What Vedic Maths Tricks Actually Are
Vedic maths is a collection of mental-calculation methods organised around 16 sutras (short word-formulae) and 13 sub-sutras, compiled by the Indian scholar Bharati Krishna Tirtha in the early twentieth century. The word "Vedic" points to the ancient Vedas, but the system as written is modern. Each trick is a re-packaging of the same place-value and distributive rules a student already meets in school arithmetic, arranged so the steps run in the head instead of down a column on paper.
That framing matters before the list starts. These are not a different mathematics. Multiplying by 11 by "splitting and adding the middle" is the distributive law wearing a costume. Knowing the costume is fast. Knowing what is underneath is what lets a student adapt when the numbers do not cooperate.
Below are 15 tricks worth knowing, each with what it does well. After the list, the honest part: where a bag of tricks reaches its ceiling, and what carries a learner past it.
15 Vedic Maths Tricks With Worked Examples
1. Multiplying Any Two-Digit Number by 11
Split the two digits apart and drop their sum into the middle.
For $35 \times 11$: keep the 3 and the 5 on the outside, add $3 + 5 = 8$, place it between them.
$$35 \times 11 = 3\,\underline{8}\,5 = 385$$
When the middle sum passes 9, carry it left. For $68 \times 11$: $6 + 8 = 14$, so write the 4 and carry the 1 into the 6.
$$68 \times 11 = (6+1)\,4\,8 = 748$$
Good for: any quick ×11, and a first feel for why place value lets you "insert" a digit.
Where it falls short: the clean "split and drop the sum" version only holds for two-digit numbers; a three-digit number needs a chained carry across several middle positions and the mental load climbs fast.
2. Squaring Numbers That End in 5 (Ekadhikena Purvena)
Take the digits before the 5, multiply that number by the next whole number up, and tack 25 on the end.
For $75^2$: the front part is 7, so compute $7 \times 8 = 56$, then append 25.
$$75^2 = 56\,|\,25 = 5625$$
Good for: instant squares of 15, 25, 35, up through 95. It is the cleanest single trick in the system.
Where it falls short: it applies only to numbers whose last digit is 5; a number ending in any other digit gets nothing from it and needs a general squaring method instead.
3. Nikhilam — Multiplying Numbers Just Below a Base
"Nikhilam" means "all from 9 and the last from 10." For two numbers near 100, write each as base minus a deficit. Cross-subtract one deficit from the other number for the left part; multiply the two deficits for the right part.
For $97 \times 96$ (base 100): deficits are $-3$ and $-4$.
$$\text{Left: } 97 - 4 = 93 \qquad \text{Right: } 3 \times 4 = 12$$ $$97 \times 96 = 93\,|\,12 = 9312$$
Good for: products where both numbers hug a power of 10 (98×97, 994×996). Outside that neighbourhood it stops being a shortcut.
Where it falls short: it is only efficient when both numbers sit close to a base like 100 or 1000; as the deficits grow (say $72 \times 61$) the cross-subtraction and multi-digit deficit product become clumsier than the standard method.
4. Nikhilam Above the Base
Same idea, but for numbers just above the base, add the surpluses instead of subtracting.
For $103 \times 104$ (base 100): surpluses are $+3$ and $+4$.
$$\text{Left: } 103 + 4 = 107 \qquad \text{Right: } 3 \times 4 = 12$$ $$103 \times 104 = 107\,|\,12 = 10712$$
Good for: near-base products on the high side. Same narrow window as Trick 3.
Where it falls short: it only helps when both numbers sit just above the same base; mix a number above the base with one below it and the surplus-and-deficit signs no longer line up cleanly.
5. Urdhva-Tiryagbhyam — Vertically and Crosswise
This is the general multiplication sutra — the one that works for any two numbers, not just special cases. For two two-digit numbers, you compute three column-products: units times units, the cross-sum, and tens times tens.
For $23 \times 21$:
$$\text{Units: } 3 \times 1 = 3$$ $$\text{Cross: } (2 \times 1) + (3 \times 2) = 2 + 6 = 8$$ $$\text{Tens: } 2 \times 2 = 4$$ $$23 \times 21 = 4\,|\,8\,|\,3 = 483$$
Good for: any multiplication, and it scales to bigger numbers. It is also the trick most students get wrong under pressure, because the carries between columns stack up.
Where it falls short: it stays reliable only while the running carries are small; on three- and four-digit numbers the crosswise products multiply and the mental bookkeeping outgrows what most people can hold, so accuracy drops before speed does.
6. Multiplying by 5
Since $5 = \tfrac{10}{2}$, multiply by 10 and halve.
For $48 \times 5$: $48 \times 10 = 480$, then $480 \div 2 = 240$.
$$48 \times 5 = 240$$
Good for: any ×5 without reaching for the column method.
Where it falls short: the halving stays clean only while the number is even; an odd number leaves a trailing $.5$ to track (for example $\tfrac{470}{2} = 235$ is fine, but $47 \times 5$ forces a decimal midway), which slows the mental version.
7. Multiplying by 25
Since $25 = \tfrac{100}{4}$, append two zeros and halve twice.
For $36 \times 25$: $3600 \div 2 = 1800$, then $1800 \div 2 = 900$.
$$36 \times 25 = 900$$
Good for: money-style sums and quarters (25, 50, 75 all have cousins of this).
Where it falls short: it only stays tidy when the number divides evenly by 4; otherwise the two successive halvings leave a fractional remainder to carry, and the shortcut loses its edge over just multiplying.
8. Multiplying by 9, 99, 999 (Ekanyunena Purvena)
To multiply a number by a string of nines equal in length, subtract 1 from the number for the left part, then take the "all from 9, last from 10" complement of the number for the right part.
For $46 \times 99$: left is $46 - 1 = 45$; right is $100 - 46 = 54$.
$$46 \times 99 = 45\,|\,54 = 4554$$
Good for: any ×9, ×99, ×999. It goes quiet the moment the multiplier is not a run of nines.
Where it falls short: it depends on the multiplier being a run of nines exactly as long as the number; a mismatch in length (like $46 \times 999$) breaks the neat "subtract 1, complement the rest" split and needs a correction step.
9. Subtracting From a Power of 10 (Nikhilam Complement)
Subtract every digit from 9, and the last digit from 10.
For $10000 - 7478$: $9-7=2$, $9-4=5$, $9-7=2$, $10-8=2$.
$$10000 - 7478 = 2522$$
Good for: change from round amounts, and clean borrowing without a chain of crossed-out digits.
Where it falls short: it only applies when the number you subtract from is an exact power of 10; subtract from something like $9500$ instead of $10000$ and the "all from 9, last from 10" rule no longer maps directly.
10. Squaring Numbers Near a Base (Yavadunam)
For a number near 100, adjust by the deficit for the left part and square the deficit for the right.
For $98^2$ (base 100, deficit $-2$): left is $98 - 2 = 96$; right is $2^2 = 04$.
$$98^2 = 96\,|\,04 = 9604$$
Good for: squares of 96–104 and similar near-base values.
Where it falls short: it only pays off within a tight band around the base; once the deficit grows past a few units the squared-deficit part needs its own carry and the method stops being faster than a direct square.
11. The Digit-Sum (Digital Root) Check
Add a number's digits repeatedly until one digit remains. The digit sums must stay consistent across an operation — a fast way to catch a slip.
For $23 \times 21 = 483$: digit sum of 23 is 5, of 21 is 3, and $5 \times 3 = 15 \to 6$. Digit sum of 483 is $4+8+3=15 \to 6$. They match, so the answer survives the check.
Good for: catching arithmetic errors. It confirms an answer is plausible, not that it is correct (it will not catch a digit-swap like 483 vs 438).
Where it falls short: it catches many slips but is blind to any error that preserves the digit sum, so digit-transposition mistakes (483 written as 438) pass the check untouched.
12. The Vinculum — Turning Big Digits Into Small Ones
A vinculum rewrites an awkward digit as a small negative one, so $9$ becomes $1\bar{1}$ (ten minus one). It makes messy multiplications tidier.
Write $19$ as $2\bar{1}$ (that is $20 - 1$). Then $19 \times 19 = (2\bar{1})^2$ works with the small digits 2 and 1 instead of 9s, and converts back at the end to $361$.
Good for: simplifying numbers full of 7s, 8s, and 9s before another trick runs.
Where it falls short: it only earns its keep when a number is dominated by large digits; on numbers already made of small digits the conversion to and from negative digits adds steps rather than removing them.
13. Multiplying Two Numbers Whose Tens Match and Units Add to 10 (Antyayordashake'pi)
When the tens digits are equal and the units add to 10, multiply the tens digit by one more than itself for the left, and multiply the units for the right.
For $43 \times 47$ (tens both 4, units $3+7=10$): left is $4 \times 5 = 20$; right is $3 \times 7 = 21$.
$$43 \times 47 = 20\,|\,21 = 2021$$
Good for: a very specific pair shape. When it fits, it is delightful; it fits rarely.
Where it falls short: it needs both conditions at once — equal tens digits and units that add to exactly 10 — so it applies to only a thin slice of pairs and does nothing for the vast majority of products.
14. Dividing by 9 (Quick Quotient and Remainder)
For a two-digit number divided by 9, the first digit is the quotient start and the digit sum gives the remainder.
For $23 \div 9$: bring down the 2 as the running quotient, add it to the next digit $2 + 3 = 5$ for the remainder.
$$23 \div 9 = 2 \text{ remainder } 5$$
Good for: fast division by 9 and a neat link to the digit-sum idea in Trick 11.
Where it falls short: the simple "carry the first digit, add for the remainder" form only holds for small two-digit dividends; once the digit sum reaches 9 or more it spills a further carry, and longer dividends need repeated adjustment.
15. Multiplying Numbers Near Different Bases (Anurupyena)
When the two numbers sit near a convenient working base and its multiple, scale the deficits proportionally. For numbers near 50, use 50 as a working base (half of 100) and adjust.
For $48 \times 47$ (working base 50, which is $\tfrac{100}{2}$): deficits are $-2$ and $-3$. Cross-subtract for the raw left part: $48 - 3 = 45$. Because the base is $\tfrac{100}{2}$, halve it: $45 \div 2 = 22.5$. The whole part 22 is the left; the leftover half carries 50 into the right. The deficit product is $2 \times 3 = 6$, and $6 + 50 = 56$.
$$48 \times 47 = 22\,|\,56 = 2256$$
Good for: numbers near 50, 200, 500. It is powerful and also the trick where students most often lose the thread, because the scaling step has to be reasoned, not recited.
Where it falls short: it only works when both numbers sit near the same convenient multiple of a base, and the proportional scaling step (halving or doubling mid-calculation) has to be reasoned each time, so it stops being reliable the moment a learner tries to run it on autopilot.
Where Vedic Maths Tricks Are Genuinely Strong
Two things are true and worth saying plainly. First, these tricks are fast. A student who has drilled ×11, squaring-ending-in-5, and Nikhilam will beat a calculator to certain answers, and that speed feels good — it lowers the friction of arithmetic and can rebuild a shaky learner's confidence with numbers. Second, a few of them teach something real. Trick 1 and Trick 9 quietly reveal how place value works. The digit-sum check in Trick 11 is a genuine mathematical idea (modular arithmetic in disguise) that shows up again in divisibility rules and later in number theory.
So this is not a takedown. Speed on the operations the tricks cover is a real skill, and for mental arithmetic drills, competition rounds, or shaking off calculator dependence, the tricks earn their place.
Where the Tricks Hit a Ceiling
Here is the honest part. A trick fires when the numbers fit its pattern — and goes quiet when they do not. That is the structural limit of the whole approach.
Pattern-specific by design. Nikhilam wants numbers near a power of 10. The ×9 trick wants a run of nines. Trick 13 wants matching tens and complementary units. Give a student $63 \times 58$ and most of the specialised tricks have nothing to say — they are back to Urdhva-Tiryagbhyam or the column method.
Speed, not understanding. A learner can execute "square a number ending in 5" perfectly and still not be able to explain why $75^2$ works, because the recipe never asked them to. The memoriser archetype — the student who can recite the steps but freezes when the problem looks slightly different — is exactly who a trick-only diet produces.
No transfer to the math that matters most. Speed on $97 \times 96$ does nothing for solving $x^2 - 5x + 6 = 0$, proving two triangles congruent, or reading a word problem and deciding what to compute. Those need reasoning about structure, not faster arithmetic.
Fragile under variation. Because the tricks are keyed to surface patterns, a small change in the numbers can break the shortcut a student reached for, and they often notice too late.
None of this makes Vedic tricks bad. It makes them narrow. They are a set of fast tools for a set of specific jobs — not a foundation you can build the rest of mathematics on.
How Bhanzu Approaches the Same Goal Differently
The instinct behind Vedic maths — "I want to be quick and confident with numbers" — is a good one. Bhanzu chases the same outcome from the other direction: it builds mental agility out of understanding number structure, not out of memorised shortcuts.
The difference shows up in a single question. A trick answers "what steps get the answer fastest?" Understanding answers "why does this work, and what else does it let me do?" When a student sees that $75^2$ is really $(70+5)^2 = 70^2 + 2 \cdot 70 \cdot 5 + 25$, they have not learned one square — they have met the identity that later powers algebra, factoring, and completing the square. The same mental quickness a trick promises comes for free, and it transfers.
That is Bhanzu's whole stance: patterns over memorising, WHY before the shortcut. Fast arithmetic is a byproduct of understanding, and understanding is what carries into the algebra, geometry, and reasoning that arithmetic tricks never reach.
Conclusion
Vedic maths tricks like ×11, Nikhilam, and squaring numbers ending in 5 genuinely speed up specific arithmetic once you recognise the pattern.
A few tricks — place-value splits and the digit-sum check — carry real mathematical ideas worth keeping.
Each trick is pattern-specific: it works for the numbers it was built for and goes quiet on the rest.
The tricks build speed on known patterns, not the reasoning that algebra, geometry, and word problems demand.
Understanding why a shortcut works gives the same speed and, unlike the trick, transfers to the rest of math.
To build number sense that lasts rather than a set of one-off shortcuts, explore Bhanzu's math classes online or a structured approach to mental maths for kids that starts from understanding. You can also book a free demo class to see the understanding-first model in action.
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