Parents ask us about the Trachtenberg method and Vedic maths in the same breath, usually because a child has seen a video of someone multiplying four-digit numbers in their head and wants to know the trick. Both systems are real, both work, and both are sold with more mystique than they need. This post does the same multiplications both ways so you can see exactly what each one is, where each is faster, where each falls apart, and which parts are worth teaching a child of 8 to 12. Every example is worked in full; check them with a calculator if you like, that is rather the point.
What the two systems are
The Trachtenberg system was worked out by Jakow Trachtenberg, a Russian-born engineer, while he was held in Nazi camps in the 1940s, and published as The Trachtenberg Speed System of Basic Mathematics in 1960. Its heart is a set of rules for multiplying any number by each of 2 to 12, working from the right, one digit at a time, using only the digit, its right-hand neighbour and a carry. There is also a general method for any multiplier and methods for division and squaring, but the by-2-to-12 rules are what people mean by "Trachtenberg".
Vedic mathematics is a set of sixteen sutras (short rules) published in 1965 by Bharati Krishna Tirthaji, an Indian monk and scholar, in a book of the same name. The most used are Urdhva Tiryagbhyam (vertically and crosswise, a general multiplication), Nikhilam (multiplying numbers near a base such as 10, 100 or 1 000), and Ekadhikena Purvena (one more than the previous one, for squares ending in 5). The sutras cover more ground than Trachtenberg, including division, square roots and some algebra, but the multiplication shortcuts are what schools and coaching centres teach.
Multiplying by 11: both ways
Trachtenberg rule for 11: write a 0 in front of the number. Working from the right, each answer digit is the digit plus its right-hand neighbour.
3 254 × 11: write 0 3 2 5 4. - 4 has no neighbour: 4 - 5 + 4 = 9 - 2 + 5 = 7 - 3 + 2 = 5 - 0 + 3 = 3
Answer: 35 794.
Vedic (vertically and crosswise), treating 11 as 1 and 1: the same operations appear as "write the first digit, add each pair of neighbours, write the last digit", which for 11 is identical to the Trachtenberg rule. 43 × 11: 4, then 4 + 3 = 7, then 3: 473.
For 11 the two systems are the same trick; Trachtenberg's leading 0 just makes the carry tidier when a pair adds to more than 9 (67 × 11: 7, 6 + 7 = 13 write 3 carry 1, 0 + 6 + 1 = 7: 737).
Multiplying by 12: where Trachtenberg shines
Trachtenberg rule for 12: double each digit and add its right-hand neighbour, from the right, with a 0 in front.
3 254 × 12: write 0 3 2 5 4. - 4 × 2 = 8, no neighbour: 8 - 5 × 2 + 4 = 14: write 4, carry 1 - 2 × 2 + 5 + 1 = 10: write 0, carry 1 - 3 × 2 + 2 + 1 = 9: 9 - 0 × 2 + 3 = 3
Answer: 39 048. Check: 3 254 × 10 = 32 540, plus 3 254 × 2 = 6 508, total 39 048.
Vedic vertically and crosswise, 32 × 12 (two digits, to keep it readable): - Right: 2 × 2 = 4 - Cross: 3 × 2 + 2 × 1 = 6 + 2 = 8 - Left: 3 × 1 = 3
Answer: 384. The same by Trachtenberg: 0 3 2 → 2 × 2 = 4; 3 × 2 + 2 = 8; 0 × 2 + 3 = 3: 384.
Both get there. The difference is what the child holds in their head: Trachtenberg holds one digit, one neighbour and a carry; vertically-and-crosswise holds two products at once in the middle step, three in the middle of a three-digit product. For a multiplier of 12, Trachtenberg is lighter.
Multiplying by 6: the rule that puts people off Trachtenberg
Trachtenberg rule for 6: to each digit add half of its right-hand neighbour (ignore remainders); if the digit itself is odd, add 5 as well.
3 254 × 6: write 0 3 2 5 4. - 4: even, no neighbour: 4 - 5: odd, so + 5; half of 4 is 2: 5 + 5 + 2 = 12: write 2, carry 1 - 2: even; half of 5 is 2 (ignore the half); 2 + 2 + 1 = 5 - 3: odd, so + 5; half of 2 is 1: 3 + 5 + 1 = 9 - 0: half of 3 is 1: 1
Answer: 19 524. Check: 3 254 × 6 = 19 524.
It works every time, and it is genuinely fast once learned, but a child has to memorise a different rule for each of 5, 6, 7, 8 and 9, and the "add 5 if odd" rule has no visible reason at the child's level. This is the honest cost of Trachtenberg: a dozen rules, learned by rote, that feel like magic because the reason (each rule is long multiplication rearranged) is hidden.
Numbers near a base: where Vedic wins outright
Nikhilam, 97 × 96: both numbers are close to 100. Write the deficits: 97 is 3 below, 96 is 4 below. - Left part: 97 − 4 (or 96 − 3, same thing) = 93 - Right part: 3 × 4 = 12
Answer: 9 312. Check: 97 × 96 = 9 312.
Squares ending in 5, Ekadhikena Purvena, 65²: multiply the tens digit by one more than itself, then write 25. 6 × 7 = 42, then 25: 4 225.
There is no Trachtenberg equivalent of either; a child would multiply 97 × 96 by the general method, which is slower than column multiplication. For anything near 10, 100 or 1 000, and for squares of numbers ending in 5, Vedic is the faster system by a wide margin.
Where each one breaks down
| Trachtenberg | Vedic | |
|---|---|---|
| Fastest for | Any number × 2 to 12 | Numbers near a base; two-digit × two-digit; squares ending in 5 |
| Slowest for | Two-digit × two-digit with neither near a base (the general method) | Long numbers by long numbers (vertically-and-crosswise has too many middle terms) |
| Rules to memorise | About 12, one per multiplier, unrelated to each other | 3 or 4 sutras cover most school use; the full 16 are for enthusiasts |
| Shows the child why | No; the rules are opaque at primary level | Partly; vertically-and-crosswise is visibly the distributive law |
| Fits the school method | Yes for times tables 2 to 12; no for column multiplication | Vertically-and-crosswise is column multiplication rearranged, so yes |
| Risk | Child learns the rule and cannot do 7 × 8 without it | Child learns tricks for special cases and slows down on ordinary ones |
Which one to teach a child, and when
For a child of 8 to 10 whose times tables are not yet automatic: neither. Both systems assume the tables, and a shortcut on top of a gap makes the gap permanent. Get the tables to 12 automatic first; ten minutes a day for a term.
For a child of 8 to 10 whose tables are automatic: the Trachtenberg rules for 11 and 12 (two rules, both obvious, both used at school), and the Vedic vertically-and-crosswise method for two-digit by two-digit, taught alongside column multiplication so the child sees they are the same thing written differently. That is a fortnight's work and it is the part of both systems that a Year 5 NAPLAN or a Grade 5 state test rewards.
For a child of 10 to 12 who enjoys mental maths: add Nikhilam and the squares-ending-in-5 rule, and the Trachtenberg rules for 5, 6 and 7 if the child likes the challenge. Treat it as a puzzle, not a syllabus.
For any child: never let the trick replace the question "why does this work". We teach the Vedic vertically-and-crosswise method by showing 23 × 41 as (20 + 3)(40 + 1) first, and the Trachtenberg by-12 rule as "10 times plus 2 times", so the shortcut sits on top of understanding. Whether a child needs abacus, Vedic maths or just the school method is a separate question we answered in abacus, Vedic maths or school maths.
A drill sheet
Do these both ways, then check with a calculator. Trachtenberg by 11 and 12: 4 231 × 11, 5 678 × 11, 2 143 × 12, 6 789 × 12. Vedic vertically and crosswise: 23 × 41, 34 × 52, 67 × 89. Nikhilam: 98 × 97, 94 × 93, 996 × 995. Squares ending in 5: 35², 75², 115². Answers: 46 541, 62 458, 25 716, 81 468; 943, 1 768, 5 963; 9 506, 8 742, 991 020; 1 225, 5 625, 13 225.
What we do
Our maths classes are one tutor with one child, mapped to the school's curriculum, and mental-maths shortcuts are taught where they help the child at school, not as a separate course. If your child has seen the videos and wants the tricks, bring them to a free 30-minute demo class and we will show which ones to keep.
Questions parents ask
1Is the Trachtenberg method better than Vedic maths?
2Which should a child learn first?
3Does the Trachtenberg method work for any number?
4Is Vedic maths really from the Vedas?
See how we teach this, 1-on-1 online →
Shobha
Founder and lead tutor, Science with Shobha
Teaching maths, science and English to children online since 2018. 500+ students, 20+ tutors, families in nine countries.