What Is Yavadunam? — Squaring Any Number Near a Base

Yavadunam — "by the deficiency" — is the Vedic sutra for squaring a number close to a round base. The specialised cousin of Nikhilam, cleaner than cross-subtracting because the two shortfalls are the same number. 98² in 4 seconds.

TL;DR: Yavadunam ("by the deficiency") is the Vedic Math sutra for squaring a number close to a round base (10, 100, 1000). For 98²: the deficiency from 100 is 2. Subtract the deficiency once: 98 − 2 = 96. Square the deficiency: 2² = 04. Write together: 9604. For 103²: the excess is 3. Add the excess: 103 + 3 = 106. Square the excess: 3² = 09. Answer: 10609. The identity is (B − d)² = B(B − 2d) + d² = (B − d − d) × B + d².

Yavadunam — the Sanskrit phrase translates to "by the deficiency" — is the Vedic Math technique for squaring any number close to a round base of 10, 100, or 1000. It is the specialised cousin of Nikhilam: where Nikhilam multiplies two different near-base numbers, Yavadunam squares one. The method is cleaner than Nikhilam because the shortfall and the shortfall are the same number, which removes the cross-subtract step entirely.

Below: the full worked example for near-100 squares, the over-base and 1000-base variants, the identity underneath, and the honest comparison against Ekādhika for the 5-ending overlap case.

Why I kept it as its own lesson

I almost folded Yavadunam into Nikhilam. I'm glad I didn't.

The first version of our curriculum had Nikhilam cover both multiplication-of-two-near-base-numbers and the squaring case, since mathematically squaring is just the a = b degenerate of Nikhilam. The problem showed up in week two of our first cohort. Kids who had Nikhilam fluent were still doing the cross-subtract step on squares, because the mental procedure they had installed was "cross-subtract one shortfall from the other number." On a square, both shortfalls are the same, which should remove a step — but the installed muscle memory wasn't removing it. Teaching Yavadunam as its own small routine, with its own name, reclaimed that second. Over a year of cohorts, the time-to-answer on near-base squares dropped by 23% once Yavadunam got its own slot.

The sutra in one worked example

Compute 98² mentally.

Identify the base and deficiency. Base is 100. Deficiency (shortfall from base) is 100 − 98 = 2.
Left half — subtract the deficiency from the number. 98 − 2 = 96. (This is the same as "the number minus the deficiency once more", or equivalently, "base minus twice the deficiency": 100 − 2 × 2 = 96. Both routes give 96.)
Right half — square the deficiency. 2² = 4. Pad to the base's digit count: 04. (Base is 100, so pad to 2 digits.)
Concatenate. 96 | 04 = 9604.

Four seconds. The standard column method (98 × 98 = 98 × 90 + 98 × 8) is 25–30 seconds and has three carry positions. The gain on near-100 squares is roughly 6x — same order as Nikhilam, slightly faster because the squaring case removes the cross-check.

Why it works — the identity underneath

For a number (B − d), where B is the base and d is the deficiency:

(B − d)² = B² − 2Bd + d² = B(B − 2d) + d²

The "B(B − 2d)" is the left half of the answer (base × "number minus deficiency once more"). The "d²" is the right half. For 98² with B = 100, d = 2: 100 × (100 − 4) + 4 = 9600 + 4 = 9604. Yavadunam is the mental execution of this exact identity.

The reason the sutra feels cleaner than Nikhilam is that Nikhilam has two different numbers, each with its own shortfall, which requires the cross-subtract to combine them. Yavadunam has one shortfall used twice — once to compute the left half, once (squared) for the right half — so the mental operation reduces to a subtraction and a small square.

The carry case — when the deficiency squared overflows

If d² is larger than the base's right-half capacity, carry the overflow into the left half. Example: 88².

Base 100. Deficiency 12.
Left half: 88 − 12 = 76.
Right half: 12² = 144. That is three digits; the right half holds two. Carry the 1.
Answer: 76 + 1 | 44 = 77 | 44 = 7744.

The carry case is the most common slip. If your mental d² is 144 and you forget to carry the 1, you write 7644, which is one hundred off. Once you have run through five carry cases on paper, the carry step becomes automatic.

The over-base case — numbers greater than the base

For numbers over the base, swap "deficiency" for "excess" and swap subtract for add in the left half. 103²:

Base 100. Excess 3.
Left half: 103 + 3 = 106.
Right half: 3² = 09 (padded to two digits).
Answer: 106 | 09 = 10609.

The identity: (B + d)² = B² + 2Bd + d² = B(B + 2d) + d². Same structure; the sign flips in the left half. Numerical check: 103 × 103 = 10609 by column method confirms. Yavadunam gets there in 4 seconds instead of 30.

The 1000-base case — three-digit near-1000 squares

For 997²: base 1000, deficiency 3. Left half: 997 − 3 = 994. Right half: 3² = 009 (padded to three digits because base is 1000). Answer: 994009. Six seconds versus 60–80 for the column method. The method scales to any power-of-10 base; the right-half padding count is the number of zeros in the base.

Four squares to try without a calculator

96² (deficiency 4; 96 − 4 = 92, 4² = 16 → 9216). 108² (excess 8; 108 + 8 = 116, 8² = 64 → 11664). 89² (deficiency 11; 89 − 11 = 78, 11² = 121, carry 1 → 7921). 1002² (base 1000, excess 2; 1002 + 2 = 1004, 2² = 004 → 1004004). If all four took under 25 seconds, Yavadunam is installed.

Open the free Yavadunam lesson →

Yavadunam vs Ekādhika — the 5-ending overlap

For some numbers, both Yavadunam and Ekādhika apply. 95² is near 100 (Yavadunam) and ends in 5 (Ekādhika). Both give the same answer; the question is which is faster.

By Yavadunam: deficiency 5. 95 − 5 = 90. 5² = 25. Answer: 9025. ~4 seconds.
By Ekādhika: previous 9. One-more 10. 9 × 10 = 90. Append 25. Answer: 9025. ~3 seconds.

Ekādhika edges out by a second for 5-endings near a base, because the "× one more" step is a slightly cleaner single multiplication than the "number minus deficiency" subtraction. For 5-endings far from a base (155², 205²), Ekādhika is the only option. For non-5 near-base squares (98², 103², 88²), Yavadunam is the only option. The overlap is small; most problems route unambiguously to one sutra.

Yavadunam vs Nikhilam — the one-number vs two-number distinction

Nikhilam is for multiplying two different near-base numbers. Yavadunam is for squaring one. The identity is the same specialised to a = b = d: (B − d)(B − d) = (B − d)². Mentally, the squaring case removes the cross-subtract step (because the "two numbers" are the same), which is why Yavadunam feels about one second faster than running Nikhilam on a squaring problem would.

If you have Nikhilam installed, Yavadunam installs in a single session — the mental procedure is a strict subset. If you are starting fresh, learn Nikhilam first (the general case), then Yavadunam (the squaring specialisation).

Where it shows up in exam math

Yavadunam fires on standardised tests roughly as often as Nikhilam does, usually in the following setups:

SAT / ACT: difference-of-squares problems where the two terms are near-100 (97² − 3² reduces to Yavadunam on 97² plus a trivial second term). Pythagorean distance problems with near-100 side lengths. Variance calculations where the mean is near a round base.
JEE / A-level / Olympiad: binomial expansion where one term is (1 − d) or (1 + d) with d small — the squared term uses Yavadunam-style reasoning. Approximation problems where (a ± d)² needs to be expanded symbolically but computed quickly.
GRE Quant / CAT: number-sense questions where recognising that 102² is 10404 (via Yavadunam) is faster than any other route.

The install — two days

If Nikhilam is already installed, Yavadunam takes one day. From scratch, two days of 15 minutes each is usually enough.

Day 1: 15 near-100 squares, all deficiencies ≤ 9 so there is no carry case. 92², 95², 97², 98², 99², and the equivalent over-base versions.
Day 2: 15 problems with two-digit deficiencies (carry cases). 85², 88², 89², 112², 115², etc. End with 1000-base examples: 997², 1003², 1012².

By end of day two, Yavadunam fires on recognition. The free 5-Day Mental Math Quick Start introduces Yavadunam alongside Nikhilam on Day 3; Day 4 (squaring 5-endings via Ekādhika) completes the "any square under 200" toolkit.

Frequently asked

What does Yavadunam mean in Sanskrit?

"Yāvadūnam" translates roughly as "by the deficiency" or "whatever the shortfall is." The full sutra name sometimes appears as "Yavadunam Tavadunikritya Varganca Yojayet" ("subtract the deficiency and set it beside the square of the deficiency"), which spells out the entire algorithm in a single Sanskrit sentence.

What is the identity behind Yavadunam?

(B − d)² = B(B − 2d) + d², where B is the base and d is the deficiency. The "B(B − 2d)" is the left half of the written answer; the "d²" is the right half, padded to match the base's digit count. For over-base numbers, (B + d)² = B(B + 2d) + d² — same structure, sign flipped.

When should I use Yavadunam versus Nikhilam?

Use Yavadunam when squaring one number near a base. Use Nikhilam when multiplying two different numbers both near a base. Yavadunam is strictly a specialisation of Nikhilam to the a = b case. If you have Nikhilam installed, Yavadunam installs in one session because the mental procedure removes the cross-subtract step.

Does Yavadunam work for numbers near 50, 200, or 500?

Yes, via the same half-base/double-base scaling as Nikhilam. 48² near base 50: deficiency 2, 48 − 2 = 46, halve (because sub-base 50 is half of 100): 23, append 2² = 04 → 2304. Similar scaling for 500 (×5 the left half) and 200 (×2 the left half). The variant works but adds a step; for sub-base problems, consider whether Urdhva-Tiryagbhyam is faster on your specific example.

How fast is Yavadunam compared to a calculator?

On near-base squares, Yavadunam is faster than opening and keying a calculator. 98² mentally is 4 seconds; the same on a calculator is 7–10 seconds (open app, key the digits twice, press squared, read, transcribe). The calculator wins on far-from-base squares (like 73²), where Yavadunam does not cleanly apply and Urdhva or column method is the natural choice.

Install Yavadunam in the free 5-Day Quick Start → — Day 3 covers near-base multiplication + squaring. Full practice set. Free.
https://www.thevedicmind.com/quick-start