Speed maths

Where these shortcuts come from

Most of these tricks have names. The Indian set is called Vedic Mathematics: 16 short Sanskrit rules (sutras) published by Bharati Krishna Tirthaji in 1965. The German-Swiss set is the Trachtenberg system. Knowing the name tells students it is a real, general method, not a one-off party trick.

Practise (6 questions) Present as slides

The idea

Formula summary

Urdhva-Tiryagbhyam - 'vertically and crosswise'

Criss-cross multiplication of ANY two numbers. Each answer column = sum of the digit pairs whose places add up to that column.

Nikhilam Navatashcaramam Dashatah - 'all from 9 and the last from 10'

The base method for numbers near 10, 100, 1000. Also instant subtraction from a power of 10: 1000 - 357 → 9-3, 9-5, 10-7 = 643.

Ekadhikena Purvena - 'by one more than the one before'

Squares ending in 5: 85² → 8 × (8+1) | 25 = 7225.

Antyayor Dashake'π - 'last digits adding to 10'

63 × 67: same front, units add to 10 → 6 × 7 | 3 × 7 = 4221.

Yavadunam - 'whatever the deficiency'

Squares near a base: 96² → 96 - 4 = 92 | 4² = 16 → 9216.

Anurupyena - 'proportionately'

Working base: for numbers near 50 use 100/2, near 300 use 100 × 3.

Ekanyunena Purvena - 'by one less than the one before'

Times 9, 99, 999: 47 × 99 → 47 - 1 = 46 | 99 - 46 = 53 → 4653.

Trachtenberg system (Jakow Trachtenberg, 1960)

Digit-by-digit rules for x11, x12, x5, x9. The 'add the neighbour' rule for x11 is his.

Lattice (gelosia) multiplication

Medieval grid method, brought to Europe by Fibonacci. The same partial products as criss-cross, drawn in a grid.

Casting out nines

Centuries-old check: digit sums must agree. Catches most slips in 3 seconds.

Watch it step by step

Urdhva-Tiryagbhyam: vertically and crosswise, 43 × 27

Nikhilam: 97 × 96 near base 100

Shortcuts and tricks

Nikhilam for subtraction

Use it when: Any 'power of ten minus a number'. Change counters, bills, 10000 - x.

Take every digit from 9, and the last digit from 10.
10000 - 3786: 9-3, 9-7, 9-8, 10-6 = 6214.

Example10000 - 3786

9-3 = 6, 9-7 = 2, 9-8 = 1, 10-6 = 4.

Answer: 6214

Ekanyunena for × 99

Use it when: Multiplying by 9, 99, 999 with the same number of digits.

Left: the number minus 1.
Right: the 9s minus the left part.

Example47 × 99

Left 46.
Right 99 - 46 = 53.

Answer: 4653

Honest footnote for curious students: historians point out these sutras are not found in the actual Vedas; Tirthaji wrote them in the 20th century. The arithmetic is correct either way - every method here is ordinary algebra, which is why it always works.

Practice questions (6)

Try each one before opening the solution. Or practise them one by one so your score is saved.

easy 1000 - 478 (Nikhilam)

9-4, 9-7, 10-8.

Answer: 522

medium 100000 - 27394

9-2, 9-7, 9-3, 9-9, 10-4 = 72606.

Answer: 72,606

medium 738 × 999 (Ekanyunena)

Left 737.
Right 999 - 737 = 262.

Answer: 7,37,262

hard Which sutra gives 996 × 993 fastest?

Both are just below 1000.

Answer: Nikhilam (base method)

hard Which sutra gives 75² fastest?

Ends in 5: 7 × 8 | 25 = 5625.

Answer: Ekadhikena Purvena

hard Which sutra gives 58 × 52 fastest?

Same front 5, units 8 + 2 = 10: 5 × 6 | 16 = 3016.

Answer: Antyayor Dashake'π

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