The Sūtras

Every technique includes its Sanskrit name, English meaning, when to use it, and at least one fully worked example.

Main Sūtras

Sūtra 1

Ekādhikena Pūrveṇa

By one more than the previous one

Squaring numbers ending in 5, and converting fractions like 1/19, 1/29.

Sūtra 2

Nikhilaṁ Navataścaramaṁ Daśataḥ

All from 9 and the last from 10

Multiplication of numbers near a power of 10 (like 98 × 97).

Sūtra 3

Ūrdhva-Tiryagbhyām

Vertically and crosswise

General multiplication for any two numbers — the 'Vedic multiplication' people know.

Sūtra 4

Parāvartya Yojayet

Transpose and adjust

Division by numbers slightly above a power of 10, and solving simple equations.

Sūtra 5

Śūnyaṁ Sāmyasamuccaye

When the sum is the same, that sum is zero

Solving equations where a particular sum equals zero.

Sūtra 6

Ānurūpyeṇa Śūnyamanyat

If one is in ratio, the other is zero

Used in solving simultaneous equations when coefficients of one variable are in ratio.

Sūtra 7

Saṅkalana-Vyavakalanābhyām

By addition and by subtraction

Solving simultaneous equations by adding and subtracting them.

Sūtra 8

Pūraṇāpūraṇābhyām

By the completion or non-completion

Completing a number to the next round figure (like making 97 into 100).

Sūtra 9

Calanā-Kalanābhyām

Differences and similarities

Used in calculus (differentiation), and in finding roots of quadratics by discriminant.

Sūtra 10

Yāvadūnam

Whatever the deficiency

Squaring numbers near a power of 10.

Sūtra 11

Vyaṣṭisamaṣṭiḥ

Part and Whole

Average / mean methods; factorising via averages.

Sūtra 12

Śeṣāṇyaṅkena Carameṇa

The remainders by the last digit

Computing recurring decimals of fractions like 1/7.

Sūtra 13

Sopāntyadvayamantyam

The ultimate and twice the penultimate

Solving certain equation types instantly.

Sūtra 14

Ekanyūnena Pūrveṇa

By one less than the previous one

Multiplying by a number consisting of all 9s.

Sūtra 15

Guṇitasamuccayaḥ

The product of the sum of coefficients

Verification of polynomial multiplication.

Sūtra 16

Guṇakasamuccayaḥ

The factors of the sum is equal to the sum of the factors

Verifying factorisation of polynomials.

Sub-Sūtras

Sub 1

Ānurūpyeṇa

Proportionately

Multiplication using a working base proportional to a power of 10.

Sub 2

Śiṣyate Śeṣasaṁjñaḥ

The remainder remains

Quick divisibility tests using remainders.

Sub 3

Ādyamādyenāntyamantyena

First by the first, last by the last

Factorising quadratics quickly.

Sub 4

Yāvadūnaṁ Tāvadūnam

Lessen by the deficiency

Cubing numbers near a base.

Sub 5

Yāvadūnaṁ Tāvadūnīkṛtya Vargaṁ ca Yojayet

What deficit, lessen by that and set up the square

Squaring near a base, expanded form of Yāvadūnam.

Sub 6

Antyayoḥ Daśake'pi

When the final digits add to 10

Multiplication when last digits sum to 10 and other digits are equal.

Sub 7

Antyayoreva

Only the last terms

Algebraic identity solutions using only end terms.

Sub 8

Lopanasthāpanābhyām

By alternate elimination and retention

Factorising expressions in two variables (HCF of polynomials).

Sub 9

Vilokanam

By mere observation

Solving by inspection.

Sub 10

Guṇitasamuccayaḥ Samuccayaguṇitaḥ

The product of the sum equals the sum of the product

Verification rule (cross-check with Sutra 15).

Sub 11

Dhvajāṅka

On the flag

Division by any number using a 'flag digit'.

Sub 12

Samuccayaguṇitaḥ

The sum of the products

Sum-of-products check for factorisations.

Sub 13

Ekanyūnena

By one less (sub-form)

Variant of Sutra 14 for general all-9 multiplication.