Vedic Maths is a collection of ancient mathematical techniques and principles originating from the Vedas, the ancient Indian scriptures. It offers a unique approach to solving complex calculations efficiently and accurately. This is an ancient wisdom-based technique that transforms the complex calculations of numbers into an easy process. These techniques were mentioned in ancient Vedas by the then saints. Hence, the name is given as "Vedic Maths".
Vedic Maths is an ancient system of Indian mathematics that simplifies arithmetic operations and problem-solving. It is based on techniques derived from the Vedas, ancient Indian scriptures, and emphasizes mental calculation methods, enabling faster and more efficient computations.
.webp)
While the principles of Vedic maths go back thousands of years, the system was rediscovered in the early 20th century by swami Bharati Krishna Trithaji, a brilliant scholar and spiritual leader. After a deep study of the Vedas, he extracted 16 Sutras and 13 sub-sutras that form the backbone of Vedic mathematics. These formulas aren't limited to just addition or multiplication they extend to algebra, geometry, trigonometry, calculus, and more.
Table of Content
Origin of Vedic Mathematics
Vedic Mathematics was rediscovered by Indian mathematician Jagadguru Shri Bharati Krishna Tirthaji between 1911 and 1918. He was a Sanskrit, math, history, and philosophy expert. From 1925 until 1960, he was also the Shankaracharya (spiritual head) of Puri. He spent several years researching the Vedas and other ancient manuscripts, claiming to have discovered a cohesive mathematical system concealed within them.
The Sulba Sutras, which are part of the Kalpa Sutras (ritual manuals), contain the majority of the Vedic mathematical knowledge. The Sulba Sutras are concerned with the building of altars and geometric shapes for sacrifice rites. They also include early instances of algebraic equations, the Pythagorean theorem, irrational numbers, square roots, and pi.
These formulae were later published in a book called Vedic Mathematics in 1965. The system is based on 16 sutras (formulae) and 13 sub sutras.
Vedic Maths Sutras
The sixteen sutras (word-formulas), and thirteen sub-sutras that constitute the foundation of Vedic mathematics each offer precise solutions for a variety of mathematical problems.These approaches are applicable to addition, subtraction, multiplication, and division, as well as other mathematical operations.
Here are all the main Sutras (word formulae) and sub-Sutras from Vedic Mathematics discussed below:
Vedic Maths Main Sutras
There are sixteen main sutras in Vedic Maths. These Vedic Maths Sutras are discussed below in the tabular form.
No | Sutras | Meaning | Uses |
|---|---|---|---|
1 | Ekadinkena Purvena | By one more than the one before | This Sutra simplifies squaring numbers close to base values |
2 | Nikhilam Navatashcaramam Dashatah | All from 9 and the last from 10. | A powerful technique for subtraction, especially useful when dealing with numbers close to multiples of 10. |
3 | Urdhva Tiryak | Vertically and Crosswise. | This Sutra streamlines multiplication, especially useful for multiplying large numbers. |
4 | Paraavartya Yojayet | Transpose and adjust | This technique aids in simplifying complex mathematical problems involving equations and variables. |
5 | Shunyam Saamyasamuccaye | When the sum is the same, that sum is zero. | An effective approach for solving algebraic equations with equal sums on both sides. |
6 | (Anurupye) Shunyamanyat | If one is in ratio, the other is zero | This Sutra is indispensable for solving proportionality problems. |
7 | Yavadunam Tavadunikritya Varga Samam | Whatever the extent of its deficiency, lessen that deficiency to form a square. | Simplifies division and finding square roots. |
8 | Vilokanam | By mere observation | A technique that encourages quick, intuitive solutions based on patterns and observations. |
9 | Sankalana-vyavakalanabhyam | By addition and subtraction | This Sutra offers techniques for both addition and subtraction, enabling quick calculations |
10 | Puranapuranabhyam | By the completion or non-completion. | This Sutra aids in finding fractions and complements, simplifying various mathematical operations. |
11 | Chalana-kalanabyham | Differences and Similarities | Useful for problems involving ratios and proportions |
12 | Yaavadunam | Partial Products | This Sutra facilitates the multiplication of large numbers by breaking them down into smaller, more manageable parts |
13 | Vestanam | Specific and General | This Sutra helps in solving problems where a specific value is derived from a general one |
14 | Yavadvividham Vyashtih | Separately the particular from the general | This Sutra is handy for finding individual components from a group |
15 | Samuccaye | Collective addition. | Useful for quick summations, especially when dealing with a series of numbers |
16 | Ekanyunena Purvena | By one less than the previous one | This Sutra provides a technique for division and helps in finding quotients efficiently. |
Sub-Sutras of Vedic Maths
Vedic maths tricks are also known as sub-sutras or corollaries. They are derived from the main sutras and provide additional methods or shortcuts to solve problems faster and easier. There are 13 sub sutras. These sub-sutras are discussed below in the table.
No | Sub-Sutras | Meaning | Uses |
|---|---|---|---|
1 | Antyayordashakepi | The last digit remains the same | This sub-sutra aids in quickly determining the last digit of a product. |
2 | Sopantyadvayamantyam | The last two of the last | Useful for solving problems where the last two digits are required. |
3 | Ekaadhikena Purvena | One more than the previous | This sub-Sutra extends the "Ekadhikena Purvena" technique for squaring numbers closer to the base |
4 | Paravartya Sutra | Transposition and adjustment | Helps in solving linear equations and balance problems |
5 | Calana-Kalanabhyam | Differences and Similarities | Offers additional methods for solving ratio and proportion problems. |
6 | Gunakasamuccayah | The product of the sum | Useful for solving problems involving the product of two sums. |
7 | Gunita Samuccayah | The product of the sum is the sum of products | Aids in simplifying algebraic expressions. |
8 | Yavadunam Tavatirekena Varga Yojayet | By one less than the one so much is the square | Provides an alternative approach for finding squares. |
9 | Antyayordasake'pi | The last digit is as it is | Useful for quick calculations involving the last digit of numbers |
10 | Antyayorekadhikaduhitayor | On the last two digits | Enables efficient calculations when focusing on the last two digits. |
11 | Ardhasamuccayah Samuccayoh | The sum of the half-sums is the sum | A technique for adding fractions with common denominators |
12 | Ekanyunena Sesena | One less than the one followed by the last | Facilitates quick division. |
13 | Sesanyankena Caramena | The last by the last, and the ultimate by one less than the last | A technique for division, especially when dealing with recurring decimals. |
These Sutras and sub-Sutras together constitute the comprehensive system of Vedic Mathematics, offering a multitude of strategies and techniques for mental calculations and problem-solving. Mastery of these principles can significantly enhance one's mathematical prowess and efficiency.
Advantages of Vedic Maths
Vedic math offers several benefits over traditional computation methods. Some of the advantages of Vedic maths are listed below:
- It is basic and straightforward to learn and remember.
- It is quick and accurate, reducing the possibility of mistakes.
- It's entertaining and pleasant, and it encourages innovation and lateral thinking.
- It is adaptable and versatile, and it can be used in any discipline of mathematics.
- It is comprehensive and global, promoting mental and spiritual growth.
Vedic Maths Tricks with Examples
Vedic Maths is known for improving calculation speed while promoting mental calculation without using pen and paper. Let's learn some of the strategies of Vedic Maths for enhancing mental calculation:
1. The Vertically and Crosswise Technique (Nikhilam Sutra)
Nikhilam Sutra is used for multiplication, especially with large numbers.
Let's consider the example of multiplying 87 by 93
To solve this let's understand the steps.
Step 1: Identify the Base
Both 87 and 93 are close to 100, so we'll take 100 as our base.
Step 2: Find the Differences between the Number and the Base
- 87 is 13 less than 100, so its difference is -13.
- 93 is 7 less than 100, so its difference is -7.
Step 3: Cross Subtract or Add the Differences
The differences are then either cross subtracted or added. This means we subtract or add the difference between the first and second numbers, or vice versa.
- In this situation, (87 +(-7)) or (93+(-13)) equals 80.
Step 4: Multiply the Differences
After that, we multiply the differences (-13 × -7), which gives us 91.
Step 5: Combine the Results
Finally, we combine the results from steps 3 and 4 to get our answer:
- The result from step 3 (80) becomes the left part of our answer.
- The result from step 4 (91) becomes the right part of our answer.
So, combining these together, we find that 87 × 93 = 8091
2. The All from 9 and the Last from 10 Technique (Urdhva Tiryak Sutra)
The Urdhva Tiryak Sutra (The All from 9 and the Last from 10 Technique) is a Vedic Mathematics method for quickly subtracting a number from a power of ten (for example, 10, 100, 1000). This is how it works:
Let's consider the example of subtracting 78 from 100.
To solve this we follow these steps:
Step 1. Identify the Base:
We start by choosing a base that is larger than the amount we're subtracting and has a power of 10. In this instance, our base is 100 as we are deducting 78 hi from 100.
Step 2. Subtract Each Digit from 9 and the Last Digit from 10:
- For the first digit of 78 (which is 7), we subtract it from 9: 9 - 7 = 2
- For the last digit of 78 (which is 8), we subtract it from 10: 10 - 8 = 2
Step 3. Combine:
Finally, we add the results from steps 2 and 3 to get our answer. The result of subtracting the first digit becomes our answer's tens place, and the result of subtracting the last digit becomes our answer's ones place.
- So, when we add these two together, we get 100-78=22.
This technique can make mental calculations faster and easier, especially when dealing with large numbers.
3. The By One More than the One Before Technique (Ekadhikena Purvena Sutra)
The By One More than the One Before technique simplifies squaring numbers that end in 5 or numbers close to a power of 10.
For example, Let's find the square of 12 :
Step 1. Identify the Base:
First, we identify a base that's close to the number we're squaring. In this case, we're squaring 12, so our base is 10.
Step 2. Find the Difference:
12 is 2 more than 10, so its difference is +2.
Step 3. Add One to the Base:
Now, we add one to the base: 10+1=11
Step 4. Multiply and Add:
After that, we multiply the base by the difference and add the square of the difference:(11×(+2)+(+2)^2=(22)+(4)=26.
Step 5. Combine:
Combine the results from steps 3 and 4 to get our answer:
- The result from step 3 (11) becomes the left part of our answer.
- The result from step 4 (26) becomes the right part of our answer.
So, combining these together, we find that: 122 = 144.
This technique can make mental calculations faster and easier, especially when dealing with large numbers.
4. The Proportionality Rule (Anurupye Sutra)
This methodology is a Vedic Mathematics method for solving ratio and proportion problems.
This is how it works:
For example: Assume you want to know how many liters of water are in an 80% full tank, and you know the tank can contain 500 liters when filled. You can answer this problem with the Anurupye Sutra as follows:
Step 1. Express the Problem as a Proportion:
First, we define the issue as a proportion. We know that 80% of the tank's capacity (let's call it x) equals the amount of water in the tank (let's call it y) in this scenario. This can be expressed as follows: (80)/(100)=y/x
Step 2. Cross-Multiply:
Then, to solve for y, we cross-multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the product of the first fraction's denominator and the numerator of the second fraction:80×x=10×y
Step 3. Solve for y:
Finally, we solve for y by dividing both sides of the equation by 100:
- y= (80 × x)/(100)=0.8 × x
- Since we know that x is equal to 500 liters, we can plug it in and find out that y is equal to 0.8×500=400
So, if a tank can hold 500 liters when it is completely full, and it is 80% full, then it contains 400 liters of water.
5. The Digital Root Method (Yavadunam Tavadunikritya Varga Samam)
This is a Vedic Mathematics approach for simplifying calculations by determining the digital root of a number. The digital root of a number is the single digit obtained by continually adding all of the digits of a number until only one digit remains.
For example, let's determine if the number 387 is divisible by 3 by using Yavadunam Tavadunikritya Varga Samam.
Here's how it works:
Step 1. Find the Sum of the Digits:
First, we find the sum of the digits of the number. In this case, we're dealing with 387, so we add up its digits: 3+7+8=18
Step 2. Calculate the Digital Root:
Then, we calculate the digital root by again adding up the digits of the result from step 1: 1+8=9
Step 3. Check Divisibility:
Finally, we check if the digital root is divisible by 3. If it is, then the original number is also divisible by 3. In this case, since 9 is divisible by 3, so is 387.
Hence, 387 divisible by 9.
This technique can make checking divisibility faster and easier, especially when dealing with large numbers.
People Also View:
Vedic Maths Tricks for Arithmetic Operations
Here are some examples of how to use Vedic maths formulas and tricks to solve various problems
Division Tricks
Type: When Divisor is closer to but less than a Power of 10
Example: Divide 243 by 9
Solution:
"The Nikhilam Sutra is a vedic arithmetic division technique that may be employed when the divisor is near to but less than a power of ten."
Divide 243 by 9,we may do it as follows:
Determine the divisor's delays from the nearest power of ten. In this situation, 9 is one less than ten, thus the deficit is one.
Divide the payout in two: quotient and remainder. The residual should have the same length as the divisor. In this situation, 243 may be divided into 24 and 3.
Leave the first digit of the quotient alone. In this situation, the answer is 2.
Multiply the shortfall by the quotient's first digit and write the result underneath the quotient's second digit. Then add them up column by column. In this situation, we divide 1 by 2 and write the result below 4. Then we add 4 to 2 to obtain 6.
Repeat the preceding process for the remaining digits of the quotient. In this situation, we multiply 1 by 6 and write it below 3. Then we put 3 and 6 together to get 9.
The solution is the sum of the columns. In this example, 27.
The result is checked by ensuring that the residual is either zero or equal to the divisor. The residual is zero in this situation, hence the solution is accurate.
Subtraction Tricks
Trick 1: When Subtraction is done with the Power of 10
Example 1: Subtract 47 from 100
Solution:
"Ekadhiken Purvena is one of the Vedic math sutras for the subtraction technique, which implies "one more than the previous". This sutra may be used to subtract a number from a power of ten, for example, 10, 100, 1000, and so on."
Use this sutra as follows to substract 47 from 100:
Add a line with the number 47 below the number 100.
All save the final digit of 47 should be subtracted from 9. Below the line, type the outcome.
Subtract 10 from 47's last digit. Below the line, type the outcome.
The sum of the digits below the line yields the final result, which is the number.
This is how the calculation appears:
Subtract each of 47's digits from 9, except the final one, and then subtract the last digit from 10. We get 10 - 7 = 3 and 9 - 4 = 5. As a result, the answer is 53.
Trick 2: When a Number has a Symmetrical Component
Example 2: Subtract 72 from 88
Solution:
The Dwandwa Yoga technique is a Vedic Math subtraction trick. When integers contain symmetrical portions, this approach is utilized. This is how it works:
Step 1. Split: Divide the numbers into two halves.
Step 2. Subtract: Take the first portion and subtract it from the second part.
Let us use this way to substract 72 from 88:
1. 72 and 88 may be divided into two parts: 7 and 2 for 72, and 8 and 8 for 88.
2. Subtract the first component (7 from 8) to get one, and the second part (2 from 8) to obtain six.
So, when we subtract 72 from 88 utilizing the Dwandwa Yoga technique, our final result is 16.
This strategy can make subtraction faster and easier, especially when dealing with huge numbers with symmetrical portions!.This technique can speed up and simplify subtraction, especially for bigger numbers with symmetrical portions. It's crucial to remember that this approach functions best when each portion of the minuend is greater than or equal to the corresponding part of the subtrahend.
Multiplication Tricks
Trick 1: Multiplication by 11
Example 1: Multiply
(i) 23 by 11
(ii) 47 by 11
To multiply any two-digit number by 11, use the sutra Ekadhikena Purvena. This means that we add one to the previous digit and write it in between.
- 23×11 = 2(2+3)3 = 253
- 47×11 = 4(4+7)7 = 517
Example 2: Multiply 68 by 11
If the sum of the digits is more than 9, we carry over the extra digit to the left
68×11 = 6(6+8)8 = 6(14)8 = (6+1)48 = 748
Trick 2: Squaring numbers ending in 5
Example 3: Find a Square of (i) 65 (ii) 95
To square any number ending in 5, we can use the sutra Yavadunam Tavadunikritya Varganca Yojayet. This means that we multiply the first part of the number by one more than itself and 25 at the end.
(i) 652 = (6×(6+1))∣25∣ = (6×7)∣25∣=4225
(ii) 952 = (9×(9+1))∣25| = (9×10)∣25∣=9025
Trick 3: Multiplication by 9
Example 4: Multiply (i) 23 × 9 (ii) 47 × 9
Solution:
To multiply any number by 9, use the sutra Ekanyunena Purvena. This means that we subtract one from the previous digit and write it in between.
(i) 23 × 9 = 2 (2-1) 3 = 207
(ii) 47 × 9 = 4 (4-1) 7 = 423
Example 5: Multiply (i) 68 × 9 (ii) 95 × 9
Solution:
If the sum of the digits is more than 9, so we carry over the extra digit to the left.
(i) 68 × 9 = 6 (6-1) 8 = 6 (5) 8 = (6+5) 8 = 612
(ii) 95 × 9 = 9 (9-1) 5 = 9 (8) 5 = (9+8) 5 = 855
Trick 4: Multiplication by a near power of 10
Example 6: Multiply (i) 23 × 99 (ii) 47 × 999 (iii) 68×9999
Solution:
To multiply any number by a near power of 10, such as 99, 9999, or 99999, we can use the sutra Nikhilam Navatashcaramam Dashatah. This means that we have to subtract each digit from 9, except the last one, which we subtract from 10. The result is the complement of the number. Then we multiply the number by the next higher power of 10 and subtract the complement from it.
(i) 23 × 99 = (9-2) |(10-3)| = (7) |(7)|
Complement = 77
23 × (100) - (77) = 2300 - 77 = 2277
(ii) 47 × 999 = (9−4)∣(9−7)∣∣(10−7)∣ = (5)∣(2)∣∣(3)∣
Complement = 523
47 × (1000) − (523) = 47000 − 523 = 46953
(iii) 68 × 9999 = (9−6)∣(9−8)∣∣(10−8)∣∣(10−8)∣ = (3)∣(1)∣∣(2)∣∣(2)∣
Complement = 3122
68 × (10000) − (3122) = 680000 − 3122 = 679932
Type 5: Multiplication by a Series of Ones
Example 7: Multiply 23 by 11
Solution:
To multiply any number by a series of ones, such as 11, 111,1111 or 11111, use the sutra Sankalana Vyavakalanabhyam. This means that we add or subtract the digits alternately from right to left and write them in between.
23 × 11 = (2+3) |(3-2)| |3| = (5) |(1)| |3| = 253
Trick 6: Multiplication by 8
Example 8: Multiply (i) 68 × 8 (ii) 23 × 8
Solution:
To multiply any number by 8, use the sub-sutra Kevalaih Saptakam Gunyat. This means that we multiply the number by 7 and add the original number to it.
(i) 68 × 8 = (68 × 7) + 68 = 476 + 68 = 544
(ii) 23 × 8 = (23 × 7) + 23 = 161 + 23 = 184
Trick 7: Multiplication by a number close to a power of 10
Example 9: Solve (i) 47×1001 (ii) 68×9999
Solution:
To multiply any number by a number close to a power of 10, such as 1001, 10001, or 100001, we can use the sub-sutra Anurupyena. This means that we multiply the number by the next higher power of 10 and add or subtract the difference between the multiplier and the power of 10.
(i) 47 × 1001 = (47 × 1000) + (1001 − 1000) × 47 = (47000) + (1) × (47) = (47000) + (47) = 47047
(ii) 68 × 9999 = (68 × 10000) − (10000 − 9999) × 68 = (680000) − (1) × (68) = (680000) − (68) = 679932
Practice Questions on Vedic Maths
Question 1: Multiply 34 by 11 by Using the sutra Ekadhikena Purvena
Question 2: Use the sutra Nikhilam Navatashcaramam Dashatah to divide 243 by 9.
Question 3: Use the sutra Urdhva Tiryagbhyam for multiplying 23 by 17.
Question 4: Use the sutra Paravartya Yojayet to solve the given equation
Question 5: Use the sutra Shunyam Saamyasamuccaye to find the value of x for the equation
Question 6: Use the sutra Anurupye Shunyamanyat and find the ratio of two numbers if their sum is 15 and their product is 56.
Question 7: Use the sub-sutra Anurupyena to multiply 101 by 56.
Answer Sheet
1) 374
2) 27
3) 391
4) x =1
5) x = - 1, - 2, - 3
6) 7 : 8
7) 5656
Conclusion
Vedic Mathematics presents multiple interesting techniques rooted in ancient Indian wisdom that can revolutionize our approach to mathematical computations. This system integrates a range of sutras and sub-sutras designed to simplify and speed up arithmetic operations. From rapid multiplication and division to efficient squaring and subtraction, Vedic Mathematics offers methods that enhance mental calculation speed and accuracy.