# 16 Vedic Maths Sutras:- English Translation and Mathematical Meaning

The Vedic Mathematics Sutras were used by ancient scholars to make mathematical calculations faster when calculators and computers were not available. They consist of one-line Vedic sutras. which can enrich one's computational skills using formulas that provide a new way of resolving problems relating to Arithmetic calculations and a few Algebraic methods.

The name "VEDIC MATHEMATICS" was given by the late Swami Shri Bharati Krishna Tirthaji Maharaj (1884-1960). Also known as "The Father of Vedic Maths". An outstanding scholar of Mathematics, History, Sanskrit, Philosophy, and English. It took him 8 long years in deep silence to study these texts called "Ganit Sutras" in the forests of Shingeri. As a result of his deep meditation, he collected the lost formulae from the Atharva Vedas and wrote them down as 16 sutras or formulae or Vedic tricks, followed by 13 sub-sutras + 3 (corollary), which derived from these sutras.

## Vedic Mathematics sutras are divided into 2 categories:-

### 1- Vedic Sutras that make Arithmetic computation easy. 2- Vedic Sutras that makes Algebra easy.

With the help of these sutras or tricks, we can effortlessly calculate Arithematic computations like Addition, Subtraction, Multiplication, Division, Fractions, HCF/LCM, Squares, cubes, and its roots etc... Along with a few Algebraic computations like Algebratic multiplication, Factorizing quadratic equations, Linear equations, Simultaneous equations, etc.. In addition, these sutras help us dodge times tables up to 99 very easily, the multiplication tables tricks are time-saving. We have Vedic tricks for calculating the 500-year calendar, which aids in mental calculations.

Below are the names and English translations of 16 Vedic sutras and 16 sub-sutras (corollary), and additionally, I have given the Main Vedic Sutras Names which are being followed with English Translation, and details on Mathematical Operations wherever applicable are mentioned clearly.

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**The Main 16 Vedic Sutras and its English Translation:- **

**Ekadhikena Purvena:-**By One More than the Previous One.**Nikhilam Navatashcaramam Dashatah:-**All From Nine, and Last From Ten.**Urdhva - Tiryagbhyam:-**Vertically and Crosswise.**Yavadhunam:-**Whatever the Extent of the Deficiency.**Paraavartya Yojayet:-**Transpose and Apply.**Ekanyunena Purvena:-**By One Less than the Previous One.**Vyasti-Samasti:-**Individuality and Totality.**Gunakasamuccaya Samucchaya-Gunaka:-**Total of expression equals the total in the product.**Gunitasamuchyah:-**The product of the sum**Shunyam Saamyasamuccaye:-**Whenever the expression is the same, that expression is Zero.**Anurupye Sunyamanyat:-**If one is in ratio, the other is 0.**Sankalana Vyavakalanabhyam:-**By Addition and By Subtraction.**Purana-apurnabhyam:-**The completion and non-completion**Chalana-Kalanabhyam:-**Differential Calculus**Sopaantyadvayamantyam:-**The Ultimate and the Twice the Penultimate.**Shesanyankena Charamena:-**The Reminders by the Last Digit.

## The Corollary of the Sub-Sutras:-

**Anurupyena:-**Proportionately**Sisyate Sesasamjnah:-**The Remainder Remains Constant**Adyamadyenantyamantyena:-**The First by the First and the Last by the Last**Kevalaih Saptakam Gunyat:-**For 7 the multiplicand is 143**Vestanam:-**By Osculation**Yavadunam Tavadunam:-**Lessen by the Deficiency**Yavadunam Tavadunikritya Vargancha Yojayet:-**Whatever the Deficiency lessen by that amount and set up the square of the deficiency.**Antyayordashake’pi:-**Last Totalling 10**Antyayoreva Differences and Similarities:-**Only the Last Terms**Samuccayagunitah:-**The Sum of the Products**Lopanasthapanabhyam:-**By Alternative Elimination and Retention**Vilokanam:-**By Mere Observation**Gunitasamuccayah Samuccayagunitah:-**The Product of the Sum is the Sum of the Products**Dhvajanka:-**On the Flag**Dwandwa Yoga:-**Duplex combination**Adyam Antyam Madhyam:-**The Factors of the Sum is Equal to the Sum of Factors.

## The following is a frequently used list of the Vedic Sutras with Names, Translations, and Mathematical Applications that are used in Arithmetic and Algebra:-

### Nikhilam Navatascaramam Dasatah:- All from nine and last from ten.

Mathematical Application:- Complements, Subtraction, Multiplication by 9 and base method, Division by 9 and numbers less than the base, Squares, Cubes.

To find the complement (deficiency) of any number to the next base (nearest base, also called working base), subtract all the other digits (except last) from 9 and the last digit from 10.

### Antyayoreva:- Only the last 2 digits (absolute term).

Mathematical Application:- Multiplication by 11 and also used in Algebra.

Consider the equations, where the numerator and denominator on one side barring the independent terms have the same ratio to each other as the entire numerator and the entire denominator to another side. Then the ratio is equal to the ratio of the absolute term.

### Sopantyadvayamantyam:- The ultimate and twice the penultimate.

Mathematical Application:- Multiplication by 12 and also used in Algebra.

If factors A, B, C, D are in arithmetic progression and we have the equation of the form

1/AB + 1/AC = 1/AD + 1/BC then D + 2C = 0 gives the solution.

### Ekanyunena Purvena:- By one less than the one before.

Mathematical Application:- Multiplication by 9.

### Anurupyena:- Proportionately

Mathematical Application:- Multiplication by working base and cubes, Factorising quadratics.

### Urdhvatiryagbhyam:- Vertically and crosswise.

Mathematical Application:- Multiplication, Division, Fractions, Equations, and Squares.

i- Vertical product of numbers I

ii- Cross product of numbers X

iii- Vertical and cross product of numbers *

### Paravartya Yojayet:- Transpose and apply.

Mathematical Application:- Division of numbers, more than the base, Equations.

The deficiency of any number to the previous base (nearest base) is surplus. This deficiency (surplus) is here a negative number and surplus is a positive.

Surplus = number - base

Deficiency = base - number.

### Ekadhikena Purvena:- By one more than the one before.

Mathematical Application:- Squares of numbers ending in 5, Osculators, Subtraction.

### Antyayordasakepi:- Last totalling to 10.

Mathematical Application:- Multiplication of numbers whose units add to 10.

### Yavadunam Tavadunikrtya Vargancha Yojayet:- Whatever the extent of its deficiency lessen by that amount and set the square of the deficiency.

Mathematical Application:- Squares of numbers close to the base.

### DwandwaYoga:- Duplex combination.

Mathematical Application:- Squares of all numbers.

### Yavadunam:- By the deficiency.

Mathematical Application:- Cubes of numbers close to the base.

### Vilokanam:- By mere observation.

Mathematical Application:- Square roots and Cube roots of exact squares and cubes.

**Dvajanka:-** On top of the flag.

Mathematical Application:- Straight division.

**Sankalana Vyavakalanabhyam:-** By addition and subtraction.

Mathematical Application:- Alternate remainders, simultaneous linear equation.

In simultaneous linear equations, if x - coefficients and y - coefficients are found interchanged, to solve the equations, add the equations and also subtract the equations.

**Vestanam:-** By Osculation.

Mathematical Application:- Checking divisibility by prime numbers.

**Adyamadyenantyamantyena:- **The First by the first and the last by last.

Mathematical Application:- Factorizing quadratics, Algebraic multiplication.

**Lopanasthapanabhyam:-** By alternate elimination and retention.

Mathematical Application:- Factorizing Harder Quadratics.

In factorization, the expression required are retained and those that are not needed are neglected.

**Shunyam Saamyasamuccaye:-** If the samuccaya is the same, it is zero.

Mathematical Application:- Linear Equations

**Anurupye Shunyamanyat:-** If one is in ratio, the other is zero.

Mathematical Application:- Simultaneous Linear Equations.

In simultaneous linear equations, if the ratio of the coefficients of one of the unknown quantities is the same as that of independent terms. The other unknown quantities are equal to zero.

**Conclusion:-
**

By now, I hope you understand that Vedic mathematics is a collection of sutras that have numerous benefits. During our research, we were able to come up with easy ways to remember and understand all the sutras and sub-sutras, which can assist you when solving math problems. Although it is not that easy to remember the sutra names, once you get started practicing the topics and analyzing them, and solving the sums using interesting Vedic maths tricks, you will grow to love them. They are easy to understand once you start implementing them. The English translation and mathematical meaning are the best parts for all ages to understand. As soon as you start applying these maths formulae you will love math and your math phobia will disappear.

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