make me smarter Play

Vedic maths, explained

Vedic maths is a set of methods for working sums out in your head, built around 16 short Sanskrit rules called sutras. Some are shortcuts for special numbers, like squaring 75 or multiplying 97 by 96; others, like vertically and crosswise, work for any numbers. Each page below explains one method step by step, with worked examples, why it works, and questions to try.

The methods

Squares ending in 575²: 7 × 8 = 56, then 25: 5625. All from 9, the last from 101000 − 357: 9 − 3, 9 − 5, 10 − 7: 643. Multiplying near 10097 × 96: 97 − 4 = 93, 3 × 4 = 12: 9312. Squaring near 10096²: 96 − 4 = 92, 4² = 16: 9216. Vertically and crosswise23 × 41: 2 × 4 | 2 × 1 + 3 × 4 | 3 × 1: 943. Multiplying by 9, 99, 999437 × 999: 436, then 999 − 436 = 563: 436563. Last digits adding to 1063 × 67: 6 × 7 = 42, 3 × 7 = 21: 4221. Multiplying by 11253 × 11: 2 | 2 + 5 | 5 + 3 | 3: 2783. Checking with digit sums437 × 26 = 11362? 5 × 8 → 4, and 11362 → 4: it passes. Dividing by 92312 ÷ 9: 2, 2 + 3, 5 + 1: 256, remainder 6 + 2 = 8. Squaring with the duplex43²: 4² | 2 × 4 × 3 | 3² = 16 | 24 | 9: 1849.

Where it comes from

Vedic maths comes from one book, Vedic Mathematics, by Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaja (1884–1960), Shankaracharya of the Govardhan Math in Puri from 1925 until his death. He taught the methods for decades, and wrote the book in 1957 as the first volume of a planned series. Banaras Hindu University published it in 1965, five years after his death. It sets out 16 sutras and 13 sub-sutras, each a few Sanskrit words, and shows how they apply to arithmetic, algebra and more. It has been reprinted many times since, and its methods are taught in India and around the world.

In his preface, Tirthaji says the sutras come from an appendix (pariśiṣṭa) of the Atharvaveda. Historians of mathematics and Sanskrit scholars have not found them in any known Vedic text. The book's own general editor, V. S. Agrawala, wrote in his foreword that they are not in the known appendices, and that their style points to Tirthaji's own discovery. So the sutras are best seen as Tirthaji's own work, from the 20th century, called Vedic in his broad sense of the word.

The methods don't depend on that question: they are sound arithmetic, built on place value and a little algebra, and each page here shows why its method works. Some have a long history in India of their own: Śrīdhara and Mahāvīra taught cross-multiplication like vertically and crosswise more than a thousand years ago, and Indian mathematicians checked their sums by casting out nines by the 10th century.

Is it faster?

For numbers that suit a special method, like 97 × 96 or 75², yes: the answer comes in one or two small steps. The general methods, like vertically and crosswise, do the same multiplications as long multiplication in a different order; what they save is writing, so the working fits on one line or in your head. Either way, they are a good way to see how place value works.

The 16 sutras

As the book lists them, with what each says and what the book uses it for. The list at the front of the book was put together by its editors from references scattered through the text. The meanings are short English renderings of short Sanskrit phrases, and books word them in different ways.

  1. एकाधिकेन पूर्वेण Ekādhikena Pūrveṇa: “by one more than the one before”. Used for squares of numbers ending in 5; recurring decimals such as 1/19. How it works
  2. निखिलं नवतश्चरमं दशतः Nikhilaṃ Navataścaramaṃ Daśataḥ: “all from 9 and the last from 10”. Used for subtracting from 100, 1000 and so on; multiplying and dividing near a base. How it works
  3. ऊर्ध्वतिर्यग्भ्याम् Ūrdhva-tiryagbhyām: “vertically and crosswise”. Used for multiplying any two numbers; squares, with the duplex. How it works
  4. परावर्त्य योजयेत् Parāvartya Yojayet: “transpose and apply”. Used for dividing by numbers just above a base; solving equations.
  5. शून्यं साम्यसमुच्चये Śūnyaṃ Sāmyasamuccaye: “when the total (samuccaya) is the same, it is zero”. Used for solving equations of special forms.
  6. (आनुरूप्ये) शून्यमन्यत् (Ānurūpye) Śūnyamanyat: “if one is in ratio, the other is zero”. Used for simultaneous equations of a special form.
  7. संकलनव्यवकलनाभ्याम् Saṅkalana-vyavakalanābhyām: “by addition and by subtraction”. Used for simultaneous equations whose coefficients are swapped.
  8. पूरणापूरणाभ्याम् Pūraṇāpūraṇābhyām: “by completion or non-completion”. Used for completing the square or the cube to solve equations.
  9. चलनकलनाभ्याम् Calana-Kalanābhyām: “by motion and reckoning”. Used for quadratic equations, and calculus.
  10. यावदूनम् Yāvadūnam: “by the deficiency”. Used for squares and cubes of numbers near a base. How it works
  11. व्यष्टिसमष्टिः Vyaṣṭisamaṣṭiḥ: “the part and the whole”. Used for some fourth-degree equations, solved through an average.
  12. शेषाण्यङ्केन चरमेण Śeṣāṇyaṅkena Carameṇa: “the remainders by the last digit”. Used for recurring decimals such as 1/7.
  13. सोपान्त्यद्वयमन्त्यम् Sopāntyadvayamantyam: “the last and twice the second-last”. Used for special equations whose factors are in arithmetic progression.
  14. एकन्यूनेन पूर्वेण Ekanyūnena Pūrveṇa: “by one less than the one before”. Used for multiplying by 9, 99, 999 and so on. How it works
  15. गुणितसमुच्चयः Guṇitasamuccayaḥ: “the product of the sums”. Used for checking algebra: the coefficients of a product add up to the product of its factors’ coefficient sums. How it works
  16. गुणकसमुच्चयः Guṇakasamuccayaḥ: “the sum of the factors”. Used for factorising, and in calculus.

The 13 sub-sutras

  1. आनुरूप्येण Ānurūpyeṇa: “proportionately”. Used for working bases such as 50, which is 100 ÷ 2; factorising quadratics.
  2. शिष्यते शेषसंज्ञः Śiṣyate Śeṣasaṃjñaḥ: “the remainder stays the same”. Used for working bases: why the right-hand part is not scaled.
  3. आद्यमाद्येनान्त्यमन्त्येन Ādyamādyenāntyamantyena: “the first by the first and the last by the last”. Used for factorising quadratics and cubics.
  4. केवलैः सप्तकं गुण्यात् Kevalaiḥ Saptakaṃ Guṇyāt: “for 7, use 143”. Used for the recurring decimal of 1/7: in the book’s letter code “kevalaiḥ” is 143, and 143 × 999 = 142857.
  5. वेष्टनम् Veṣṭanam: “by osculation”. Used for testing whether one number divides another.
  6. यावदूनं तावदूनम् Yāvadūnaṃ Tāvadūnam: “lessen by the deficiency”. The book lists it but gives no worked example of it.
  7. यावदूनं तावदूनीकृत्य वर्गं च योजयेत् Yāvadūnaṃ Tāvadūnīkṛtya Vargañca Yojayet: “whatever the deficiency, lessen by it, and set up its square”. Used for squares of numbers near a base. How it works
  8. अन्त्ययोर्दशकेऽपि Antyayordaśake’pi: “when the last digits add up to ten”. Used for products like 63 × 67, where the last digits add up to 10. How it works
  9. अन्त्ययोरेव Antyayoreva: “only the last terms”. Used for equations with fractions of a special form.
  10. समुच्चयगुणितः Samuccayaguṇitaḥ: “the sum of the products”. The book lists it but gives no worked example of it.
  11. लोपनस्थापनाभ्याम् Lopanasthāpanābhyām: “by alternate elimination and retention”. Used for factorising expressions in several letters, and finding HCFs.
  12. विलोकनम् Vilokanam: “by mere observation”. Used for equations solved by looking at them.
  13. गुणितसमुच्चयः समुच्चयगुणितः Guṇitasamuccayaḥ Samuccayaguṇitaḥ: “the product of the sums is the sum of the products”. Used for checking multiplications and factorisations.

Practise them

Reading a method takes a minute; getting quick at it takes practice. The game’s Speed Tricks has levels for several of these methods, each with a short lesson first, free and with no sign-up.

Sources

The methods and the sutras are from Tirthaji's book. Every explanation, example and question on these pages is our own: nothing is copied from the book, or from other books, courses or websites.