Nikhilam Explained: Multiplying Numbers Near a Base

Ask most people to multiply 97 × 96 and they will reach for paper or a phone. A student who knows Nikhilam does it in about four seconds, in their head.

It is the most impressive of the Vedic Maths techniques and, once understood, one of the simplest. Here is how it works and why.

The sutra

Nikhilam Navatashcaramam Dashatah — "all from 9 and the last from 10".

That phrase describes how to find how far a number is from a base like 100. For 97: take each digit from 9, and the last from 10 → 9−9 = 0, 10−7 = 3 → deficiency 3. Which is just 100 − 97, arrived at without borrowing.

The method for numbers below 100

Three steps.

1. Write how far each number is below 100 (the deficiency)
2. Left half = either number minus the other number's deficiency
3. Right half = product of the two deficiencies (as two digits)

97 × 96

• Deficiencies: 3 and 4
• Left: 97 − 4 = 93 (or equivalently 96 − 3 = 93 — both work, which is a useful self-check)
• Right: 3 × 4 = 12
• Answer: 9312

98 × 94

• Deficiencies: 2 and 6
• Left: 98 − 6 = 92
• Right: 2 × 6 = 12
• Answer: 9212

When the right half overflows

The right half must occupy exactly as many digits as the base has zeros. For base 100, that is two digits.

88 × 96

• Deficiencies: 12 and 4
• Left: 88 − 4 = 84
• Right: 12 × 4 = 48 → fits in two digits
• Answer: 8448

88 × 86

• Deficiencies: 12 and 14
• Left: 88 − 14 = 74
• Right: 12 × 14 = 168 → three digits, one too many
• Carry the extra 1 into the left: 74 + 1 = 75, keep 68
• Answer: 7568

This carry step is where students slip. The rule: count the zeros in the base, keep that many digits on the right, carry the rest.

Numbers above the base

The same method works, with additions instead of subtractions.

103 × 104

• Surpluses: 3 and 4
• Left: 103 + 4 = 107
• Right: 3 × 4 = 12
• Answer: 10712

112 × 108

• Surpluses: 12 and 8
• Left: 112 + 8 = 120
• Right: 12 × 8 = 96
• Answer: 12096

One above, one below

Here the right half becomes negative, which needs one extra move.

103 × 97

• 103 is +3, 97 is −3
• Left: 103 − 3 = 100
• Right: 3 × (−3) = −9
• So the answer is 10000 − 9 = 9991

Rule: when the right half is negative, reduce the left half by 1 and take the complement of the right half. 100 − 1 = 99, and the complement of 9 within 100 is 91 → 9991 ✓

Other bases

Nikhilam is not limited to 100.

994 × 998

• Deficiencies: 6 and 2
• Left: 994 − 2 = 992
• Right: 6 × 2 = 12, padded to three digits = 012
• Answer: 992012

Padding is essential. Forgetting it is the second most common error after the carry.

Why it works

For numbers near 100, write them as (100 − a) and (100 − b).

(100 − a)(100 − b) = 10000 − 100a − 100b + ab = 100(100 − a − b) + ab

The bracket (100 − a − b) is exactly "one number minus the other's deficiency", and ab is the product of deficiencies. The method is that identity, arranged for mental use.

Showing this derivation to a Class 8 student is worthwhile — it turns a trick into mathematics.

Practising it

1. Day 1–3: both numbers below 100, small deficiencies, no carry
2. Day 4–6: larger deficiencies that force a carry
3. Day 7–9: both above the base
4. Day 10–12: one above and one below
5. Day 13–15: bases of 10 and 1000

Our free Vedic Maths worksheet on multiplying near 100 is built for exactly this sequence, with 24 practice questions and answers.

Frequently asked questions

Which class can learn this? Class 4 and above once tables are fluent. Understanding why it works suits Class 8 upwards.

Does it only work near 100? No — any base works, including 50 and 200 with a small adjustment. Bases of 10, 100 and 1000 are the most useful.

Is it faster than normal multiplication? For numbers near a base, dramatically. For 47 × 63 it is not the right tool — use vertically-and-crosswise instead.

Can I use it in exams? Yes for the answer. In board exams write the standard working as well.

What is the most common mistake? Forgetting to pad the right half to the correct number of digits, and mishandling the carry when the product overflows.

Learn the whole system

Nikhilam is one of sixteen sutras, each solving a particular shape of problem. Our Vedic Maths course covers them in order from Class 4 upwards.

Book a free demo class — most children have two or three tricks working in the first month.

Cover image: Jeuwre — CC BY-SA 4.0.