What Is the Nikhilam Sutra? — The Full Worked Example

Nikhilam Navataścaramaṃ Daśataḥ — "all from 9, last from 10" — is the Vedic sutra for multiplying two numbers close to a round base. 97 × 98 in four seconds. The full worked example, the identity underneath, and the variants.

TL;DR: Nikhilam Navataścaramaṃ Daśataḥ, "all from 9, last from 10", is the Vedic Math sutra for multiplying two numbers that are both close to a round base (10, 100, or 1000). For 97 × 98: both fall short of 100 by 3 and 2. Cross-subtract one shortfall from the other number (97 − 2 = 95), multiply the shortfalls (3 × 2 = 06), write together: 9506. The algebraic identity is (100 − a)(100 − b) = 100(100 − a − b) + ab.

Nikhilam Navataścaramaṃ Daśataḥ — the first full name is unwieldy, but the sutra is three steps and about four seconds of work. It is the Vedic Math technique for multiplying two numbers that are both close to a round base: 10, 100, or 1000. The literal translation is "all from 9, last from 10", which describes how you compute the complement of a number from its base. It is the first sutra most Vedic Math students learn because it produces the biggest visible speed gain against the standard column method.

Below: the full worked example, the algebraic identity underneath, the variants that extend it to bases of 50, 200, and 500, and the honest limits of where the sutra applies.

From the author

If I only got to teach one sutra, it would be this one.

Nikhilam is the sutra I open with because it produces the loudest first reaction. In the Charter Cohort I keep a tally — out of the last 90 students, 71 said something out loud at the end of their first 97 × 98 attempt. The specific sentences varied ("wait, that's it?", "do another one", "my mom's going to lose her mind"), but the pattern is reliable enough that we design the first week around it. The ×11 trick on Day 1 warms them up. Nikhilam on Day 3 is the moment the resistance to practising breaks. Everything else gets taught in the gap that opens there.

The sutra in one worked example

Compute 97 × 98 using Nikhilam.

Identify the base. Both numbers are close to 100. The base is 100.
Compute the shortfalls. 97 is short of 100 by 3. 98 is short of 100 by 2. Write them: 97 (−3), 98 (−2).
Left half — cross-subtract. Take either number and subtract the other shortfall. 97 − 2 = 95. (You would get the same answer from 98 − 3 = 95. This is a check — if the two values disagree, you have made an arithmetic error.)
Right half — multiply the shortfalls. 3 × 2 = 06. Pad to two digits because the base is 100 (two zeros).
Write together. Left half, right half, concatenated: 95 | 06 → 9506.

The standard column method takes about 35 seconds and has three places to drop a carry. Nikhilam takes about 4 seconds with one subtraction and one multiplication. The gain is 8–9x on this problem, which is typical for near-100 pairs.

Why it works — the identity underneath

Nikhilam is the algebraic identity (B − a)(B − b) = B(B − a − b) + ab, applied mentally with B equal to the base. For 97 × 98 with B = 100, a = 3, b = 2:

(100 − 3)(100 − 2) = 100(100 − 3 − 2) + 3 × 2 = 100 × 95 + 6 = 9506.

That is it. Nothing hidden. The "cross-subtract" step is just the 100 − a − b in the identity, computed by subtracting one shortfall from the other number because 100 − a − b = (100 − a) − b. The "multiply the shortfalls" step is the ab term. The sutra reorders the identity into something a 10-year-old can execute without writing anything down. Your child's algebra teacher would recognise it immediately.

The carry case — when the right half overflows

If the two shortfalls multiply to a number too big for the right-half padding, carry the overflow left. Example: 88 × 92.

Shortfalls: 88 (−12), 92 (−8).
Cross-subtract: 88 − 8 = 80. (Check: 92 − 12 = 80. ✓)
Multiply shortfalls: 12 × 8 = 96. That fits in two digits — no carry.
Answer: 80 | 96 = 8096.

Now a case that does overflow: 83 × 87.

Shortfalls: 83 (−17), 87 (−13).
Cross-subtract: 83 − 13 = 70. (Check: 87 − 17 = 70. ✓)
Multiply shortfalls: 17 × 13 = 221. That is three digits; the right half holds two. Carry the 2 to the left half.
Answer: 70 + 2 | 21 = 72 | 21 = 7221.

The carry case is the one place most self-taught learners slip. Once you have run through it on paper five or six times, it becomes a non-event. If you miss it the first time, the sutra feels wrong — do not blame yourself; the carry case is the only subtle bit.

Try these three right now

Without a calculator or paper. 96 × 94. 103 × 108. 76 × 74. (Answers: 9024. 11124. 5624. The third one is a carry case — 24 × 26 = 624 would be the right-half; the 6 carries. 76 − 26 = 50, add the carry → 56, so 56 | 24 = 5624.) If all three took under 20 seconds total, Nikhilam is installed. If not, see the free lesson below for the stepped practice.

Open the free Nikhilam lesson →

The over-base case — when both numbers exceed the base

Nikhilam also works when both numbers are above the base. 103 × 108 with base 100.

Excesses (not shortfalls): 103 (+3), 108 (+8).
Cross-add (not subtract): 103 + 8 = 111. (Check: 108 + 3 = 111. ✓)
Multiply excesses: 3 × 8 = 24.
Answer: 111 | 24 = 11124.

The identity: (B + a)(B + b) = B(B + a + b) + ab. Same structure; the signs flip on the shortfall side.

The mixed case — one above, one below the base

98 × 103 with base 100. One is short by 2, one is over by 3. This is the awkward case; the right half becomes negative, which the standard write-together method cannot display.

Shortfall 2, excess 3. Mixed signs.
Cross-combine: 98 + 3 = 101. (Check: 103 − 2 = 101. ✓)
Multiply shortfall-and-excess: 2 × 3 = 6, but the sign is negative. Right half = −6.
Answer: 101 | −6 = 10100 − 6 = 10094.

The mixed case is the entry to vinculum notation, where negative digits sit inline for cleaner mental execution. See the vinculum post for the full treatment.

Variants — bases of 50, 200, 500

Nikhilam's extension (Ānurūpyeṇa, "proportionally") lets you use non-round bases by scaling. For 48 × 46 with sub-base 50 (treating 50 as half of 100):

Shortfalls from 50: 48 (−2), 46 (−4).
Cross-subtract: 48 − 4 = 44.
Multiply shortfalls: 2 × 4 = 08.
Halve the left half (because sub-base 50 is half of 100): 44 ÷ 2 = 22.
Answer: 22 | 08 = 2208.

Similar scaling applies for sub-bases of 200 (double the left half), 500 (× 5), and 1000 (base extends to three right-half digits). The variant table triples the range of problems Nikhilam covers.

Where Nikhilam is not the right tool

Three cases:

Numbers that are not close to a base. 73 × 46 — 73 is 27 off from 100, 46 is 54 off from 100. The shortfalls are so large that the "mental shortcut" becomes two multi-digit multiplications of its own. For problems like this, use Urdhva-Tiryagbhyam (general vertically-and-crosswise multiplication) or just the standard column method.
Single-digit multiplication. 7 × 8 should be known. Nikhilam on 7 × 8 is technically possible with base 10 (shortfalls 3 and 2 → 7 − 2 | 3 × 2 = 56), but if a child needs Nikhilam for single-digit facts, the problem is multiplication tables, not a shortcut method.
Numbers that are close to a base, but not to each other. 97 × 11 — 97 is close to 100, 11 is close to 10. Different bases. Not a Nikhilam problem. Use ×11 instead for the 11-factor; Nikhilam needs both numbers near the same base.

Where it shows up in exam math

Nikhilam appears on standardised tests more often than people expect. SAT difference-of-squares problems are a Nikhilam setup in disguise (51² − 49² reduces to (100)(2) = 200 instantly). ACT geometry area problems with near-100 sides fire Nikhilam. JEE Main and CAT routinely include two-digit multiplication of near-100 pairs, where the sutra saves 30 seconds per problem over the column method.

For the exam-specific framings, see the 5 Vedic shortcuts for SAT and ACT and JEE Main math without a calculator.

The practice ladder — install Nikhilam in three days

If you have never done this before, three days of 15 minutes installs it reliably. The free 5-Day Vedic Math Quick Start builds this out as Day 3; Days 1, 2, 4, and 5 cover the surrounding sutras that make Nikhilam compound.

Day 1: 15 near-100 pairs where both shortfalls are single digits (97 × 93, 96 × 94, etc.). No carry case yet.
Day 2: 15 pairs with two-digit shortfalls, including the carry case (83 × 87, 76 × 72).
Day 3: 15 mixed problems including the over-base case (108 × 103) and the mixed case (98 × 103).

By the end of day three, Nikhilam fires on recognition — you see the pair, the shortfalls appear in your head, the answer forms before you have reached for a pencil. That is the install complete. From there, Nikhilam becomes a tool you reach for whenever two numbers are standing near the same round base, which is more often than you expect.

Frequently asked

What does Nikhilam Navataścaramaṃ Daśataḥ literally translate to?

"All from 9, last from 10." The phrase describes how to compute the complement of a number from its base: subtract each digit except the last from 9, and subtract the last digit from 10. For example, the complement of 736 from 1000 is 264 (9−7=2, 9−3=6, 10−6=4). This complement-computation step is embedded in every Nikhilam calculation.

What is the formula or identity behind Nikhilam?

(B − a)(B − b) = B(B − a − b) + ab, where B is the base. For bases 10, 100, 1000, this is just a mental factoring of the standard FOIL expansion. The "cross-subtract" step is the B − a − b term; the "multiply shortfalls" step is the ab term. Every Vedic sutra unpacks to an identity like this one; nothing is hidden.

When should I use Nikhilam versus Urdhva-Tiryagbhyam?

Use Nikhilam when both numbers are close to the same round base (10, 100, 1000, or variants 50, 200, 500). Use Urdhva-Tiryagbhyam ("vertically and crosswise") when numbers are not close to a base — it is the general-purpose Vedic multiplication method that works on anything, including polynomials. Nikhilam is faster when it applies; Urdhva is more flexible.

Does Nikhilam work for three-digit numbers?

Yes — base 1000 for near-1000 pairs (e.g., 998 × 997). Shortfalls are 2 and 3. Cross-subtract: 998 − 3 = 995. Multiply shortfalls: 2 × 3 = 006 (padded to three digits because base is 1000). Answer: 995006. The method scales to any power-of-10 base; the only thing that changes is the right-half padding.

Is Nikhilam faster than a calculator?

On near-base problems, yes. Our timed measurements: 97 × 98 by Nikhilam is around 4 seconds; the same problem on a calculator (open app, key digits, read display, transcribe) is 7–10 seconds in practice. The calculator still wins on problems where no base is close (73 × 46). The honest rule: pattern-match first, pick the tool that is faster for the pattern, then execute.

Try Nikhilam in the free 5-Day Quick Start → — Day 3 is Nikhilam. 15 practice problems. Full answer key. Free.
https://www.thevedicmind.com/quick-start