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Vedic Mathematics

The 16 sutras, how each method works, and an interactive trainer.

What is Vedic Mathematics?

AI-drafted · awaiting admin review

A system of 16 sutras (word-formulas) and 13 sub-sutras for fast mental calculation, presented by Jagadguru Shankaracharya Swami Sri Bharati Krishna Tirtha of Govardhan Math, Puri.

Where it comes from

Swami Sri Bharati Krishna Tirtha (1884–1960), Shankaracharya of the Govardhan Math at Puri, was a scholar of Sanskrit, mathematics, science and philosophy. After years of study of the Vedic texts, he said he had reconstructed sixteen sutras from the Atharvaveda that together cover arithmetic, algebra, geometry and calculus. His book Vedic Mathematics was published in 1965, after his passing.

How it works

Each sutra is a short Sanskrit phrase that names a way of thinking about numbers, for example 'vertically and crosswise' or 'all from 9 and the last from 10'. The same sutra applies to many kinds of problems. Most calculations can be done in one line, often mentally, and answers can be checked quickly with digit sums.

Why it matters

  • It builds speed, confidence and a feel for numbers, which helps in school and competitive exams.
  • It shows several ways to solve the same problem, encouraging flexible thinking.
  • The National Education Policy 2020 calls for bringing Indian Knowledge Systems into education, and many schools and states now teach Vedic Mathematics.

Try the trainer below: pick a method, solve the problem, and see each step.

Sources

Try it: Vedic Maths trainer

Sutra: Ekādhikena Pūrveṇa

15² =

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Methods

  • Squares of numbers ending in 5

    5 पर समाप्त होने वाली संख्याओं का वर्ग

    Ekādhikena Pūrveṇa: by one more than the previous one.

    Ekādhikena Pūrveṇa
  • Multiplying numbers near 100

    100 के पास की संख्याओं का गुणा

    Nikhilaṁ Navataścaramaṁ Daśataḥ: all from 9 and the last from 10.

    Nikhilaṁ Navataścaramaṁ Daśataḥ
  • Any multiplication: vertically and crosswise

    कोई भी गुणा: ऊर्ध्व-तिर्यक

    Ūrdhva-tiryagbhyām: the general method that works for every multiplication.

    Ūrdhva-tiryagbhyām
  • Multiplying by 9, 99, 999

    9, 99, 999 से गुणा

    Ekanyūnena Pūrveṇa: by one less than the previous one.

    Ekanyūnena Pūrveṇa
  • Squares of numbers near 100

    100 के पास की संख्याओं का वर्ग

    Yāvadūnam: whatever the extent of its deficiency.

    Yāvadūnam
  • Same tens digit, units add to 10

    दहाई समान, इकाइयों का योग 10

    Antyayor Daśake'pi (a sub-sutra): when the last digits add up to 10.

    Antyayor Daśake'pi
  • Checking answers with digit sums

    अंकों के योग से उत्तर की जाँच

    Guṇitasamuccayaḥ: the product of the sums equals the sum of the product.

    Guṇitasamuccayaḥ

The 16 sutras

#SutraMeaning
1एकाधिकेन पूर्वेणEkādhikena PūrveṇaBy one more than the previous one
2निखिलं नवतश्चरमं दशतःNikhilaṁ Navataścaramaṁ DaśataḥAll from 9 and the last from 10
3ऊर्ध्वतिर्यग्भ्याम्Ūrdhva-tiryagbhyāmVertically and crosswise
4परावर्त्य योजयेत्Parāvartya YojayetTranspose and adjust
5शून्यं साम्यसमुच्चयेŚūnyaṁ SāmyasamuccayeWhen the sum is the same, that sum is zero
6(आनुरूप्ये) शून्यमन्यत्(Ānurūpye) ŚūnyamanyatIf one is in ratio, the other is zero
7संकलनव्यवकलनाभ्याम्Saṅkalana-vyavakalanābhyāmBy addition and by subtraction
8पूरणापूरणाभ्याम्PūraṇāpūraṇābhyāmBy completion or non-completion
9चलनकलनाभ्याम्Calana-kalanābhyāmDifferences and similarities
10यावदूनम्YāvadūnamWhatever the extent of its deficiency
11व्यष्टिसमष्टिःVyaṣṭisamaṣṭiḥPart and whole
12शेषाण्यङ्केन चरमेणŚeṣāṇyaṅkena CarameṇaThe remainders by the last digit
13सोपान्त्यद्वयमन्त्यम्SopāntyadvayamantyamThe ultimate and twice the penultimate
14एकन्यूनेन पूर्वेणEkanyūnena PūrveṇaBy one less than the previous one
15गुणितसमुच्चयःGuṇitasamuccayaḥThe product of the sum equals the sum of the product
16गुणकसमुच्चयःGuṇakasamuccayaḥThe factors of the sum equal the sum of the factors

India's mathematical heritage

AI-drafted · awaiting admin review

From the geometry of the Sulba Sutras to zero, the decimal system, infinite series and Ramanujan.

  1. c. 800–500 BCE

    The Sulba Sutras of Baudhayana, Apastamba and Katyayana give exact constructions for fire altars, including the rule for the diagonal of a rectangle (later called the Pythagorean theorem) and a value of √2 correct to five decimal places.

  2. c. 3rd–2nd century BCE

    Pingala's Chandahshastra counts poetic metres with a method that resembles binary numbers and arranges results in the Meru-prastara (later known in Europe as Pascal's triangle).

  3. 499 CE

    Aryabhata's Aryabhatiya uses place value, gives π ≈ 3.1416, a table of sines, and explains the rotation of the Earth.

  4. 628 CE

    Brahmagupta's Brahmasphutasiddhanta gives the rules of arithmetic with zero and negative numbers, and formulas for cyclic quadrilaterals.

  5. c. 825 CE

    Al-Khwarizmi writes on calculation with the Hindu numerals; through the Arab world, the Indian decimal system with zero reaches Europe.

  6. 1150 CE

    Bhaskaracharya (Bhaskara II) writes Lilavati and Bijaganita. The chakravala method solves equations that Europe tackled only centuries later.

  7. c. 1340–1425

    Madhava of Sangamagrama founds the Kerala school, finding infinite series for π, sine and cosine about 250 years before they appeared in Europe.

  8. 1887–1920

    Srinivasa Ramanujan transforms number theory and infinite series. India celebrates his birthday, 22 December, as National Mathematics Day.

Sources

Pages marked “AI-drafted” were written with AI from the official sources listed on each page and have not yet been checked by our team. Spot an error? Click the ✎ next to any entry, or “Suggest a correction” at the top of its page.