Exploring the applications and efficiency of Vedic mathematics

Introduction

Vedic Mathematics, derived from ancient Indian scriptures known as the Vedas, is recognized for its efficiency and applicability across various Mathematical domains. This system, built on concise and easy-to-apply Sūtras and Sub-Sūtras, offers alternative methods for arithmetic, algebra, and calculus, often surpassing conventional techniques in efficiency. This study explores the applications and efficiency of Vedic Mathematics, evaluating its impact on computational efficiency and problem-solving through comparative analysis with conventional methods. We aim to understand its relevance in modern Mathematical education and practice [4, 11] . This research underscores the importance of incorporating Vedic Mathematics into higher education, acknowledging its historical significance in classical Indian sciences, particularly in texts like Vedanga Jyoutisha [12] . Discovered by Jagadguru Shankaracharya Swami Bharati Krishna Tirtha between 1911 and 1918. Vedic Mathematics is built on sixteen Sutras that enhance pattern recognition and accelerate computations. These methods offer faster computation techniques compared to traditional methods typically taught in schools [2] .

Objectives of Research

To explore the applications and efficiency of Vedic Mathematics in Algebra, Arithmetic and Calculus.

Research Methodology

This research employs a historical research method to examine and study various works of higher Mathematics and advanced calculus that are based on Vedic Mathematics Sutras. The methodology involves a comprehensive review of literature on Vedic Mathematics Sūtras and Sub-Sūtras to understand their historical significance and meaning. This examination highlights a significant research gap, as limited work has been conducted in exploring the applications of Vedic Mathematics in higher Mathematics and advanced calculus. After an extensive literature review, two primary references were identified as particularly informative and useful for the research: "Vedic Mathematics" by Swami Sri Bharati Kranti Ji Maharaja, Shankaracharya S.N of Govardhana Matha, Puri, edited by Dr. V. S. Agrawala, and "Vertically and Crosswise: Applications of Vedic Mathematics Sūtras" by Pickles, Nicholas & Williams. These books serve as essential resources for understanding and analyzing Vedic Mathematics Sūtras and their historical significance in the context of higher Mathematics and advanced calculus. The link between Vedic Sūtras and Sub-Sūtras to arithmetic, algebra, and calculus was explored by analyzing examples solved using conventional methods and Vedic Mathematics techniques.

Literature Review

This research relies on secondary data, including past research papers, journals, books, and theses. A review of these materials revealed that much of the computer science research has applied Vedic Sūtras and Sub-Sūtras for binary operations like subtraction, multiplication, division, as well as binary squaring and cubing. In the article "Mathematics Hundred Years Before and Now," Acharya, [1] , looks at the history of Mathematics and how ancient manuscripts have been preserved. Ratna connects old and new ways of thinking about math, discussing how mathematical ideas have changed over time. The article explains how derivatives can be used to solve quadratic equations and shows how the Vertical and Crosswise Sūtra can be used for algebra and exponential functions. It highlights that although the core mathematical concepts have stayed the same, the methods and algorithms used to solve problems have changed. Babaji [3] , in "Solving Systems of Linear Equations using the Parāvartya Rule," demonstrates solving linear equations with the Parāvartya Yojayet Sūtra, extending it to systems of n linear equations. Babaji's research shows that this method is comparable to Cramer's Rule for two equations and more efficient than Gauss elimination and LU Decomposition for three equations. The paper also suggests using the Parāvartya rule multiple times to enhance efficiency. Swami Sri Bharati Kranti Ji Maharaja's book "Vedic Mathematics" [4] , edited by V.S. Agrawala, provides practical implementations of Vedic Sūtras across Mathematical disciplines. The book covers multiplication using the Ūrdhva-Tiryagbhyām Sūtra, division with the Parāvartya and Nikhilaṁ Sūtras, and factorization of quadratic equations. It also explains solving cubic equations and simultaneous equations, and the relationship between differential calculus and polynomial factors.

In "Applications of Vedic Mathematics Sūtra -Vertically and Crosswise," Nicholas, [8] , focus on the Vertically & Crosswise Sūtra. The book shows how to quickly solve arithmetic operations, expand determinants, and simplify matrix operations. The authors provide methods for evaluating determinants and finding matrix inversions, demonstrating the practical efficiency of Vedic Sūtras in simplifying complex computations.

Results and Discussion

Exploring the importance of Vedic Mathematics Sūtras for teaching classical Mathematics revealed their effectiveness in simplifying and expediting problem-solving. These Sūtras streamline approaches, emphasizing mental calculations for increased efficiency. Comparisons with conventional methods showed that Vedic Sūtras offer a more intuitive and concise methodology, aiding quicker understanding and application. Participants preferred the clarity and efficiency of Vedic Sūtras. The findings suggest that incorporating Vedic Sūtras into classical Mathematics teaching could provide a more accessible and efficient learning experience, warranting further research to enhance Mathematics education methodologies.

The Sixteen main Sūtras and Thirteen Sub-Sūtras of Vedic Mathematics are as follows

Vedic Mathematics, based on sixteen sutras from the Atharva Veda, offers an efficient system of computation and problemsolving.

Table 1: Sixteen Shutras and Sub Shutras [4,5] . In this study, the researcher endeavors to elucidate various methods using Sūtras and Sub-Sūtras, demonstrating their connection with Algebra, Arithmetic, and Calculus.

Algebra: [4] 1. Ekādhikena Purvena (By one more than the previous one):

The Mathematical rule you provided is called "Ekādhikena Purvena" which translates to "By one more than the previous one." It's a method used for factorization and solving algebraic equations. Example: Given the quadratic equation: 𝑥 2 + 6𝑥 + 8 = 0 Using the Ekādhīkena Purvena rule, we rewrite the equation as follows:

(𝑥+ 4) (𝑥+2) = 0 This rewriting comes from noticing that the constant term in the original equation is 8, and we are looking for two numbers whose product is 8 and sum is 6. These numbers are 4 and 2, so we rewrite the middle term of the quadratic equation as 4x + 2x, which factors into (x + 4) (x + 2). Now, to find the solutions, we set each factor equal to zero: 𝑥 + 4 = 0 ⇒ 𝑥= -4

x + 2 = 0 ⇒ x= -2 So, the solutions to the equation 𝑥 2 + 6x + 8 = 0 are x = -4 and x = -2.

2-Urdhva-Tıryagbhyām (Vertically and Crosswise): [8] . This rule is particularly useful for algebraic multiplication, especially for polynomials.

Example 1

Given the expression: (x+5) (x+7) Using the "URDHVA-TIRYAGBHYĀM" rule, we multiply the expressions as follows:

So, applying the "URDHVA-TIRYAGBHYĀM" rule to (x+5) (x+7) yields 𝑥 2 + 12x +35.

Example 2 Urdhava-Tiryakbhyam

The Urdhava-Triyakbhyam Sutra, meaning "Vertically and crosswise," is a universal multiplication formula. It emphasizes the parallel process of generating and summing partial products, as illustrated in Fig 1 . For example, using this sutra, the multiplication of 546 by 324 results in 176,904.

Illustration of Vertically and Crosswise multiplication [7,11] .

Anurūpye Sunyamanyat (If one is in ratio, the other is zero):

This Sūtra is particularly useful for solving quadratic equations where one root is the reciprocal of the other.

Example 1

Given the quadratic equation: 𝑥 2 -𝑝𝑥 + 𝑞 = 0 And let a and 𝑏 be the roots of this equation. According to the properties of the roots:

The sum of the roots, a + b, is equal to the coefficient of the linear term with the opposite sign (here p).

The product of the roots, ab, is equal to the constant term (here q).

Given that a is twice b, we have a = 2b.

From the sum of roots property: a

From the product of roots property: ab = q, 2𝑏 2 = 𝑞, 2

So, the values of 𝑎 and 𝑏 satisfying the given conditions are: a

Hence, if 𝑎 is twice b, then 𝑎 =

Puranapuranābhyām (Completion and Non-Completion):

This Sūtra is useful for factorization and completing the square.

Example: Example: 𝑥 2 + 10x + 24 Using the Sūtra: 𝑥 2 + 10x + 25-1 (𝑥 + 5) 2 -1 [6] .

Arithmetic: [4] . = 91 -80 = 11 [2] .

Example 2

For example, to determine the complement of the number 543268, subtract it from 1,000,000. To find a number's complement, subtract each digit from 9, and for the final digit, subtract it from 10.

Complement of 543268 can be calculate as 9 9 9 9 9 10-3 5 8 7 3 2= 456732 9 9 9 9 9 1 0 ↓ ↓ ↓ ↓ ↓ ↓ 5 4 3 2 6 8 Subtract = 4 5 6 7 3 2

The Formula is "All from 9 and the last from 10'' [4] . Similarly, the Complement of 5 4 8 6 can be calculate as, 9 9 9 10-5 4 8 6 = 4 5 1 4 [9]

Chalana-Kalanābhyām (Differences and Similarities):

This Sūtra is used for addition and subtraction of fractions. Example:

Let's add Using the Sūtra "Chalana-Kalanābhyām":

Step 1: Find the common denominator, which is 3 × 5 = 15.

Step 2: Adjust the fractions to have the same

Step 3: Now, we can add the fractions: Example: If the product of two numbers is constant, i.e., x y = k, then if x increases, y decreases, and vice versa. Suppose we have two numbers x and y such that their product is constant, denoted as k, i.e., 𝑥 𝑦 = 𝑘 Now, if we increase x while keeping y constant, the product x y would exceed 𝑘 To maintain 𝑥 𝑦 = k, y must decrease. Conversely, if we increase y while keeping x constant, the product x y would exceed k. Again, to maintain x y = k, x must decrease. This principle holds true because of the inverse relationship between x and y when their product is constant. As one variable increases, the other variable must decrease proportionally to keep their product constant. This concept is essential in optimization problems in calculus, where we seek to maximize or minimize a function subject to certain constraints. In such problems, understanding how changes in one variable affect another helps in finding the optimal solution.

Example: If the product of two numbers is constant, i.e., x y = 10, and we want to maximize the sum x + y, we can use calculus.

Express 𝑦 in terms of 𝑥: 𝑦 = Sub-Sutras for Calculus: [4] . Lopanasthāpanabhyām (By Alternately Lessening and Restoring): This Sub-Sutra is useful for integration. Example:

The given example demonstrates the application of the Sub-Sutra "Lopanasthāpanabhyām (By alternately lessening and restoring)" which is useful for integration.

Where C is the constant of Integration.

Vilokanam (The method of observation):

This Sub-Sutra is useful for integration.

The given example demonstrates the application of the Sub-Sutra "Vilokanam (The method of observation)" which is useful for integration.

Example: ∫ (𝑥

Here C is the constant of Integration. These Sutras and Sub-Sutras provide efficient methods for differentiation and integration, making calculus computations faster and more straightforward. [4,6] 1. Anurūpyena (Proportionately): This Sutra is useful for matrix multiplication. Example:

Matrices and Determinants:

Kevalaih Saptakam Gūņyat (By one less than the previous one and the one below)

This Sutra is useful for finding the determinant of higherorder matrices. These Sutras and Sub-Sutras provide efficient methods for performing matrix multiplication and finding determinants, making matrix computations faster and more straightforward. These step-by-step solutions demonstrate how Vedic Mathematics Sutras and Sub-Sutras can be used to solve matrix multiplication and determinant problems efficiently.

Example

For Linear Equation [3,4] Ekādhikena Purvena (By one more than the previous one) Formula Let's solve the equation 2x+3 = 0 using the Vedic Mathematics sutra "Ekādhikena Pūrvēṇa" (By one more than the previous one).

Sutra Explanation

The sutra "Ekādhikena Pūrvēṇa" suggests solving problems using a method that involves incrementing by one more than the previous value. This sutra can be applied in various mathematical contexts. Here, we are using it to solve a simple linear equation of the form ax + b = 0.

Formula

The general form of the equation is ax + b = 0.

To solve for x: = -𝑏/𝑎 We need to solve for x.

The left-hand side equals the right-hand side, confirming that 𝑥 = - is indeed the correct solution.

Anurūpye Sunyamanyat (If one is in ratio, the other is zero):

Let's explore the sutra "Anurūpye Sunyamanyat" (If one is in ratio, the other is zero) and apply it to solving the equation 3x=6y.

Sutra Explanation

The sutra "Anurūpye Sunyamanyat" implies that if two quantities are proportional to each other, then one of them must be zero if the ratio cannot be balanced otherwise. In mathematical terms, if ax=by, then either a=0 or b=0.

Example

Given the equation:3x = 6y

We need to solve this using the sutra.

Analyze the given equation:3x = 6y

Rewrite the equation to express the proportional relationship: This means x = 2y. Apply the sutra "Anurūpye Sunyamanyat": According to the sutra, either 3 = 0 or 6 = 0, which is clearly not possible since both 3 and 6 are non-zero constants. Since both conditions are not possible, we can interpret the sutra in this context to mean that the given proportional relationship x = 2y must hold for non-zero values of x and y.

Paravartya Yojayet (Transpose and Apply)

If 𝑎𝑥 + 𝑏𝑦 = 𝑐, then 𝑥 = These formulas provide efficient methods for solving linear equations, making the process faster and more straightforward.

Conclusions

The study provides a comprehensive overview of the efficiency and applications of Vedic Mathematics in algebra, arithmetic, and calculus. Through a historical research method and comparative analysis with conventional methods, we have demonstrated the unique advantages of Vedic Mathematics in simplifying computations and problem-solving techniques.

The practical examples and literature review underscore the significance of incorporating Vedic Mathematics into higher education to enhance computational efficiency and problemsolving skills among students. By integrating Vedic Mathematics into the curriculum, educators can provide students with valuable tools to tackle complex mathematical problems with ease.

Acknowledgement

AcknowledgementI would like to express my heartfelt thanks to Prof. Dr. Chet Raj Bhatt, Prof. N. Pahari, and Prof. Dr. Ajay Singh for their invaluable guidance and insights that greatly influenced this research article. I also appreciate Shyam Raj Subedi for his assistance and encouragement throughout the research process.

References

1. Acharya ER. Mathematics hundred years before and now. History Research. 2015;3(3):41-47. Available from: http://www.sciencepublishinggroup.com/j/history
2. Babaji DK. Solving systems of linear equations using the Parāvartya rule. International Journal of Mathematics Trends and Technology. 2012;3(1):37-41.
3. Bharati Krishna Tirtha. Vedic Mathematics. Delhi: Motilal Banarsidass Publishers; 1965.
4. Kenneth RW. Discover Vedic Mathematics. Delhi: Motilal Banarsidass Publishers; 1984.
5. Kulshreshtha H. Revisiting Vedic math for engineering applications. International Journal of Advanced Research, Ideas, and Innovations in Technology. 2022;8(1):1392.
6. Nicholas A, et al. Applications of Vedic Mathematics Sūtra -Vertically and Crosswise. Publisher; 2010.
7. Rajaraman V. Historical significance of Vedic mathematics in classical Indian sciences. Indian Journal of History of Science. 1992;27(1):1-13.
8. Raksha, et al. Review on Vedic mathematics. International Journal of Advance Research in Science Communication and Technology (IJARSCT). 2022;3(1).