Vedic Maths: 16 Powerful Vedic Maths Sutras to Unlock Effortless Math Mastery | EKADEMY | Online Courses, Live Classes, Lessons & Quizzes

Introduction to Vedic Maths

Mathematics can often seem like a daunting subject, filled with complex formulas and tedious calculations. However, there exists an ancient system of math that simplifies these processes—Vedic Maths. Rooted in the Vedas, the sacred scriptures of ancient India, this technique was rediscovered by Jagadguru Bharati Krishna Tirthaji in the early 20th century. He uncovered a set of formulas, or Sutras, that allow for faster, more efficient mathematical computations. These Sutras are not just shortcuts but deeply intuitive methods for performing arithmetic, algebra, geometry, and even calculus in a much quicker and easier manner.

Vedic Maths is a powerful tool for students and professionals alike, as it makes learning mathematics more fun, less stressful, and, most importantly, faster.

The Origin and History of Vedic Maths

The word “Vedic” comes from the Sanskrit term “Veda,” which means “knowledge.” While the Vedas themselves do not explicitly detail these mathematical techniques, Tirthaji interpreted certain mathematical ideas embedded within them. In his 1965 publication, Vedic Mathematics, Tirthaji explained the 16 core Sutras (formulas) and 13 additional Sub-Sutras (sub-formulas), which form the backbone of the system.

These techniques were not only designed to simplify calculations but were also intended to enhance the mathematical capabilities of the human mind. For centuries, Vedic Maths was largely overlooked, but since its rediscovery, it has become a major method for teaching mathematics in schools, particularly in India, and has gained global recognition.

The Core Sutras of Vedic Maths

Vedic Maths is built upon 16 Sutras (main formulas) and 13 Sub-Sutras (corollaries). These Sutras are versatile, intuitive, and can be applied to a wide variety of mathematical problems—from simple arithmetic to complex algebra.

Let’s dive deeper into some of the most useful Sutras with practical explanations:

1. Ekadhikena Purvena

(By One More Than the Previous One)

🔸 Use: Squaring numbers ending in 5
🔸 Formula: If a number ends in 5, say x5, then its square is: (x×(x+1)) followed by 25(x \times (x + 1)) \; \text{followed by} \; 25(x×(x+1))followed by25

✅ Example:
75² = ?
Take the digit before 5 → 7
Multiply 7 × 8 = 56
Attach 25 → 5625
So, 75² = 5625

2. Nikhilam Navatashcaramam Dashatah

(All from 9 and the Last from 10)

🔸 Use: Fast multiplication of numbers close to powers of 10 (e.g. 100, 1000)

✅ Example: 98 × 97
Base = 100
98 is 2 less → -2
97 is 3 less → -3
Cross-subtract: 98 – 3 = 95
Multiply: 2 × 3 = 6
Answer = 9506

✅ Why it works:
You’re subtracting from 100, using deviations to calculate both the base result and the correction.

3. Urdhva-Tiryagbhyam

(Vertically and Crosswise)

🔸 Use: General multiplication formula for numbers of any size.
This is a visual method where you multiply digits vertically and crosswise and add them accordingly.

✅ Example: 23 × 12
Break the numbers into digits:

Multiply: 3 × 2 = 6
Cross multiply and add: (2×2 + 3×1) = 4 + 3 = 7
Multiply: 2 × 1 = 2

Answer (from bottom to top): 276

4. Paravartya Yojayet

(Transpose and Adjust)

🔸 Use: Division, particularly with divisors slightly above or below a base like 10, 100.

✅ Example: Divide 1234 by 9
We transpose 9 as -1 (since 10 – 9 = 1)
Apply repeated addition/multiplication to get the answer using Vedic digit sum logic.

🧠 While this needs practice, it turns division into mental math without long division steps.

5. Shunyam Saamyasamuccaye

(When the Samuccaya is the Same, It is Zero)

🔸 Use: Solving algebraic expressions and factorization where terms cancel out.

✅ Example: (x+1)(x+2)−(x+2)(x+1)(x + 1)(x + 2) – (x + 2)(x + 1)(x+1)(x+2)−(x+2)(x+1)

Since the expressions are identical and being subtracted, the result = 0
Hence, Samuccaya (common sum) = 0

6. Anurupyena

(Proportionately)

🔸 Use: Scaling methods, used when numbers are not near base numbers like 100 or 1000 but proportionally close.

✅ Example:
Multiply 42 × 98 using base 100, but adjusting proportionately (Anurupya) because 98 is close to 100 but 42 is not. Use ratios or scaling.

7. Sankalana-Vyavakalanabhyam

(By Addition and By Subtraction)

🔸 Use: Solving simultaneous equations

✅ Example:
Solve:
x + y = 10
x – y = 2
Add: 2x = 12 → x = 6
Subtract: 2y = 8 → y = 4

8. Puranapuranabhyam

(By the Completion or Non-Completion)

🔸 Use: Completing numbers to the nearest base

✅ Example:
To find 1000 – 678 quickly, complete 678 to 1000 by adding 2 to 678 (to make 680), then 320 more → 322

9. Sopaantyadvayamantyam

(The Ultimate and Twice the Penultimate)

🔸 Use: Useful in algebra and polynomial identities

✅ Example:
In sequences like (x² + 5x + 6), this Sutra helps you spot that it’s (x + 2)(x + 3)

10. Yavadunam

(Whatever the Extent of Deficiency)

🔸 Use: Fast multiplication of numbers slightly less than a base

✅ Example:
Multiply 96 × 96
Base = 100, deficiency = 4
Square the deficiency = 16
Subtract 4 from 96 = 92
Final answer = 9216

11. Vyastisamashtih

(Part and Whole)

🔸 Use: Solving algebraic expressions and simplifying equations.
This Sutra suggests breaking a number or expression into parts and recombining them intelligently.

✅ Example:
For simplifying an expression like (a+b)²−(a−b)²

Use part and whole thinking: 4ab

It helps in mental algebraic manipulation.

12. Shesanyankena Charamena

(Remainder by the Last Digit)

🔸 Use: Shortcut for finding remainders, especially in divisibility problems.

✅ Example:
To find if 207 is divisible by 9, sum the digits: 2 + 0 + 7 = 9
Since 9 is divisible by 9 → so is 207.
This principle extends to other numbers using modular arithmetic.

13. Sopantyadvayamantyam

(The Last and Twice the Penultimate)

🔸 Use: In complex algebra and trigonometric simplifications, especially in polynomial factorization.
It focuses on identifying patterns by analyzing the last and second-last terms.

✅ Example:
In certain cubic expressions, this Sutra helps detect symmetry or balance in coefficients for factorization.

14. Ekanyunena Purvena

(By One Less than the Previous One)

🔸 Use: Multiplying numbers composed of 9s, like 999 × 7, quickly.

✅ Example:
999 × 7
Take one less than 7 → 6
Subtract 6 from 999 → 993
Attach (1000 – 999 × 7 = 999 – 6 = 993)
Answer = 6993

This is a reverse-style trick compared to “Ekadhikena Purvena.”

15. Gunitasamuccayah

(The Product of the Sum is Equal to the Sum of the Product)

🔸 Use: Factorization and simplification
This is often used to verify whether a factorization is correct.

✅ Example:
Check: (x+2)(x+3)=x²+5x+6

Sum of coefficients: (2 + 3) = 5
Product = 6
Matches the expanded form → Sutra holds.

16. Gunakasamuccayah

(The Factors of the Sum is Equal to the Sum of the Factors)

🔸 Use: Helps identify errors in complex algebra.
This Sutra is similar in spirit to the above but emphasizes the equality of factorized expressions and expanded forms.

✅ Example:
Used when checking or deriving complex factorized expressions like:

a²+ab+ac+bc = (a+c)(a+b)

You can verify by expanding and comparing.

🧠 Summary of Key Vedic Maths Sutras

Use: Squaring numbers ending in 5
Example: 75² = (7 × 8) → 56 + 25 = 5625

Use: Multiplication near base 10, 100, etc.
Example: 98 × 97 → 100 - 2 & 3 → 95 | 06 → 9506

Use: General multiplication
Example: 23 × 12 → 2×1 = 2, 2×2 + 3×1 = 7, 3×2 = 6 → 276

Use: Division
Example: 1234 ÷ 9 (transposition method)

Use: Algebra simplification
Example: (x + 1)(x + 2) - (x + 2)(x + 1) = 0

Use: Multiplication near non-bases using scaling

Use: Solving simultaneous equations
Example: x + y = 10, x - y = 2 → x = 6, y = 4

Use: Subtraction using completion
Example: 1000 - 678 = 322

Use: Factor patterns in polynomials

Use: Multiplication with base deficiency
Example: 96 × 96 → (96 - 4) | (4²) = 9216

Use: Algebraic manipulations using parts of expressions

Use: Quick remainder check using digit sums
Example: 207 → 2 + 0 + 7 = 9 → divisible by 9

Use: Identifying symmetry in equations

Use: Multiplying numbers made of 9s
Example: 999 × 7 = (7-1) | (1000 - 7) = 6993

Use: Verify identities like (x+2)(x+3) = x² + 5x + 6

Use: Checking factorizations for correctness

Applications of Vedic Maths

1. Arithmetic Calculations

Vedic Maths dramatically simplifies everyday calculations. For example, addition, subtraction, multiplication, and division can be performed mentally, leading to faster results and a deeper understanding of numbers.

2. Algebra

Vedic techniques can be applied to factorization, solving equations, and simplifying algebraic expressions. By using the Sankalana-Vyavakalanabhyam Sutra, complex polynomial equations become easier to solve.

3. Competitive Exams

For students preparing for competitive exams like SSC, UPSC, GRE, or banking exams, Vedic Maths can significantly improve both speed and accuracy. In exams with time constraints, these quick calculation techniques can be the difference between success and failure.

4. Geometry and Trigonometry

Vedic methods can also be extended to geometry and trigonometry. For example, the Shunyam Saamyasamuccaye Sutra can simplify problems involving angles and areas.

Vedic Maths vs. Conventional Maths

One of the most striking features of Vedic Maths is how it contrasts with conventional mathematical methods. Let’s compare:

Feature	Vedic Maths	Conventional Maths
Speed	Extremely fast, especially for large calculations	Slower due to step-by-step approach
Mental Calculation	Strong emphasis on mental math	Heavy reliance on paper and pen
Complexity	Very simple and intuitive	Complex and rule-based
Flexibility	Multiple approaches to the same problem	Usually a single standard approach
Error Reduction	Fewer errors due to logical steps	Greater chances of mistakes in multi-step problems

How to Start Learning Vedic Maths

Starting with Vedic Maths can be a rewarding experience for anyone, whether you’re a student or a professional. Here are some steps to get started:

1. Master the Basics

Begin with the simpler Sutras like Nikhilam and Ekadhikena Purvena. Practice with small numbers, gradually increasing the complexity as you gain confidence.

2. Practice Regularly

Vedic Maths is all about pattern recognition. The more you practice, the faster and more intuitive you’ll become. Use flashcards, online quizzes, or even apps to make it fun!

3. Apply in Real-Life Scenarios

Try using Vedic techniques in real-life situations—grocery shopping, mental calculations, budgeting, etc. You’ll soon begin to realize how much faster you can perform tasks.

4. Join a Course

Consider enrolling in a structured Vedic Maths course to get comprehensive learning and guidance.

5. Teach Others

One of the best ways to reinforce your learning is by teaching others. Share the techniques with your friends, family, or students, and see how quickly it becomes second nature.

Who Can Benefit from Vedic Maths?

Students: Grades 4 and above can build strong mathematical foundations.
Competitive Exam Aspirants: Perfect for cracking time-sensitive exams like GRE, CAT, or SSC.
Teachers: Help engage students in creative and fun ways.
Parents: Ideal for improving children's number sense and math confidence.
Math Enthusiasts: Those looking for new, exciting ways to look at numbers.

Real-Life Examples: Try it Yourself!

Here are a couple of simple yet powerful examples to show how Vedic Maths can change your approach to calculations:

Example 1: Multiplying 95 × 96 using Nikhilam Sutra

Base = 100
95 is 5 less than 100, and 96 is 4 less than 100.
Multiply: 5 × 4 = 20 (write 20)
Subtract: 10000 - (5 + 4) × 100 = 10000 - 900 = 9100
So, 95 × 96 = 9100 + 20 = 9120.

Example 2: Squaring 85 using Ekadhikena Purvena

85 → First digit = 8, and 8 + 1 = 9
Multiply: 8 × 9 = 72
Add: 72 and then append 25 at the end.
So, 85 × 85 = 7225.

Conclusion: Why Choose Vedic Maths?

In a world where speed and accuracy matter, especially in competitive exams and daily life, Vedic Maths offers a systematic yet highly efficient way to perform calculations. The techniques are easy to learn, fun to practice, and incredibly effective.

Want to Learn Vedic Maths?

At EKADEMY, we offer engaging and comprehensive Vedic Maths courses, workshops, and resources designed for every age group. Join us today to unlock the true potential of your mathematical skills!