The 16 sutras
Each word-formula and what it does
The system is built on 16 short ‘sutras’ (word-formulas) and 13 sub-sutras. Below is each sutra, its plain-English meaning, and where you meet it in practice. These methods are in the public domain; the wording and examples here are our own.
| Sutra & meaning | Where it is used |
|---|---|
| 1. Ekadhikena Purvena By one more than the one before | Squaring numbers ending in 5: 25² → 2×3|25 = 625. |
| 2. Nikhilam Navatashcaramam Dashatah All from 9 and the last from 10 | Fast subtraction and near-base multiplication. |
| 3. Urdhva-Tiryagbhyam Vertically and crosswise | General multiplication of any two numbers. |
| 4. Paravartya Yojayet Transpose and adjust | Division and solving linear equations. |
| 5. Shunyam Samyasamuccaye When the sum is the same, it is zero | Solving certain equations by inspection. |
| 6. Anurupye Shunyamanyat If one is in ratio, the other is zero | Special simultaneous equations. |
| 7. Sankalana-Vyavakalanabhyam By addition and by subtraction | Solving simultaneous equations quickly. |
| 8. Puranapuranabhyam By completion and non-completion | Completing shapes/expressions, e.g. quadratics. |
| 9. Chalana-Kalanabhyam Differences and similarities | Calculus-style step reasoning. |
| 10. Yavadunam By the deficiency | Squares of numbers near a base. |
| 11. Vyashtisamashtih Part and whole | Moving between specific and general. |
| 12. Shesanyankena Charamena Remainders by the last digit | Recurring decimals and divisibility. |
| 13. Sopantyadvayamantyam Ultimate and twice the penultimate | Certain algebraic identities. |
| 14. Ekanyunena Purvena By one less than the one before | Multiplying by a row of nines (e.g. ×99). |
| 15. Gunitasamuchyah The product of the sum | A factorisation check. |
| 16. Gunakasamuchyah The factors of the sum | A companion factorisation check. |
How to actually learn them
Do not memorise all 16 at once
- Start with three that give instant wins: Ekadhikena (squares ending in 5), Nikhilam (near-base) and Urdhva-Tiryagbhyam (general multiplication).
- Learn a sutra by using it on 10–15 problems, not by reciting the Sanskrit.
- Add the next sutra only once the previous one is automatic.
- Keep the goal in mind: understanding first, speed second.
Vedic Maths — Veda Topper
The 16 Vedic Maths Sutras (and 13 Sub-Sutras)
All of Vedic Mathematics is built on 16 short word-formulae called sutras and 13 supporting sub-sutras. Here is the complete list with plain-English meanings and what each one is used for — plus links to step-by-step tutorials.
Learn all of this live. Veda Topper runs online Vedic Maths classes for kids and adults, worldwide. Book a free demo class → · WhatsApp us →
The 16 Sutras
| Sutra | Meaning | Mainly used for |
|---|---|---|
| 1. Ekadhikena Purvena | By one more than the one before | Squaring numbers ending in 5; certain recurring decimals |
| 2. Nikhilam Navatashcaramam Dashatah | All from 9 and the last from 10 | Subtraction from a base; base multiplication |
| 3. Urdhva-Tiryagbhyam | Vertically and crosswise | General multiplication of any two numbers |
| 4. Paravartya Yojayet | Transpose and adjust | Division and solving equations |
| 5. Shunyam Saamyasamuccaye | If the Samuccaya is the same, it is zero | Solving special equations |
| 6. Anurupye Shunyamanyat | If one is in ratio, the other is zero | Simultaneous equations |
| 7. Sankalana-Vyavakalanabhyam | By addition and by subtraction | Simultaneous equations |
| 8. Puranapuranabhyam | By the completion or non-completion | Factorising and quadratics |
| 9. Chalana-Kalanabhyam | Differences and similarities | Calculus and equations |
| 10. Yavadunam | Whatever the extent of its deficiency | Squaring numbers near a base |
| 11. Vyashtisamashtih | Part and whole | Averages and factorisation |
| 12. Sheshanyankena Charamena | The remainders by the last digit | Recurring decimals |
| 13. Sopantyadvayamantyam | The ultimate and twice the penultimate | Special divisions |
| 14. Ekanyunena Purvena | By one less than the one before | Multiplying by a row of 9s |
| 15. Gunitasamuchyah | The product of the sum | Checking factors of expressions |
| 16. Gunakasamuchyah | All the multipliers | Factorisation checks |
The 13 Sub-Sutras
| Sub-Sutra | Meaning |
|---|---|
| 1. Anurupyena | Proportionately |
| 2. Shishyate Sheshasamjnah | The remainder remains constant |
| 3. Adyamadyenantyamantyena | First by first and last by last |
| 4. Kevalaih Saptakam Gunyat | For 7 the multiplier is 143 |
| 5. Vestanam | By osculation |
| 6. Yavadunam Tavadunam | Lessen by the deficiency |
| 7. Yavadunam Tavadunikritya Varganca Yojayet | Square the deficiency and add |
| 8. Antyayordashake\u2019pi | When the last digits total 10 |
| 9. Antyayoreva | Only the last terms |
| 10. Samuccayagunitah | The sum of the products |
| 11. Lopanasthapanabhyam | By elimination and retention |
| 12. Vilokanam | By observation |
| 13. Gunitasamuccayah Samuccayagunitah | The product of the sum is the sum of the products |
Learn them with worked examples
Reading the sutras is one thing; using them is another. Start with these hands-on tutorials, each with step-by-step examples and practice:
- Multiply by 11 in one line
- Square any number ending in 5 (Ekadhikena Purvena)
- Base multiplication near 100 (Nikhilam)
- Vertically and crosswise (Urdhva-Tiryagbhyam)
- Square numbers near a base (Yavadunam)
- Divide by 9 instantly
- Subtract from 100/1000 (all from 9)
More Vedic Maths tricks
← Vedic Maths home Multiply by 11 in one lineSquare numbers ending in 5Base multiplication (Nikhilam)Vertically & crosswiseSquares near a base (Yavadunam)Instant division by 9Subtraction: all from 9
Frequently asked questions
How many sutras are there in Vedic Maths?
There are 16 main sutras (word-formulae) and 13 sub-sutras that support them, covering multiplication, division, squares, roots, fractions, algebra and more.
Who compiled the 16 Vedic Maths sutras?
They were presented by Swami Bharati Krishna Tirtha in his book on Vedic Mathematics, and are now taught worldwide as fast mental-calculation methods.
Which sutra is the most useful for beginners?
Ekadhikena Purvena (squaring numbers ending in 5) and Nikhilam (subtraction and base multiplication) are the easiest wins for beginners.
Learn all of this live. Veda Topper runs online Vedic Maths classes for kids and adults, worldwide. Book a free demo class → · WhatsApp us →