Vedic Maths Sutras — All 16 Sutras Explained with Examples

What Are Vedic Maths Sutras?

The word 'Sutra' comes from Sanskrit and means a thread or aphorism — a compact, memorable formula that encapsulates a broader principle. In Vedic Maths, the 16 Sutras (and 13 Sub-Sutras) are the complete ruleset systematised by Sri Bharati Krishna Tirthaji in his work 'Vedic Mathematics' (1965). Each Sutra is a short Sanskrit phrase that, when understood in mathematical context, describes a specific computational technique.

The power of the Sutras is that each applies to multiple areas of mathematics. Urdhva-Tiryagbhyam, for example, is primarily taught as a multiplication method but also applies to polynomial multiplication, algebraic expansion, and even some division scenarios. This generality is what makes Vedic Maths a system rather than a collection of unrelated tricks.

Sutras 1–4 — The Four Most Used

1. Ekadhikena Purvena — 'By one more than the previous one.' Used for squaring numbers ending in 5 (e.g., 65² = 6×7 | 25 = 4225), and for the recurring decimal representation of fractions like 1/19, 1/29. It is one of the first Sutras students learn because the squaring application is immediately useful.

2. Nikhilam Navatascaramam Dasatah — 'All from 9, last from 10.' This is the complement method: to find the complement of any number from a base of 10, 100, 1000, etc., subtract each digit from 9 except the last which is subtracted from 10. Complement of 97 from 100: (9−9)=0 for tens, (10−7)=3 for units → complement = 03. This powers the base multiplication method (Nikhilam multiplication) covered in the tricks guide.

3. Urdhva-Tiryagbhyam — 'Vertically and crosswise.' The general multiplication Sutra, applicable to any multiplication. Products are formed by multiplying vertically (same position) and crosswise (across positions) and summing. This is the most universally applicable Sutra and handles 2-digit, 3-digit, and polynomial multiplication.

4. Paravartya Yojayet — 'Transpose and apply.' Used for division, particularly dividing by numbers close to a base. When dividing by 97, treat the divisor as 100 − 3 and apply the transposed 3 as a multiplier in the synthetic division process. Also used for solving equations where terms are transposed to the other side.

Sutras 5–8 — Equations and Special Cases

5. Shunyam Saamyasamuccaye — 'When the sum is the same, that sum is zero.' If the sum of numerator coefficients equals the sum of denominator coefficients in an equation, that common sum is a root. Example: (3x+5)/(6x+7) = (3x−5)/(6x−7) → both sides have numerator + denominator = 9x+12 and 9x+2. When sums match, 9x+something=0. Applied to equations like x+2/x+5 = x+1/x+4 — the sum of numerators and denominators both equal 2x+6, so 2x+6=0 → x=−3.

6. Anurupyena — 'Proportionality.' If the ratio of one pair of factors equals the ratio of another, proportional simplification applies. Used in multiplication when numbers have a convenient common ratio to a base. Example: 42 × 48 — these are equidistant from 45. Result = 45² − 3² = 2025 − 9 = 2016.

7. Sankalana-Vyavakalanabhyam — 'By addition and subtraction.' When two equations are set up such that addition or subtraction simplifies the system, this Sutra is applied. Used in simultaneous equations. Example: If 7x+5y=49 and 5x+7y=43, adding gives 12(x+y)=92 → x+y=7.67. Subtracting gives 2(x−y)=6 → x−y=3. Solving: x=5.33, y=2.33.

8. Puranapuranabhyam — 'By completion or non-completion.' The method of completing the square, extended to polynomials and equations. Used to factorise polynomials and solve quadratics by finding the completing terms that make a perfect square.

Sutras 9–12 — Roots, Verification and Remainders

9. Calana-Kalanabhyam — 'Sequential motion' or 'differential calculus.' Applied to finding roots of polynomials iteratively — if f(a)=0 is a root, Newton's method-style iteration using this Sutra can find the next root. Also has applications in calculus-related problems.

10. Yavadunam — 'Whatever the extent of its deficiency.' Used for finding squares and cubes of numbers near a base. For 97² near base 100: deficiency is 3. Left part: 97−3=94. Right part: 3²=09. Answer: 9409. For cubes near a base, the method extends to include the cube of the deficiency.

11. Vyashtisamashtih — 'Part and whole.' When a polynomial can be expressed as a product of factors where one is the sum (whole) and the other is part of the expression. Used in factorisation.

12. Shesanyankena Charamena — 'The remainders by the last digit.' Used in converting fractions to decimals by tracking the remainders in the division process, particularly for recurring decimals. If you divide 1 by 7 (1/7 = 0.142857...), the Sutra describes the cyclic nature of the remainders.

Sutras 13–16 — Polynomial and Advanced Applications

13. Sopaantyadvayamantyam — 'The ultimate and twice the penultimate.' Applied to equations of the form 1/(a×b) + 1/(b×c) = 1/(c×d). The Sutra provides a shortcut for solving certain types of algebraic fraction equations that appear in higher mathematics.

14. Ekanyunena Purvena — 'By one less than the previous one.' The complement of Ekadhikena Purvena — where the first Sutra adds one more, this subtracts one less. Used for multiplication by all-9s numbers (e.g., 99, 999): n × 999 = n × 1000 − n, where the method produces the result directly. Example: 456 × 999 = 456000 − 456 = 455544.

15. Gunitasamuchyah — 'The product of the sum is the sum of the products.' Used as a check for polynomial multiplication and factorisation. The digit sum of the product of two numbers should equal the digit sum of the product of their digit sums — a verification principle similar to the casting-out-nines check.

16. Gunakasamuchyah — 'The factors of the sum is the sum of the factors.' Applied to verify factorisation results. If a polynomial has been factorised, substituting x=1 into the original and both factors should give consistent results. This Sutra formalises the numerical verification method for algebraic work.

Which Sutras to Learn First — A Practical Sequence

For students in Class 8–12 preparing for board exams and competitive exams, the recommended learning sequence by practical value is: First priority (highest exam impact): Sutras 2 (Nikhilam — base multiplication), 3 (Urdhva-Tiryagbhyam — general multiplication), 1 (Ekadhikena Purvena — squaring ending in 5), and 10 (Yavadunam — squaring near a base).

Second priority: Sutras 4 (Paravartya — division), 6 (Anurupyena — proportional multiplication), 14 (Ekanyunena — multiplication by 9s), and 15 (Gunitasamuchyah — verification). Third priority (advanced): Sutras 5, 7, 8 for equation solving; Sutras 9, 11, 12, 13, 16 for polynomial and higher mathematics applications. Most students in Class 8–10 benefit enormously from just Sutras 2, 3, 1, and 10 — mastering all 16 is a goal for serious Vedic Maths practitioners, not a prerequisite for exam success.