Contributed by:

This PDF contains :

Abstract,

Keywords,

Introduction,

Use of Vedic Formulas in Elementary Algebra,

Parāvartya Yojayet:

(i) Use of Paravartya Yojayet for algebraic division.

(ii) Use of Paravartya Yojayet formula to solve simple equations,

(iii) Use of Paravartya Yojayet formula for solving Simultaneous Equations,

(iv) Use of Paravartya Yojayet formula in Partial Fraction,

Sunyam Samyasamuccaye :

Meanings and Applications of Sunyam Samyasamuccaye.

Anurupye Sunyamanyat :

Antyayoreva :

Lopanasthapanabhyam:

(i) Use of Lopanasthapanabhyam for factorization,

(ii) Use of Lopanasthapanabhyam to find HCF

Abstract,

Keywords,

Introduction,

Use of Vedic Formulas in Elementary Algebra,

Parāvartya Yojayet:

(i) Use of Paravartya Yojayet for algebraic division.

(ii) Use of Paravartya Yojayet formula to solve simple equations,

(iii) Use of Paravartya Yojayet formula for solving Simultaneous Equations,

(iv) Use of Paravartya Yojayet formula in Partial Fraction,

Sunyam Samyasamuccaye :

Meanings and Applications of Sunyam Samyasamuccaye.

Anurupye Sunyamanyat :

Antyayoreva :

Lopanasthapanabhyam:

(i) Use of Lopanasthapanabhyam for factorization,

(ii) Use of Lopanasthapanabhyam to find HCF

1.
MEC

Print ISSN: 2645-8500

Elementary Algebra on Vedic Mathematics

Krishna Kanta Parajuli

Department of Mathematics, Valmeeki Campus, Nepal Sanskrit University

Email: kknmparajuli@gmail.com

Abstract

The South Asian region has a long history of discovering new ideas, ideologies,

and technologies. Since the Vedic period, the land has been known as a fertile place

for innovative discoveries. The Vedic technique used by Bharati Krishna Tirthaji is

unique among South Asian studies. The focus of this study was mostly on algebraic

topics, which are typically taught in our school level. The study also looked at how

Vedic Mathematics solves issues of elementary algebra using Vedic techniques

such as Paravartya Yojayet, Sunyam Samyasamuccaye, Anurupye Sunyamanyat,

Antyayoreva and Lopanasthapanabhyam. The comparison and discussion of the

Vedic with the conventional techniques indicate that the Vedic Mathematics and

its five unique formulas are more beneficial and realistic to those learners who are

experiencing problems with elementary level algebra utilizing conventional methods.

Keywords: Elementary Algebra, Paravartya Yojayet, Sunyam Samyasamuccaye,

Anurupye Sunyamanyat, Antyayoreva, Lopanasthapanabhyam

Introduction

The word ‘Algebra’ was originated from the corruption of the Arabic word

‘Al-jabar’al-muqabulah’ where ‘al’ means ‘the', ‘jabr’ refers to the operation of

transferring a quantity from one side of an equation to another while ‘muqabulah’

means the process of subtracting similar quantities from both sides of an equation

(Sidhu, 1990). Algebra is either a form of mathematics in which letters and symbols

are used to represent numbers or a generalization of arithmetic in which numbers are

represented by letters that are combined according to arithmetic rules (Hoad, 1996).

The numbers are often represented by the symbols called variables. Its concepts are

often required in teaching and learning to return to corresponding circumstances in

Arithmetic. As a generalized arithmetic, it can be linked to geometry by claiming

that algebra is only written geometry and geometry is only pictured algebra (Sidhu,

Generally speaking, algebra can be divided into two levels: elementary

and abstract. Elementary Algebra is the most fundamental part of algebra that

is commonly considered to be essential for any study of mathematics and

its applications, i.e., the part of algebra that is typically taught in elementary

mathematics courses (Cajori, 1919). The more abstract part of algebra is known as

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

Print ISSN: 2645-8500

Elementary Algebra on Vedic Mathematics

Krishna Kanta Parajuli

Department of Mathematics, Valmeeki Campus, Nepal Sanskrit University

Email: kknmparajuli@gmail.com

Abstract

The South Asian region has a long history of discovering new ideas, ideologies,

and technologies. Since the Vedic period, the land has been known as a fertile place

for innovative discoveries. The Vedic technique used by Bharati Krishna Tirthaji is

unique among South Asian studies. The focus of this study was mostly on algebraic

topics, which are typically taught in our school level. The study also looked at how

Vedic Mathematics solves issues of elementary algebra using Vedic techniques

such as Paravartya Yojayet, Sunyam Samyasamuccaye, Anurupye Sunyamanyat,

Antyayoreva and Lopanasthapanabhyam. The comparison and discussion of the

Vedic with the conventional techniques indicate that the Vedic Mathematics and

its five unique formulas are more beneficial and realistic to those learners who are

experiencing problems with elementary level algebra utilizing conventional methods.

Keywords: Elementary Algebra, Paravartya Yojayet, Sunyam Samyasamuccaye,

Anurupye Sunyamanyat, Antyayoreva, Lopanasthapanabhyam

Introduction

The word ‘Algebra’ was originated from the corruption of the Arabic word

‘Al-jabar’al-muqabulah’ where ‘al’ means ‘the', ‘jabr’ refers to the operation of

transferring a quantity from one side of an equation to another while ‘muqabulah’

means the process of subtracting similar quantities from both sides of an equation

(Sidhu, 1990). Algebra is either a form of mathematics in which letters and symbols

are used to represent numbers or a generalization of arithmetic in which numbers are

represented by letters that are combined according to arithmetic rules (Hoad, 1996).

The numbers are often represented by the symbols called variables. Its concepts are

often required in teaching and learning to return to corresponding circumstances in

Arithmetic. As a generalized arithmetic, it can be linked to geometry by claiming

that algebra is only written geometry and geometry is only pictured algebra (Sidhu,

Generally speaking, algebra can be divided into two levels: elementary

and abstract. Elementary Algebra is the most fundamental part of algebra that

is commonly considered to be essential for any study of mathematics and

its applications, i.e., the part of algebra that is typically taught in elementary

mathematics courses (Cajori, 1919). The more abstract part of algebra is known as

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

2.
Elementary Algebra on Vedic Mathematics 83

Abstract Algebra or Modern Algebra which extends the familiar concepts found in

elementary algebra to more general concepts. Abstract Algebra or Modern Algebra

is mainly studied by trained mathematicians, and it involves the axiomatic definition

and investigation of algebraic structures such as Groups, Rings, and Fields (Menini

& Oystaeyen, 2017).

Algebra's origins can be traced back to the ancient Babylonians, who

developed it as an advanced arithmetical device. In the same period, the Egyptians,

Greeks, and Chinese formed linear and quadratic equations (Struik, 1987; Plofker,

Imhausen, Robson, & Dauben, 2007).

It is considered that, before 300 BC the Bijganit (बीजगणित) was derived from

the Bakhshali Manuscript (वाक्शाली लिपि) (Parajuli, 2021). At that time, the Bijganit

became one of the most influential practice and widely spread throughout the

continent. In pre-medieval era (500 BC – 400 AD) the development of Bijganit

took place rapidly as mathematicians were putting their hard and continuous effort

upon it. Further development of arithmetic and Bijganit took place in the later time

of the medieval era (400 AD – 1200 AD) which then got spread to Arab and other

countries. (Parajuli, 2021; Acharya, 2015; Boyer, 1991). Brahmagupta (598 AD –

668 AD) was a famous South-Asian mathematician and astronomer. He wrote the

method of solving linear and quadratic equations in his book Brahmasphutasiddhant

(ब्राह्म्स्फ़ुटसिद्धान्त) in 628 AD (Parajuli, 2021). It is greatly to the credit of Brahmagupta

that he gave all integral solutions of the linear Diophantine equation i.e., ax + by =

c, where a, b and c are constants (Boyer, 1991). Bhaskaracarya-II (1114 AD – 1993

AD) was the famous leading mathematician of ancient Indian during 12th century.

Bhaskaracarya wrote Algebra in his Siddhantasiromani (Patwardhan, Naimpally

& Singh, 2015). Hence, talking about the development of algebra, the South Asian

region has a long tradition of inventing new ideas, principles, and inventions. Since

the Vedic period, the land has been regarded as a fertile ground for new inventions

(Parajuli, 2021; Groza, 1968).

Many eastern and western mathematicians contributed to the modernization

of mathematics. They contributed to the subject of algebra as well as other areas

of mathematics. South Asian mathematicians have made a significant contribution

to these fields (Groza, 1968; Struik, 1987). Furthermore, Bharati Krishna Tirthaji

(1884 AD – 1960 AD), a South Asian glorious and divine human, made a significant

contribution to the development of mathematics in the nineteenth century (Tirthaji,

2015). He is well-known for rediscovering and reconstructing the new ideas of

mathematics from ancient Sanskrit text Veda early last century between 1911 – 1918

is popularly known today is Vedic Mathematics (Parajuli, 2021). The Vedic method,

he claims, is based on sixteen formulas and an equal number of sub-formulas that

cover all branches of mathematics, both pure and applied (Patwardhan, Naimpally &

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

Abstract Algebra or Modern Algebra which extends the familiar concepts found in

elementary algebra to more general concepts. Abstract Algebra or Modern Algebra

is mainly studied by trained mathematicians, and it involves the axiomatic definition

and investigation of algebraic structures such as Groups, Rings, and Fields (Menini

& Oystaeyen, 2017).

Algebra's origins can be traced back to the ancient Babylonians, who

developed it as an advanced arithmetical device. In the same period, the Egyptians,

Greeks, and Chinese formed linear and quadratic equations (Struik, 1987; Plofker,

Imhausen, Robson, & Dauben, 2007).

It is considered that, before 300 BC the Bijganit (बीजगणित) was derived from

the Bakhshali Manuscript (वाक्शाली लिपि) (Parajuli, 2021). At that time, the Bijganit

became one of the most influential practice and widely spread throughout the

continent. In pre-medieval era (500 BC – 400 AD) the development of Bijganit

took place rapidly as mathematicians were putting their hard and continuous effort

upon it. Further development of arithmetic and Bijganit took place in the later time

of the medieval era (400 AD – 1200 AD) which then got spread to Arab and other

countries. (Parajuli, 2021; Acharya, 2015; Boyer, 1991). Brahmagupta (598 AD –

668 AD) was a famous South-Asian mathematician and astronomer. He wrote the

method of solving linear and quadratic equations in his book Brahmasphutasiddhant

(ब्राह्म्स्फ़ुटसिद्धान्त) in 628 AD (Parajuli, 2021). It is greatly to the credit of Brahmagupta

that he gave all integral solutions of the linear Diophantine equation i.e., ax + by =

c, where a, b and c are constants (Boyer, 1991). Bhaskaracarya-II (1114 AD – 1993

AD) was the famous leading mathematician of ancient Indian during 12th century.

Bhaskaracarya wrote Algebra in his Siddhantasiromani (Patwardhan, Naimpally

& Singh, 2015). Hence, talking about the development of algebra, the South Asian

region has a long tradition of inventing new ideas, principles, and inventions. Since

the Vedic period, the land has been regarded as a fertile ground for new inventions

(Parajuli, 2021; Groza, 1968).

Many eastern and western mathematicians contributed to the modernization

of mathematics. They contributed to the subject of algebra as well as other areas

of mathematics. South Asian mathematicians have made a significant contribution

to these fields (Groza, 1968; Struik, 1987). Furthermore, Bharati Krishna Tirthaji

(1884 AD – 1960 AD), a South Asian glorious and divine human, made a significant

contribution to the development of mathematics in the nineteenth century (Tirthaji,

2015). He is well-known for rediscovering and reconstructing the new ideas of

mathematics from ancient Sanskrit text Veda early last century between 1911 – 1918

is popularly known today is Vedic Mathematics (Parajuli, 2021). The Vedic method,

he claims, is based on sixteen formulas and an equal number of sub-formulas that

cover all branches of mathematics, both pure and applied (Patwardhan, Naimpally &

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

3.
84 K. K. Parajuli

Singh, 2015; Parajuli, 2021).

Out of two levels of algebra, abstract algebra and its problems are not

addressed in this article. This paper concentrated solely on the fundamental level

of algebra, which are generally required in elementary mathematics courses.

Furthermore, this paper focused on how Vedic Mathematics uses Vedic techniques

for effectively solving these levels of problems. There are two types of techniques in

Vedic Mathematics: specific and general. This paper consists of specific parts with

the Vedic formulas as Paravartya Yojayet -k/fjTo{+ of]ho]t\_, Sunyam Samyasamuccaye

-z"Go+ ;fDo;d'Rro]_, Anurupye Sunyamanyat -cfg'¿Ko] z"GodGot\_, Antyayoreva -GToof]/]j_

and Lopanasthapanabhyam -nf]k:yfkgfEofd\_ which are the most commonly used for

elementary algebraic solutions.

Use of Vedic Formulas in Elementary Algebra

Vedic Mathematics is a system of reasoning and mathematical working based

on ancient mathematical concept as well as modern concepts with precise unique

techniques based on its formulas with simple rules and principles (Parajuli, 2021;

Tirthaji, 2015). Many Vedic formulas can be used to solve algebraic problems. This

paper does not include all Vedic formulas. Only those that are appropriate for some

specific fundamental algebraic operations have been selected.

Parāvartya Yojayet (k/fjTo{+ of]ho]t)\

The literal meaning of the formula Paravartya Yojayet is “Transpose and

apply” (Tirthaji, 2015). This formula is used in algebraic division, solve simple

linear, quadratic, cubic equations, partial fractions of algebraic expressions etc.

(Parajuli, 2021; Tirthaji, 2015; Parajuli, Jha, Acharya, & Maske, 2020; Williams,

(i) Use of Paravartya Yojayet for algebraic division.

Let f(x) = a0 xn + a1 xn–1 + a2 xn–2 + … + an, (a0 ≠ 0). Let f(x) is divided by g(x),

then f(x) = Q(x). g(x) + R; where, Q(x) is quotient, R is remainder, f(x) is dividend

and g(x) is divisor.

In the illustrated figure below, the last row will formulate the quotient and

remainder pieces. The highest degree of Q(x) is equal to . The remainder (R) is

determined by the number of terms on g(x). If g(x) has m terms, the remainder

parts is taken from the right most m columns, and the quotient parts are taken from

the left most part. The path of arrows can be used to understand algebraic division

procedures, as seen in the illustration below (Tirthaji, 2015; Williams, 2019; Dave, et

al., 2018).

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

Singh, 2015; Parajuli, 2021).

Out of two levels of algebra, abstract algebra and its problems are not

addressed in this article. This paper concentrated solely on the fundamental level

of algebra, which are generally required in elementary mathematics courses.

Furthermore, this paper focused on how Vedic Mathematics uses Vedic techniques

for effectively solving these levels of problems. There are two types of techniques in

Vedic Mathematics: specific and general. This paper consists of specific parts with

the Vedic formulas as Paravartya Yojayet -k/fjTo{+ of]ho]t\_, Sunyam Samyasamuccaye

-z"Go+ ;fDo;d'Rro]_, Anurupye Sunyamanyat -cfg'¿Ko] z"GodGot\_, Antyayoreva -GToof]/]j_

and Lopanasthapanabhyam -nf]k:yfkgfEofd\_ which are the most commonly used for

elementary algebraic solutions.

Use of Vedic Formulas in Elementary Algebra

Vedic Mathematics is a system of reasoning and mathematical working based

on ancient mathematical concept as well as modern concepts with precise unique

techniques based on its formulas with simple rules and principles (Parajuli, 2021;

Tirthaji, 2015). Many Vedic formulas can be used to solve algebraic problems. This

paper does not include all Vedic formulas. Only those that are appropriate for some

specific fundamental algebraic operations have been selected.

Parāvartya Yojayet (k/fjTo{+ of]ho]t)\

The literal meaning of the formula Paravartya Yojayet is “Transpose and

apply” (Tirthaji, 2015). This formula is used in algebraic division, solve simple

linear, quadratic, cubic equations, partial fractions of algebraic expressions etc.

(Parajuli, 2021; Tirthaji, 2015; Parajuli, Jha, Acharya, & Maske, 2020; Williams,

(i) Use of Paravartya Yojayet for algebraic division.

Let f(x) = a0 xn + a1 xn–1 + a2 xn–2 + … + an, (a0 ≠ 0). Let f(x) is divided by g(x),

then f(x) = Q(x). g(x) + R; where, Q(x) is quotient, R is remainder, f(x) is dividend

and g(x) is divisor.

In the illustrated figure below, the last row will formulate the quotient and

remainder pieces. The highest degree of Q(x) is equal to . The remainder (R) is

determined by the number of terms on g(x). If g(x) has m terms, the remainder

parts is taken from the right most m columns, and the quotient parts are taken from

the left most part. The path of arrows can be used to understand algebraic division

procedures, as seen in the illustration below (Tirthaji, 2015; Williams, 2019; Dave, et

al., 2018).

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

4.
Elementary Algebra on Vedic Mathematics 85

Division of 5x4 – 8x2 – 15x – 6 by x – 2 can be presented as follows.

x–2 5x4 + 0.x3 – 8x2 – 15x –6

2 5 0 –8 –15 –6

10 20 24 18

5 10 12 9 12

Here, Q(x) = 5x3 + 10x2 + 12x + 9; Remainder (R) = 12

The method is easily extended to the case where the divisor is a quadratic. For example,

the division x4 – x3 + x2 + 3x + 5 by x2 – x – 1 can be presented as follows.

x2 – x – 1 x4 – x3 + x2 + 3x +5

1 1 1 –1 1 3 5

1 1

0 0

2 2

1 0 2 5 7

∴ Quotient (Q) = x2 + 2 and Remainder (R) = 5x + 7.

(ii) Use of Paravartya Yojayet formula to solve simple equations

Paravartya formula can be used for special types of simple equations by merging

RHS into LHS under the different types of headings. If the sum of the numerators

2 5 7

on LHS is equal to the single numerator on RHS, for example + = i.e.,

x+3 x+4 x+2

N1(2) + N2(5) = N (7), then we apply the formula (where N1, N2 have their usual

meaning in Mathematics).

The merging procedure of RHS into LHS (Tirthaji, 2015; Parajuli, Jha, Acharya, &

Maskey, 2020; Parajuli, Jha, Acharya, & Maskey, 2019).

• For merging RHS into LHS, we subtract the independent part of denominators

of RHS from the independent part of denominators of LHS and multiply it to the

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

Division of 5x4 – 8x2 – 15x – 6 by x – 2 can be presented as follows.

x–2 5x4 + 0.x3 – 8x2 – 15x –6

2 5 0 –8 –15 –6

10 20 24 18

5 10 12 9 12

Here, Q(x) = 5x3 + 10x2 + 12x + 9; Remainder (R) = 12

The method is easily extended to the case where the divisor is a quadratic. For example,

the division x4 – x3 + x2 + 3x + 5 by x2 – x – 1 can be presented as follows.

x2 – x – 1 x4 – x3 + x2 + 3x +5

1 1 1 –1 1 3 5

1 1

0 0

2 2

1 0 2 5 7

∴ Quotient (Q) = x2 + 2 and Remainder (R) = 5x + 7.

(ii) Use of Paravartya Yojayet formula to solve simple equations

Paravartya formula can be used for special types of simple equations by merging

RHS into LHS under the different types of headings. If the sum of the numerators

2 5 7

on LHS is equal to the single numerator on RHS, for example + = i.e.,

x+3 x+4 x+2

N1(2) + N2(5) = N (7), then we apply the formula (where N1, N2 have their usual

meaning in Mathematics).

The merging procedure of RHS into LHS (Tirthaji, 2015; Parajuli, Jha, Acharya, &

Maskey, 2020; Parajuli, Jha, Acharya, & Maskey, 2019).

• For merging RHS into LHS, we subtract the independent part of denominators

of RHS from the independent part of denominators of LHS and multiply it to the

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

5.
86 K. K. Parajuli

respective part of their numerators of LHS.

• As the work of merging has been completed, we put zero on RHS and by simple

cross-multiplication we obtain the result of x.

2 × (3 − 2) 5(4 − 2) −19

Therefore, + = 0 ⇒ 2 + 10 = 0 ⇒ x =

x+3 x+4 x+3 x+4 6

We use the same method for three terms as in two terms to merge the RHS of simple

1 3 5 9

equations if N1 + N2 + N3 of LHS = N of RHS. Consider + + =

x+2 x+3 x+5 x+4

1(2 − 4) 3(3 − 4) 5(5 − 4)

Here, N1 (1) + N2 (3) + N3 (5) = N (9). Then, + + =0

x+2 x+3 x+5

2 3 5 −5

i.e., + = . Again, using the formula as above, we get, x =

x+2 x+3 x+5 2

(iii) Use of Paravartya Yojayet formula for solving Simultaneous Equations

Paravartya Yojayet formula for solving simultaneous linear equations of general

form as a1 x + b1= y c 2 , which would be somewhat similar to Cramer's rule

y c1 ; a 2 x + b2=

(Tirthaji, 2015; Parajuli, Jha, Acharya, & Maskey, 2020). The cross-multiplication

method to solve simultaneous linear equation which is taught in our present-

day curriculum is also similar to the method Paravartya Yojayet (Tirthaji, 2015;

Williams, 2019). The Vedic formula moves in a cyclic order.

(iv) Use of Paravartya Yojayet formula in Partial Fraction (Tirthaji, 2015;

Williams, 2019).

To find the partial fractions of a proper rational function whose denominator is a

px 2 + qx + r

product of two or more linear factors. If we have to express in the

(x − a)(x − b)(x − c)

shape of partial fraction, then we write

px 2 + qx + r A B C

= + +

(x − a) (x − b) (x − c)

, where A, B and C are written as,

(x − a)(x − b)(x − c)

px 2 + qx + r px 2 + qx + r px 2 + qx + r

=A = ;B = ;C

(x − b)(x − c) (x − a)(x − c) (x − a)(x − b)

For getting the value of A:

• equate the denominator of A to zero, i.e. x – a = 0 ⇒ x = a

px 2 + qx + r

• put x = a in A =

(x − b)(x − c)

• we do similarly for B and C.

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

respective part of their numerators of LHS.

• As the work of merging has been completed, we put zero on RHS and by simple

cross-multiplication we obtain the result of x.

2 × (3 − 2) 5(4 − 2) −19

Therefore, + = 0 ⇒ 2 + 10 = 0 ⇒ x =

x+3 x+4 x+3 x+4 6

We use the same method for three terms as in two terms to merge the RHS of simple

1 3 5 9

equations if N1 + N2 + N3 of LHS = N of RHS. Consider + + =

x+2 x+3 x+5 x+4

1(2 − 4) 3(3 − 4) 5(5 − 4)

Here, N1 (1) + N2 (3) + N3 (5) = N (9). Then, + + =0

x+2 x+3 x+5

2 3 5 −5

i.e., + = . Again, using the formula as above, we get, x =

x+2 x+3 x+5 2

(iii) Use of Paravartya Yojayet formula for solving Simultaneous Equations

Paravartya Yojayet formula for solving simultaneous linear equations of general

form as a1 x + b1= y c 2 , which would be somewhat similar to Cramer's rule

y c1 ; a 2 x + b2=

(Tirthaji, 2015; Parajuli, Jha, Acharya, & Maskey, 2020). The cross-multiplication

method to solve simultaneous linear equation which is taught in our present-

day curriculum is also similar to the method Paravartya Yojayet (Tirthaji, 2015;

Williams, 2019). The Vedic formula moves in a cyclic order.

(iv) Use of Paravartya Yojayet formula in Partial Fraction (Tirthaji, 2015;

Williams, 2019).

To find the partial fractions of a proper rational function whose denominator is a

px 2 + qx + r

product of two or more linear factors. If we have to express in the

(x − a)(x − b)(x − c)

shape of partial fraction, then we write

px 2 + qx + r A B C

= + +

(x − a) (x − b) (x − c)

, where A, B and C are written as,

(x − a)(x − b)(x − c)

px 2 + qx + r px 2 + qx + r px 2 + qx + r

=A = ;B = ;C

(x − b)(x − c) (x − a)(x − c) (x − a)(x − b)

For getting the value of A:

• equate the denominator of A to zero, i.e. x – a = 0 ⇒ x = a

px 2 + qx + r

• put x = a in A =

(x − b)(x − c)

• we do similarly for B and C.

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

6.
Elementary Algebra on Vedic Mathematics 87

Sunyam Samyasamuccaye] (z"Go+ ;fDo;d'Rro])

In Sanskrit Sunnyam means zero, Sammye means equal and Samuccaye means total.

Therefore, the meaning of Sunyam Samyasamuccaye is when the Samuccaya is the

same, the Samuccaye is zero, i.e. it should be equated to zero. The word Samuccaya

has several meanings in different context. This formula is used to solve linear

equations, special type of quadratic equations, rational expressions, cubic expressions

etc. (Tirthaji, 2015; Williams, 2019; Parajuli, 2019).

Meanings and Applications of Sunyam Samyasamuccaye: The meaning and its

corresponding applications of the formula can be expressed as follows.

(i) When Samuccaye is a common factor of unknown quantities in all the terms of

linear equation then equate that factor to zero (Parajuli, 2021; Williams, 2019;

Parajuli, Jha, Acharya, & Maskey, 2019).

When 5 (x – 4) + 8 (x – 4) = 3 (x – 4) – 7 (x – 4), where (x – 4) is a common

factor, so x = 4 is the solution.

(ii) When Samuccaye is the product of independent terms in the expression like

(x + a) (x + b) = (x + c) (x + d) such that ab = cd, then equate the variable to

zero (Parajuli, 2021; Tirthaji, 2015; Parajuli, Jha, Acharya, & Maskey, 2019).

(iii) When Samuccaye is the sum of the denominators of two fractions having the

same numerator then equate the sum of the denominators to zero (Parajuli,

2021; Tirthaji, 2015; Parajuli, Jha, Acharya, & Maskey, 2019). When

p p

+ = 0 ( where, p ≠ 0 ) , then the solution is obtained by (ax + b) + (cx + d) = 0

ax + b cx + d

(iv) When Samuccaye is the sum of the numerators or denominators of the

expression like:

ax + b ax + c

= where, sum of the numerators (N1+N2) = sum of the

ax + c ax + b

denominators (D1+D2)

then the value of x is obtained by equate them to zero (Tirthaji, 2015; Parajuli,

Jha, Acharya, & Maskey, 2019).

(v) When the Samuccaye is the sum of the numerators or denominators/difference

ax + b cx + d

of numerator and denominator of the expression like: =

cx + d ax + b

where, N1 + N2 = D1 + D2 = 0, gives the 1st root of the equation

and N1 – D1 = – (N2 – D2) = 0, gives the 2nd root of the equation (Parajuli,

2021; Williams, 2019; Parajuli, Jha, Acharya, & Maskey, 2019).

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

Sunyam Samyasamuccaye] (z"Go+ ;fDo;d'Rro])

In Sanskrit Sunnyam means zero, Sammye means equal and Samuccaye means total.

Therefore, the meaning of Sunyam Samyasamuccaye is when the Samuccaya is the

same, the Samuccaye is zero, i.e. it should be equated to zero. The word Samuccaya

has several meanings in different context. This formula is used to solve linear

equations, special type of quadratic equations, rational expressions, cubic expressions

etc. (Tirthaji, 2015; Williams, 2019; Parajuli, 2019).

Meanings and Applications of Sunyam Samyasamuccaye: The meaning and its

corresponding applications of the formula can be expressed as follows.

(i) When Samuccaye is a common factor of unknown quantities in all the terms of

linear equation then equate that factor to zero (Parajuli, 2021; Williams, 2019;

Parajuli, Jha, Acharya, & Maskey, 2019).

When 5 (x – 4) + 8 (x – 4) = 3 (x – 4) – 7 (x – 4), where (x – 4) is a common

factor, so x = 4 is the solution.

(ii) When Samuccaye is the product of independent terms in the expression like

(x + a) (x + b) = (x + c) (x + d) such that ab = cd, then equate the variable to

zero (Parajuli, 2021; Tirthaji, 2015; Parajuli, Jha, Acharya, & Maskey, 2019).

(iii) When Samuccaye is the sum of the denominators of two fractions having the

same numerator then equate the sum of the denominators to zero (Parajuli,

2021; Tirthaji, 2015; Parajuli, Jha, Acharya, & Maskey, 2019). When

p p

+ = 0 ( where, p ≠ 0 ) , then the solution is obtained by (ax + b) + (cx + d) = 0

ax + b cx + d

(iv) When Samuccaye is the sum of the numerators or denominators of the

expression like:

ax + b ax + c

= where, sum of the numerators (N1+N2) = sum of the

ax + c ax + b

denominators (D1+D2)

then the value of x is obtained by equate them to zero (Tirthaji, 2015; Parajuli,

Jha, Acharya, & Maskey, 2019).

(v) When the Samuccaye is the sum of the numerators or denominators/difference

ax + b cx + d

of numerator and denominator of the expression like: =

cx + d ax + b

where, N1 + N2 = D1 + D2 = 0, gives the 1st root of the equation

and N1 – D1 = – (N2 – D2) = 0, gives the 2nd root of the equation (Parajuli,

2021; Williams, 2019; Parajuli, Jha, Acharya, & Maskey, 2019).

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

7.
88 K. K. Parajuli

When the sum-total of denominator on LHS and RHS are same (having same

numerator), then the sum-total equate to zero (Parajuli, 2021; Parajuli, Jha,

Acharya, & Maskey, 2019). For example,

1 1 1 1 1 1 1 1

when − = − i.e., + = +

x−b x−b−d x−c+d x−c x−b x−c x−c+d x−b−d

b+c

where, D1 + D2 = D3 + D4 = 2x – b – c. Then, 2x – b – c = 0 ⇒ x =

2

(vi) The expression of the form 3x − 8 + 4x − 35 =2x − 9 + 5x − 34 can be reduced to

1 1 1 1 x −3 x −9 x −5 x−7

+ = + . So, can be applied (vi)th application (Parajuli,

x −3 x −9 x −5 x−7

2021; Parajuli, Jha, Acharya, & Maskey, 2019).

Anurupye Sunyamanyat (cfg'¿Ko] z"GodGot\)

The meaning of Anurupye Sunyamanyat is “If one is in ratio, the other one

is zero”. This formula is used to solve particular type of simultaneous linear and

quadratic equations. If the ratio of coefficients of one variable is equal to the ratio of

their corresponding independent terms of the equation, then another variable is zero

(Parajuli, 2021; Tirthaji, 2015; Williams, 2019).

Consider the simultaneous linear equations of two variables: 6x + 7y = 12; 5x + 21y = 10

6 12

Here, ratio of coefficient of x = Ratio of independent terms i.e., =

5 10

∴ y = 0 then x = 2.

Consider the simultaneous equations of three variables as follows.

ax + by + cz = cm … (i)

ax + ay + fz = fm … (ii)

mx + py + qz = qm … (iii)

Here, ratio of coefficient z = Ratio of their independent terms

∴ x = 0 & y = 0 then z = m.

Antyayoreva (cGToof]/]j)

The meaning of Antyayoreva is "only the last terms". i.e. while applying the

formula, last two digits i.e. unit place and tenth place digit are observed (Tirthaji,

2015). This formula is used to solve the specific types of rational expression. The

specific type means: the type of equation of those whose numerator and denominator

on the LHS bearing the independent terms stands in the same ratio to each other

as the entire numerator and the entire denominator of the RHS stand to each other

x 2 + 3x + 6 x+3

(Parajuli, 2021; Tirthaji, 2015; Bose, 2014). Consider = where,

x 2 + 5x + 5 x+5

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

When the sum-total of denominator on LHS and RHS are same (having same

numerator), then the sum-total equate to zero (Parajuli, 2021; Parajuli, Jha,

Acharya, & Maskey, 2019). For example,

1 1 1 1 1 1 1 1

when − = − i.e., + = +

x−b x−b−d x−c+d x−c x−b x−c x−c+d x−b−d

b+c

where, D1 + D2 = D3 + D4 = 2x – b – c. Then, 2x – b – c = 0 ⇒ x =

2

(vi) The expression of the form 3x − 8 + 4x − 35 =2x − 9 + 5x − 34 can be reduced to

1 1 1 1 x −3 x −9 x −5 x−7

+ = + . So, can be applied (vi)th application (Parajuli,

x −3 x −9 x −5 x−7

2021; Parajuli, Jha, Acharya, & Maskey, 2019).

Anurupye Sunyamanyat (cfg'¿Ko] z"GodGot\)

The meaning of Anurupye Sunyamanyat is “If one is in ratio, the other one

is zero”. This formula is used to solve particular type of simultaneous linear and

quadratic equations. If the ratio of coefficients of one variable is equal to the ratio of

their corresponding independent terms of the equation, then another variable is zero

(Parajuli, 2021; Tirthaji, 2015; Williams, 2019).

Consider the simultaneous linear equations of two variables: 6x + 7y = 12; 5x + 21y = 10

6 12

Here, ratio of coefficient of x = Ratio of independent terms i.e., =

5 10

∴ y = 0 then x = 2.

Consider the simultaneous equations of three variables as follows.

ax + by + cz = cm … (i)

ax + ay + fz = fm … (ii)

mx + py + qz = qm … (iii)

Here, ratio of coefficient z = Ratio of their independent terms

∴ x = 0 & y = 0 then z = m.

Antyayoreva (cGToof]/]j)

The meaning of Antyayoreva is "only the last terms". i.e. while applying the

formula, last two digits i.e. unit place and tenth place digit are observed (Tirthaji,

2015). This formula is used to solve the specific types of rational expression. The

specific type means: the type of equation of those whose numerator and denominator

on the LHS bearing the independent terms stands in the same ratio to each other

as the entire numerator and the entire denominator of the RHS stand to each other

x 2 + 3x + 6 x+3

(Parajuli, 2021; Tirthaji, 2015; Bose, 2014). Consider = where,

x 2 + 5x + 5 x+5

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

8.
Elementary Algebra on Vedic Mathematics 89

x 2 + 3x x(x + 3) x + 3

= = = RHS, which satisfies the condition of the Antyayoreva. So,

x 2 + 5x x(x + 5) x + 5

x+3 6 , i.e., x = –15.

=

x+5 5

Lopanasthapanabhyam (nf]kg:yfkgfEofd\)

The meaning of 'Lopanasthapanabhyam' is 'By alternate elimination and

retention'. This formula is used for factorization of 2nd degree homogeneous and

non-homogeneous polynomial functions with three or more variables. It is used

for finding HCF. It is also used in solving simultaneous equations, quadratic, cubic

equations etc. (Parajuli, 2021; Tirthaji, 2015; Williams, 2019).

(i) Use of Lopanasthapanabhyam for factorization

Specially, this formula is used for factorization of 2nd degree (homogenous and non-

homogeneous) polynomials in three or more variables. Therefore, it is an important

process to find the value of any variable which removes the difficulty of factorization

and make the factorization simple. Consider the polynomial of ax2 + by2 + cz2 +

dxy + eyz + fzx, which is homogeneous of second degree in three variables x, y, z

(Tirthaji, 2015; Williams, 2019).

For the factorization, we should apply the following steps as mentioned below

(Parajuli, 2021; Tirthaji, 2015; Glover, 2013).

• Eliminate z by putting z = 0 and retain two variables x and y. Factorize the

obtained polynomial using Adyamadyena (another Vedic formula) formula.

• Similarly, eliminate y by putting y = 0 and retain x and z and factorize the

obtained polynomial.

• With these two sets of factors, fill in the gaps caused by elimination process of z

and y respectively for finding the final factors of the original expression.

Consider the homogeneous polynomial, 3x2 + y2- 4xy – yz – 2z² - zx

Putting z = 0 gives the factors (x – y) (3x – y) and putting y = 0 gives (x – z) (3x + 2z)

By filling the gaps = (x – y – z) (3x – y + 2z)

∴ 3x² + y² – 4xy – yz – 2z² – zx = (x – y – z) (3x – y + 2z)

Consider the second-degree non-homogeneous polynomial

3x² + 6y² + 2z² + 11xy + 7xz + 7yz + 16x + 20y + 12z + 16

For non-homogeneous second-degree polynomial, we eliminate two variables

at a time and retain only one variable and the independent term, each time. Then we

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

x 2 + 3x x(x + 3) x + 3

= = = RHS, which satisfies the condition of the Antyayoreva. So,

x 2 + 5x x(x + 5) x + 5

x+3 6 , i.e., x = –15.

=

x+5 5

Lopanasthapanabhyam (nf]kg:yfkgfEofd\)

The meaning of 'Lopanasthapanabhyam' is 'By alternate elimination and

retention'. This formula is used for factorization of 2nd degree homogeneous and

non-homogeneous polynomial functions with three or more variables. It is used

for finding HCF. It is also used in solving simultaneous equations, quadratic, cubic

equations etc. (Parajuli, 2021; Tirthaji, 2015; Williams, 2019).

(i) Use of Lopanasthapanabhyam for factorization

Specially, this formula is used for factorization of 2nd degree (homogenous and non-

homogeneous) polynomials in three or more variables. Therefore, it is an important

process to find the value of any variable which removes the difficulty of factorization

and make the factorization simple. Consider the polynomial of ax2 + by2 + cz2 +

dxy + eyz + fzx, which is homogeneous of second degree in three variables x, y, z

(Tirthaji, 2015; Williams, 2019).

For the factorization, we should apply the following steps as mentioned below

(Parajuli, 2021; Tirthaji, 2015; Glover, 2013).

• Eliminate z by putting z = 0 and retain two variables x and y. Factorize the

obtained polynomial using Adyamadyena (another Vedic formula) formula.

• Similarly, eliminate y by putting y = 0 and retain x and z and factorize the

obtained polynomial.

• With these two sets of factors, fill in the gaps caused by elimination process of z

and y respectively for finding the final factors of the original expression.

Consider the homogeneous polynomial, 3x2 + y2- 4xy – yz – 2z² - zx

Putting z = 0 gives the factors (x – y) (3x – y) and putting y = 0 gives (x – z) (3x + 2z)

By filling the gaps = (x – y – z) (3x – y + 2z)

∴ 3x² + y² – 4xy – yz – 2z² – zx = (x – y – z) (3x – y + 2z)

Consider the second-degree non-homogeneous polynomial

3x² + 6y² + 2z² + 11xy + 7xz + 7yz + 16x + 20y + 12z + 16

For non-homogeneous second-degree polynomial, we eliminate two variables

at a time and retain only one variable and the independent term, each time. Then we

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

9.
90 K. K. Parajuli

obtain the result as described in the above steps. (We need one step more than the

homogenous cases)

Putting x = 0 and y = 0 gives the factors (z + 4) (2z + 4); putting y = 0 and z = 0 yields

(x + 4) (3x + 4); putting x = 0 and z = 0 yields (y + 2) (6y + 8) = (2y + 4) (3y + 4).

By filling the gaps: (x + 3y + 2z + 4) (3x + 2y + z + 4)

∴ 3x² + 6y² + 2z² + 11xy + 7xz + 7yz + 16x + 20y + 12z + 16

= (x + 3y + 2z + 4) (3x + 2y + z + 4).

(ii) Use of Lopanasthapanabhyam to find HCF: (Parajul, 2021; Tirthaji, 2015;

Williams, 2019; William, 2019)

Vedic Mathematics used the method elimination and retention to find HCF. In this

method, we divide through by factors continues until the remaining numbers are co-

prime as explained the following examples. By this method, we eliminate highest

power and lowest power of the given expression by adding or subtracting the given

expression from each other. We find the HCF by removing the common factor if any

from each (Parajuli, 2021; Tirthaji, 2015). For illustration, we find the HCF of the

expressions x³ – 3x² – 4x + 12 & x³ – 7x² + 16x – 12.

In addition, and subtraction columns, we should use the following steps as follows.

Adding Subtracting

x³ – 3x² – 4x + 12 x³ – 3x² – 4x + 12

x³ – 7x² + 16x – 12 x² – 7x² + 16x – 12

– + – +

2x³ – 10x² + 12x 4x² – 20x + 24

2x (x² – 5x + 6) 4(x² – 5x + 6)

(x² – 5x + 6) (x² – 5x + 6)

∴ HCF = (x² – 5x + 6)

Discussions

The illustration of Vedic formulas in the above sub-topic 2.0 is only confined

to a small portion of basic algebraic issues. The paper expresses how it can be

expanded from specific sections to its territorial conceptions and notions of simple

algebraic instances in this discussion section.

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

obtain the result as described in the above steps. (We need one step more than the

homogenous cases)

Putting x = 0 and y = 0 gives the factors (z + 4) (2z + 4); putting y = 0 and z = 0 yields

(x + 4) (3x + 4); putting x = 0 and z = 0 yields (y + 2) (6y + 8) = (2y + 4) (3y + 4).

By filling the gaps: (x + 3y + 2z + 4) (3x + 2y + z + 4)

∴ 3x² + 6y² + 2z² + 11xy + 7xz + 7yz + 16x + 20y + 12z + 16

= (x + 3y + 2z + 4) (3x + 2y + z + 4).

(ii) Use of Lopanasthapanabhyam to find HCF: (Parajul, 2021; Tirthaji, 2015;

Williams, 2019; William, 2019)

Vedic Mathematics used the method elimination and retention to find HCF. In this

method, we divide through by factors continues until the remaining numbers are co-

prime as explained the following examples. By this method, we eliminate highest

power and lowest power of the given expression by adding or subtracting the given

expression from each other. We find the HCF by removing the common factor if any

from each (Parajuli, 2021; Tirthaji, 2015). For illustration, we find the HCF of the

expressions x³ – 3x² – 4x + 12 & x³ – 7x² + 16x – 12.

In addition, and subtraction columns, we should use the following steps as follows.

Adding Subtracting

x³ – 3x² – 4x + 12 x³ – 3x² – 4x + 12

x³ – 7x² + 16x – 12 x² – 7x² + 16x – 12

– + – +

2x³ – 10x² + 12x 4x² – 20x + 24

2x (x² – 5x + 6) 4(x² – 5x + 6)

(x² – 5x + 6) (x² – 5x + 6)

∴ HCF = (x² – 5x + 6)

Discussions

The illustration of Vedic formulas in the above sub-topic 2.0 is only confined

to a small portion of basic algebraic issues. The paper expresses how it can be

expanded from specific sections to its territorial conceptions and notions of simple

algebraic instances in this discussion section.

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

10.
Elementary Algebra on Vedic Mathematics 91

In the case of Paravartya Yojayet

• In terms of algebraic division, the Paravartya Yojayet formula is extremely

similar to the Remainder Theorem and the Horner process of synthetic division.

Horner's synthetic division method is only a minor part of the Paravartya

formula, which can be expanded to circumstances when the divisor is quadratic,

cubic, or any size polynomial.

• In the illustration part Paravartya formula of above sub-topic 2, the first

coefficient of g(x) was unity; there was no possibility of a fractional coefficient

in Q(x) being special cases. However, a lack of unity can lead to uncertainty,

redundancy, and other issues. As a result, the best option will be to divide the

divisor by its first coefficient right away and finish the calculation as in special

cases.

• The merger formula of Paravartya Yojayet can be extended to any finite

number of terms for solving equations.

• In the case of improper fractions, Paravartya Yojayet can be used by expressing

the numerator as the sum of polynomial and proper fractions.

• To find the partial fraction of the expression whose denominator is repeated i.e.

square, cube etc. there is slight variation in process is required by the formula

Paravartya Yojayet.

In the case of Sunyam Samyasamuccaye

ax + b ax + c

• It is clear that the expression = would be linear. But the

ax + b cx + d ax + c ax + b

expression = would be quadratic. It should be careful that N1 + N2

cx + d ax + b

= D1 + D2 and N1 – D1 = D2 – N2 in both cases.

2 3 1 6

• The expression of the form 2x + 3 + 3x + 2 =x + 1 + 6x + 7 when N1 × D2 = N2 ×

D1 & N3 × D4 = N4 × D3 (where N1, N2, N3, N4, D1, D2, D3, D4 have their usual

meanings) which can be changed into the (vi)th application.

In the case of Anurupye Sunyamanyat

• The formula is especially more useful for solving simultaneous equations

whose solution is complex being the large coefficients like 8906x + 45y = 73;

1953x + 31y = 63.

• The formula can be extended to any number of unknown quantities.

In the case of Antayoreva

• This formula is more valuable when the problem is unsolved by the formula

Sunyam Samyasamuccaye even the Samuccaye are equal in the case like (x+3)

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

In the case of Paravartya Yojayet

• In terms of algebraic division, the Paravartya Yojayet formula is extremely

similar to the Remainder Theorem and the Horner process of synthetic division.

Horner's synthetic division method is only a minor part of the Paravartya

formula, which can be expanded to circumstances when the divisor is quadratic,

cubic, or any size polynomial.

• In the illustration part Paravartya formula of above sub-topic 2, the first

coefficient of g(x) was unity; there was no possibility of a fractional coefficient

in Q(x) being special cases. However, a lack of unity can lead to uncertainty,

redundancy, and other issues. As a result, the best option will be to divide the

divisor by its first coefficient right away and finish the calculation as in special

cases.

• The merger formula of Paravartya Yojayet can be extended to any finite

number of terms for solving equations.

• In the case of improper fractions, Paravartya Yojayet can be used by expressing

the numerator as the sum of polynomial and proper fractions.

• To find the partial fraction of the expression whose denominator is repeated i.e.

square, cube etc. there is slight variation in process is required by the formula

Paravartya Yojayet.

In the case of Sunyam Samyasamuccaye

ax + b ax + c

• It is clear that the expression = would be linear. But the

ax + b cx + d ax + c ax + b

expression = would be quadratic. It should be careful that N1 + N2

cx + d ax + b

= D1 + D2 and N1 – D1 = D2 – N2 in both cases.

2 3 1 6

• The expression of the form 2x + 3 + 3x + 2 =x + 1 + 6x + 7 when N1 × D2 = N2 ×

D1 & N3 × D4 = N4 × D3 (where N1, N2, N3, N4, D1, D2, D3, D4 have their usual

meanings) which can be changed into the (vi)th application.

In the case of Anurupye Sunyamanyat

• The formula is especially more useful for solving simultaneous equations

whose solution is complex being the large coefficients like 8906x + 45y = 73;

1953x + 31y = 63.

• The formula can be extended to any number of unknown quantities.

In the case of Antayoreva

• This formula is more valuable when the problem is unsolved by the formula

Sunyam Samyasamuccaye even the Samuccaye are equal in the case like (x+3)

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

11.
92 K. K. Parajuli

x 2 + 3x + 2 x+3

(x+4) (x+5) = (x+1) (x+2) (x+9), which becomes 2

= then

x + 9 x + 20 x+9

−7

using antayoreva x = .

3

In the case of Lopanasthapanabhyam

• In most cases, eliminating two of the three variables is enough to factorize the

polynomials. There may arises some exceptional cases like x2 + xy – 2y2 +

2xz – 5yz – 3z2. By putting y = 0 yields (x – z) (x + 3z) and z = 0 yields (x – y)

(x +2y). Here is confusion to take right combination of factors. As a result, if

x is also removed, the factor will be (– y – z) (2y + 3z). Then there will be no

mistake about which factors to use. Hence, x2 + xy – 2y2 + 2xz - 5yz - 3z2 =

(x – y – z) (x +2y + 3z).

• Formulas from the Vedas Sankalana Vyavakalanabhyam and

Lopasthapanabhyam function together to find the HCF of an algebraic

expression using a Vedic approach, while the traditional method uses

factorization and continuous division.

Conclusions

The impression of Vedic methodologies that were more or less previously

assimilated or integrated in modern mathematical systems, such as Horner's

synthetic division process, Remainder theorem of polynomials, Crammer's rule,

cross multiplication method to solve simultaneous equations, factorizations of

polynomials, partial fractions of proper and improper rational functions. Except some

cases, conventional mathematics practitioners are utterly unaware of many of the

Vedic formulas mentioned in this paper. Even so, there are significant limitations to

Vedic formulas in specific cases involving elementary algebraic problems.

Under the presentation, demonstration, discussion, and outcomes of this paper,

the study concludes that Vedic Mathematics and its five specific Vedic formulas

are more valuable and practicable to those learners who are having difficulty with

elementary level algebra using traditional methods. Traditional students must use a

calculator to solve numerical and algebraic problems, whereas Vedic students may

solve similar problems mentally.

Suggestions

It is preferable to compare the methods to conventional approaches in order to

assess their value and effectiveness. Without really using the technique, the true beauty

and efficiency of these methods cannot be fully comprehended.

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

x 2 + 3x + 2 x+3

(x+4) (x+5) = (x+1) (x+2) (x+9), which becomes 2

= then

x + 9 x + 20 x+9

−7

using antayoreva x = .

3

In the case of Lopanasthapanabhyam

• In most cases, eliminating two of the three variables is enough to factorize the

polynomials. There may arises some exceptional cases like x2 + xy – 2y2 +

2xz – 5yz – 3z2. By putting y = 0 yields (x – z) (x + 3z) and z = 0 yields (x – y)

(x +2y). Here is confusion to take right combination of factors. As a result, if

x is also removed, the factor will be (– y – z) (2y + 3z). Then there will be no

mistake about which factors to use. Hence, x2 + xy – 2y2 + 2xz - 5yz - 3z2 =

(x – y – z) (x +2y + 3z).

• Formulas from the Vedas Sankalana Vyavakalanabhyam and

Lopasthapanabhyam function together to find the HCF of an algebraic

expression using a Vedic approach, while the traditional method uses

factorization and continuous division.

Conclusions

The impression of Vedic methodologies that were more or less previously

assimilated or integrated in modern mathematical systems, such as Horner's

synthetic division process, Remainder theorem of polynomials, Crammer's rule,

cross multiplication method to solve simultaneous equations, factorizations of

polynomials, partial fractions of proper and improper rational functions. Except some

cases, conventional mathematics practitioners are utterly unaware of many of the

Vedic formulas mentioned in this paper. Even so, there are significant limitations to

Vedic formulas in specific cases involving elementary algebraic problems.

Under the presentation, demonstration, discussion, and outcomes of this paper,

the study concludes that Vedic Mathematics and its five specific Vedic formulas

are more valuable and practicable to those learners who are having difficulty with

elementary level algebra using traditional methods. Traditional students must use a

calculator to solve numerical and algebraic problems, whereas Vedic students may

solve similar problems mentally.

Suggestions

It is preferable to compare the methods to conventional approaches in order to

assess their value and effectiveness. Without really using the technique, the true beauty

and efficiency of these methods cannot be fully comprehended.

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

12.
Elementary Algebra on Vedic Mathematics 93

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New York: CRC Press.

Parajuli, K. K. (2021). A Study on Efficacy of Teaching Vedic Mathematics for

School Level In Nepal (Unpublished Ph.D. Thesis). Kathmandu: Research

Center, Nepal Sanskrit University.

Parajuli, K. K. (2021). Three Classical Methods to Find Cube Roots: A Connective

Prospectives on Lilavati, Vedic and Pande's Procedures. Journal of Nepal

Mathematical society, 4(1), 23-32. doi:https://doi.org/10.3126/jnms.

v4i1.37110

Parajuli, K. K. (2021). Three Seperate Methods for Squaring: A Connective

Prospective on Lilavati, Vedic and Trachtenberg. International Journal of

Statistics and Applied Mathematics, 6(2), 43-47. doi:https://doi.org/10.22271/

maths.2021.v6.i2a.673

Parajuli, K. K., Jha, K., Acharya, S. R., & Maskey, S. M. (2020). Connection of

Paravartya Sutra with Vedic and Non-Vedic Mathematics. 4th. International

Vedic Mathematics Conference. Hyderabad: IAVM, UK.

Parajuli, K. K., Jha, K., Acharya, S., & Maskey, S. M. (2019). Vedic Sutra Sunnyam

Sammya samuccaya for Solving Algebriac Equations. NCMA-2019. 7th, pp.

72-78. Butwal: Nepal Mathematical Society.

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

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Prospectives on Lilavati, Vedic and Pande's Procedures. Journal of Nepal

Mathematical society, 4(1), 23-32. doi:https://doi.org/10.3126/jnms.

v4i1.37110

Parajuli, K. K. (2021). Three Seperate Methods for Squaring: A Connective

Prospective on Lilavati, Vedic and Trachtenberg. International Journal of

Statistics and Applied Mathematics, 6(2), 43-47. doi:https://doi.org/10.22271/

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Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

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Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6

Patwardhan, K. S., Naimpally, S. A., & Singh, S. L. (2015). Lilavati of

Vhaskaracarya. New Delhi: Motilal Banarasidass Pvt. Ltd.

Plofker, K., Imhausen, A., Robson, E., & Dauben, J. W. (2007). The Mathematics of

Egypt, Mesopotamiya, China, India, and Islam: A Source Book. (V. J. Katz,

Ed.) New Jersey: Princeton University Press.

Sidhu, K. S. (1990). The Teaching of Mathematics. New Delhi: Sterling Publisher's

Pvt. ltd.

Struik, D. J. (1987). A Concise History of Mathematics (Third Revised ed.). New

York: Dover Publication Inc.

Tirthaji, B. K. (2015). Vedic Mathematics. (V. S. Agrawala, Ed.) New Delhi, India:

Motilal Banarasidass Pvt. Ltd.

Williams, K. (2019). Discover Vedic Mathematics. Scotland: Inspiration Books, UK.

Retrieved from http://www.vedicmaths.org

Williams, K. (2019). Vedic Mathematics Teacher Manual, Advanced Level. Scotland,

UK: Inspiration Books. Retrieved from http://www.vedicmaths.org

Mathematics Education Forum Chitwan, September 2021, Issue 6, Year 6