Where these shortcuts come from Exit
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.

Formulas

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.

Formulas

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.

Formulas

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.

Urdhva-Tiryagbhyam: vertically and crosswise, 43 × 27

Nikhilam: 97 × 96 near base 100

Shortcut: Nikhilam for subtraction

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

  1. Take every digit from 9, and the last digit from 10.
  2. 10000 - 3786: 9-3, 9-7, 9-8, 10-6 = 6214.

Nikhilam for subtraction: try it

10000 - 3786

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

Answer: 6214

Shortcut: Ekanyunena for × 99

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

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

Ekanyunena for × 99: try it

47 × 99

  1. Left 46.
  2. Right 99 - 46 = 53.

Answer: 4653

Your turn medium

100000 - 27394

  1. 9-2, 9-7, 9-3, 9-9, 10-4 = 72606.

Answer: 72,606

Your turn medium

738 × 999 (Ekanyunena)

  1. Left 737.
  2. Right 999 - 737 = 262.

Answer: 7,37,262

Your turn hard

Which sutra gives 996 × 993 fastest?

  1. Both are just below 1000.

Answer: Nikhilam (base method)

Your turn hard

Which sutra gives 752 fastest?

  1. Ends in 5: 7 × 8 | 25 = 5625.

Answer: Ekadhikena Purvena

Now practise

6 questions with full solutions.

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