Vedic Maths

Vedic Mathematics is a system of mental calculation techniques based on 16 Sūtras (aphorisms) and 13 Upa-Sūtras (corollaries). These concise Sanskrit formulae, popularised by Jagadguru Swami Bharati Krishna Tirthaji, allow rapid and elegant solutions to arithmetic, algebraic and even higher mathematical problems.

Know about History and Origins of Vedic Maths.
Click any Sūtra or Upa-Sūtra below to explore its meaning, mathematical principle, worked examples and interactive demonstration.

16 Main Sūtras

Sūtra 1

एकाधिकेन पूर्वेण

Ekādhikena Pūrveṇa

By one more than the previous one

Sūtra 2

निखिलं नवतश्चरमं दशतः

Nikhilam Navataścaramam Daśataḥ

All from 9 and the last from 10

Sūtra 3

ऊर्ध्वतिर्यग्भ्याम्

Ūrdhva-Tiryagbhyām

Vertically and crosswise

Sūtra 4

परावर्त्य योजयेत्

Parāvartya Yojayet

Transpose and adjust

Sūtra 5

शून्यं साम्यसमुच्चये

Śūnyaṃ Sāmyasamuccaye

When the sum is the same, that sum is zero

Sūtra 6

(आनुरूप्ये) शून्यमन्यत्

(Ānurūpye) Śūnyamanyat

If one is in ratio, the other is zero

Sūtra 7

सङ्कलनव्यवकलनाभ्याम्

Saṅkalana-vyavakalanābhyām

By addition and by subtraction

Sūtra 8

पूरणापूरणाभ्याम्

Pūraṇāpūraṇābhyām

By the completion or non-completion

Sūtra 9

चलनकलनाभ्याम्

Calana-kalanābhyām

Differences and similarities

Sūtra 10

यावदूनम्

Yāvadūnam

Whatever the extent of its deficiency

Sūtra 11

व्यष्टिसमष्टिः

Vyaṣṭisamaṣṭiḥ

Part and whole

Sūtra 12

शेषाण्यङ्केन चरमेण

Śeṣāṇyaṅkena Carameṇa

The remainders by the last digit

Sūtra 13

सोपान्त्यद्वयमन्त्यम्

Sopāntyadvayamantyam

The ultimate and twice the penultimate

Sūtra 14

एकन्यूनेन पूर्वेण

Ekanyūnena Pūrveṇa

By one less than the previous one

Sūtra 15

गुणितसमुच्चयः

Guṇitasamuccayaḥ

The product of the sum is the sum of the products

Sūtra 16

गुणकसमुच्चयः

Guṇakasamuccayaḥ

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

13 Upa-Sūtras (Corollaries)

Upa-Sūtra 1

आनुरूप्येण

Ānurūpyeṇa

Proportionately

Upa-Sūtra 2

शिष्यते शेषसंज्ञः

Śiṣyate Śeṣasaṃjñaḥ

The remainder remains constant

Upa-Sūtra 3

आद्यमाद्येनान्त्यमन्त्येन

Ādyamādyenāntyamantyena

The first by the first and the last by the last

Upa-Sūtra 4

केवलैः सप्तकं गुण्यात्

Kevalaiḥ Saptakaṃ Guṇyāt

For 7 the multiplicand is 143

Upa-Sūtra 5

वेष्टनम्

Veṣṭanam

By osculation

Upa-Sūtra 6

यावदूनं तावदूनम्

Yāvadūnaṃ Tāvadūnam

Lessen by the deficiency

Upa-Sūtra 7

यावदूनं तावदूनीकृत्य वर्गं च योजयेत्

Yāvadūnaṃ Tāvadūnīkṛtya Vargañca Yojayet

Lessen by the deficiency and set up the square of the deficiency

Upa-Sūtra 8

अन्त्ययोर्दशकेऽपि

Antyayordaśake’pi

When the last digits total 10

Upa-Sūtra 9

अन्त्ययोरेव

Antyayoreva

Only the last terms

Upa-Sūtra 10

समुच्चयगुणितः

Samuccayaguṇitaḥ

The sum of the products

Upa-Sūtra 11

लोपनस्थापनाभ्याम्

Lopanasthāpanābhyām

By alternate elimination and retention

Upa-Sūtra 12

विलोकनम्

Vilokanam

By mere observation

Upa-Sūtra 13

गुणितसमुच्चयः समुच्चयगुणितः

Guṇitasamuccayaḥ Samuccayaguṇitaḥ

The product of the sum is the sum of the products