Is Vedic maths actually useful, or just a party trick?

Genuinely useful. Most of these are shortcuts for the kind of arithmetic that comes up constantly — percentages, checking a bill, estimating an answer — and they're faster than reaching for a calculator once you've practised them a little.

What is the Nikhilam sutra?

A method for multiplying two numbers that are both close to a round base (10, 100, 1000, …), using only how far each number is from that base rather than the numbers themselves.

Shunya / Vedic Math Shortcuts

Mental maths

Vedic math shortcuts

Q: Is there one method that always works, with no special conditions?

Yes: vertically and crosswise (Urdhva Tiryagbhyam). It multiplies any two numbers, digit by digit, and never depends on the numbers having a special form.

Nine shortcuts for the arithmetic you actually run into — checking a bill, estimating a total, a multiple-choice exam where a calculator isn't allowed. Each one comes from classical Indian "Vedic mathematics," with a worked example and a live practice box where you try it yourself and see exactly where you went right or wrong.

01 Multiply by 11

Add each pair of neighbouring digits and slot the sums in between the first and last digit.

43 × 11

Keep the first and last digit: 4 … 3.
Add the middle pair: 4 + 3 = 7.
43 × 11 = 473.
With a carry — 68 × 11: keep 6 … 8, add 6 + 8 = 14 (too big for one digit), so write 4 and carry the 1 into the 6: 748.

Practicebest streak 0

02 Square a number ending in 5

Multiply the leading digit(s) by one more than themselves, then stick 25 on the end.

45²

The number ends in 5; take the digit in front: 4.
Multiply it by one more than itself: 4 × 5 = 20.
Stick 25 on the end: 2025.

Practicebest streak 0

03 Same leading digit, last digits add to 10

23 × 27: same tens digit and 3 + 7 = 10 — multiply the tens digit by one more than itself, then the two units digits.

62 × 68

Same tens digit (6), and 2 + 8 = 10.
Multiply the tens digit by one more than itself: 6 × 7 = 42.
Multiply the units digits: 2 × 8 = 16.
Put them side by side: 62 × 68 = 4216.

Practicebest streak 0

04 Multiply two numbers near a base (Nikhilam)

97 × 96: both are just below 100 — work from how far each one is from the base, not from the numbers themselves.

98 × 97

Both are close to 100: 98 is 2 below it, 97 is 3 below it.
Cross-subtract: 98 − 3 = 95. Multiply by the base: 95 × 100 = 9500.
Multiply the deviations: (−2) × (−3) = 6.
Add: 9500 + 6 = 9506.

Practicebest streak 0

05 Square any number, using the nearest multiple of ten

47²: 47 is 3 below 50 — use 50 × 44 plus 3² instead of squaring 47 directly.

53²

53 is 3 above the round number 50.
Move 3 the other way too: 56 × 50 = 2800.
Add the square of the distance: 2800 + 3² = 2800 + 9 = 2809.

Practicebest streak 0

06 Multiply by 9, 99 or 999

47 × 9 = 470 − 47 — multiply by the next power of ten, then subtract the number back off.

34 × 99

99 is one less than 100.
34 × 100 = 3400.
Subtract 34: 3400 − 34 = 3366.

Practicebest streak 0

07 Multiply by 5, 25, 50 or 125

Halving (or quartering, or eighthing) and shifting the decimal point beats multiplying by an awkward number directly.

48 × 5

5 × 2 = 10, so 5 is 10 ÷ 2.
Divide 48 by 2 instead: 48 ÷ 2 = 24.
Multiply by 10 (shift the decimal point): 240.

Practicebest streak 0

08 Vertically and crosswise (any two 2-digit numbers)

The one method that always works, with no special condition on the numbers at all.

32 × 41

Units, straight down: 2 × 1 = 2. Write 2.
Crosswise, both ways, added: 3×1 + 2×4 = 11. Write 1, carry 1.
Tens, straight down, plus the carry: 3 × 4 + 1 = 13.
Read it off: 1312.

Practicebest streak 0

09 Quick divisibility checks

Add the digits to test 3 and 9; alternate adding and subtracting them (right to left) to test 11 — no division needed.

Is 4536 divisible by 9? Is 2728 divisible by 11?

4536: digits add to 4+5+3+6 = 18, a multiple of 9 — so 4536 is divisible by 9 (and by 3).
2728: alternating from the right, 8 − 2 + 7 − 2 = 11, a multiple of 11 — so 2728 is divisible by 11.

Practicebest streak 0

Why these work

Every shortcut here is ordinary algebra in disguise — (a + d)(a − d) = a² − d², or (10a + b)(10c + d) multiplied out term by term. They're not approximations: done correctly, every one gives the exact answer, not a rounded one.

Start with whichever trick matches the numbers in front of you (ends in 5? near 100? same leading digit?), and fall back to vertically-and-crosswise when nothing special applies — it always works.

Part of Shunya — see the Formula Sheet for everything else, or the quiz to test yourself.

Formulas used in this lab

Multiplication shortcuts

Multiply by 11 (2-digit): $\overline{a\,b} \times 11 = \overline{a\ (a+b)\ b}$
Same leading digit, units add to 10: $(10t+u)\bigl(10t+(10-u)\bigr) = 100\,t(t+1) + u(10-u)$
Nikhilam: two numbers near a base B: $(B+d_1)(B+d_2) = B(B+d_1+d_2) + d_1d_2$
Multiply by 9, 99, 999, …: $n \times (10^k - 1) = n\times 10^k - n$
Multiply by 5, 25, 50, 125: $n\times 5 = \dfrac{n}{2}\times 10,\qquad n\times 25 = \dfrac{n}{4}\times 100,\qquad n \times 125 = \dfrac{n}{8}\times 1000$
Vertically and crosswise (2-digit × 2-digit): $(10a+b)(10c+d) = 100ac + 10(ad+bc) + bd$

Squaring shortcuts

Square a number ending in 5: $(10a+5)^2 = 100\,a(a+1) + 25$
Square any number (base method): $n^2 = (n+d)(n-d) + d^2$

Divisibility checks

Divisible by 3 or 9: $n \bmod 3,\ n \bmod 9 \;=\; (\text{digit sum}) \bmod 3,\ (\text{digit sum}) \bmod 9$
Divisible by 11: $n \bmod 11 = (\text{alternating digit sum}) \bmod 11$