiencies). Then a×b=(B−x)(B−y)=B2−Ba\times b = (B-x)(B-y) = B^2-B(x+yx+y)+xy = B(a−ya-y)+xy = B(b−xb-x)+xy. The left part is one number minus the other's deficiency; the right part is the product of the deficiencies.

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By one more than the previous one

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Whatever the deficiency

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Ānurūpyeṇa

Proportionately