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{{Infobox person
| name              = Equations
| image              = [[File:Algebraic equation notation.svg|150px]]
}}


== Forming Equations ==
== Forming Equations ==
Before proceeding to the actual solution of an equation of any type, certain preliminary operations have necessarily to be carried out in order to prepare it for solution.
We need to carry out certain preliminary operations on equations  before moving into actual solution.


Still more preliminary work is that of forming the equation ( ''samī-karaṇa, samī-kāra'' or ''samī-kriyā; from sama, equal and kṛ ,'' to do; hence literally , making equal) from the conditions of the proposed problem. Such preliminary work may require the application of one or more fundamental operations of algebra or arithmetic.
We need to form the equation ( ''samī-karaṇa, samī-kāra'' or ''samī-kriyā; from sama, equal and kṛ ,'' to do; hence literally , making equal) from the given conditions of the proposed problem. This may require the application of one or more fundamental operations of algebra or arithmetic.


Bhāskara II says: "Let ''yāvat-tāvat'' be assumed as the value of the unknown quantity. Then doing precisely as has been specifically told-by subtracting, adding, multiplying or dividing the two equal sides of an equation should be very carefully built.
[[Bhaskara II|Bhāskara II]] says: "Let ''yāvat-tāvat'' be assumed as the value of the unknown quantity. Then doing precisely as has been specifically told-by subtracting, adding, multiplying or dividing the two equal sides of an equation should be very carefully built.


== Algebraic Expressions and Algebraic Equations ==
== Algebraic Expressions and Algebraic Equations ==
What is an algebraic expression ? Let us try to understand this with an example.
[[File:Equation illustration colour.svg|alt=Algebraic Equation|thumb|Algebraic Equation]]
Algebraic expression can be understood with the following example.<ref>{{Cite book|title=A Primer to Bhāratīya Gaṇitam , Bhāratīya-Gaṇita-Praveśa- Part-1|publisher=Samskrit Promotion Foundation|year=2021|isbn=978-81-951757-2-7}}</ref>


Geeta says that she has 10 marbles more than Mala. We do not know exactly how many marbles Mala has. She may have any number of marbles. But we know that
Ram says that he has 10 coins more than Shyam. We do not know exactly how many coins Ram has. Ram may have any number of coins. With the given information


Number of marbles of Geeta = Number of marbles of Mala = 10
Number of coins held by Ram = Number of coins held by Shyam + 10


We shall denote the ‘number of marbles of Mala' by the letter x. Here x is  unknown which could be 1, 2, 3, 4, etc.
We will denote the ‘number of coins held by  Shyam by the letter x. Here x is  unknown which could be 1, 2, 3, 4, etc.


Using x, we write,  
Using x, we write,  


Number of marbles of Geeta = x+10.
Number of coin held by Ram = x+10.


Thus 'x + 10' is an algebraic expression.
Thus 'x + 10' is an algebraic expression.


Algebra abounds in the usage of symbols. These symbols represent the unknown quantities and operations performed with them. The following table gives the symbols which were used for some basic operations by the ancient Indian Mathematicians.
Algebra utilizes the usage of symbols. These symbols represent the unknown quantities and operations performed with them. The following table gives the symbols which were used for some basic operations by the ancient Indian Mathematicians.
{| class="wikitable"
{| class="wikitable"
|+
|+
!S. No.
!S. No.
!Constituent of an algebraic expression
!Components of an algebraic expression
!Samskrit word
!Samskrit word
!Symbol
!Symbol
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नी , ........
नी , ........
|या  ३
|या  ३५


का  ४
का  १४


नी  ८
नी  ८२
|3x
|35x
4y
14y


8z
82z
|-
|-
|2
|2
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|<nowiki>-</nowiki>
|<nowiki>-</nowiki>
|या  का
|या  का
या ३ का ४
या ३५ का १४
|x + y
|x + y
3x + 4y
35x + 14y
|-
|-
|3
|3
Line 66: Line 71:
|भा
|भा
|याकाभा
|याकाभा
याकाभा  ३
याकाभा  ३२
|xy
|xy
3xy
32xy
|-
|-
|4
|4
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|रूपम्
|रूपम्
|रू
|रू
|रू  ३
|रू  ३२
|3
|32
|-
|-
|8
|8
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|dot above the quantity (.)
|dot above the quantity (.)
|'''.'''
|'''.'''
रू ४
रू ४३२
| -4
| -432
|}
|}
The letter yā (an abbreviation of yāvat-tāvat) was the most popular representation of the unknown quantity. Its square was termed yāva, the abbreviation of yāvat-tāvat-varga (varga means square). While writing equations, the constant term was denoted by the letter rū, an abbreviation of rūpa as seen in the table above. Any negative sign in the equation is denoted by a dot above the term.
The letter yā (an abbreviation of yāvat-tāvat) was the most popular representation of the unknown quantity. Its square was termed yāva, the abbreviation of yāvat-tāvat-varga (varga means square). constant term was denoted by the letter rū, an abbreviation of rūpa as shown in the above table. Any negative sign in the equation is denoted by a dot above the term.


If there are three unknown quantities in an expression, the symbols used are yā, kā and nī. These are the abbreviations for yāvat tāvat, kālaka and nīlaka. The product of first two unknown quantities is represented as yākābha where yā and kā are the two unknowns and bha stands for their product.  
If there are three unknown quantities in an expression, the symbols used are yā, kā and nī. These are the abbreviations for yāvat tāvat, kālaka and nīlaka. The product of first two unknown quantities is represented as yākābha where yā and kā are the two unknowns and bha stands for their product.  
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|-
|-
|1
|1
|x + 1
|x + 17
|या १ रू १
|या १ रू १७
|-
|-
|2
|2
|3x - 7
|7x - 17
|या ३ रू ७<sup>'''.'''</sup>
|या ७ रू १७<sup>'''.'''</sup>
|-
|-
|3
|3
|2x – 8
|18x – 8
|या २ रू ८<sup>'''.'''</sup>
|या १८ रू ८<sup>'''.'''</sup>
|-
|-
|4
|4
|15x<sup>2</sup> + 7x - 2
|15x<sup>2</sup> + 17x - 2
|याव १५  या ७ रू २<sup>'''.'''</sup>
|याव १५  या १७ रू २<sup>'''.'''</sup>
|-
|-
|5
|5
|1x<sup>4</sup> + 6x<sup>3</sup> + 25x<sup>2</sup> + 48x + 64
|1x<sup>4</sup> + 16x<sup>3</sup> + 25x<sup>2</sup> + 8x + 6
|यावव १ याघ ६ याव २५  या ४८ रू  ६४
|यावव १ याघ १६ याव २५  या ८ रू  ६
|-
|-
|6
|6
|18x<sup>2</sup> + 12xy - 6xz -6x
|8x<sup>2</sup> + 12xy - 6xz -16x
|याव १८ याकाभा १२  यानीभा ६<sup>'''.'''</sup>  या ६<sup>'''.'''</sup>
|याव ८ याकाभा १२  यानीभा ६<sup>'''.'''</sup>  या १६<sup>'''.'''</sup>
|}
|}
How algebraic expressions are written by Ancient Indian mathematicians.


== Algebraic Notations ==
Consider the equation 10 x - 18 = x<sup>2</sup> +14
* The symbols used for unknown numbers are the initial syllables yа̄ of yа̄vat-tа̄vat (as much as), kа̄ of kа̄laka (black), nī of nīlaka (blue), pī of pīta (yellow) etc.


* The product of two unknowns is denoted by the initial syllable bhā of bhāvita (product) placed after them. The powers are denoted by the initial letters ''va'' of ''varga'' (square), ''gha'' of ghana (cube); ''vava'' stands for v''argavarga'', the fourth power. Sometimes the initial syllable ''ghā'' of ''ghāta'' (product) stands for the sum of powers.
This can be written as,


* A coefficient is placed next to the symbol. The constant term is denoted by the initial symbol ''rū'' of ''rūpa'' (form).
0x<sup>2</sup> + 10 x - 18 = 1x<sup>2</sup> + 0x + 14


* A dot is placed above the negative integer
By looking at the positions of x<sup>2</sup>, x<sup>1</sup>, x<sup>0</sup> (constant term), there is some pattern ? The standard way of writing an equation starts from the highest power of x. Then the powers of x were written in the descending order up to its lowest power. This format of writing equation was followed by mathematicians from ancient times.


* The two sides of an equation are placed one below the other. Thus the equation X<sup>4</sup> - 2X<sup>2</sup> - 400x = 9999; is written as
[[Brahmagupta]] called an equation as samakaraṇa or samīkaraṇa. It means 'making equal. The two sides of an equation (LHS and RHS) were written one below the other. The symbol '=' was not used. Both the sides of an equation were made same by finding the appropriate value(s) for the unknown(s).


यावव​ १ याव २<sup>●</sup> या ४००<sup>●</sup> रू ०
Pṛthūdakasvāmin (864 CE) in his commentary on the Brāhma-sphuṭa-siddhānta  writes the equation  40x - 48 = x<sup>2</sup> + 51 as below
{| class="wikitable"
|+
!Devanāgari
!Transliteration
!
!Modern notation
|-
|याव ०  या ४०  रू ४८'''<sup>.</sup>'''
याव १  या ०    रू ५१
|Yāva  0  yā  40  rū  48'''<sup>.</sup>'''
Yāva  1  yā  0  rū  51
|⇒
|0x<sup>2</sup> + 40 x - 48 = 1x<sup>2</sup> + 0x + 51
|}
Here is an example of an equation from Bījagaṇita of Bhāskara II is:
 
X<sup>4</sup> - 2x<sup>2</sup> - 400x = 9999
 
This is represented as,


यावव​ ० याव ० या ० रू ९९९९
यावव १  याव २'''<sup>.</sup>'''    या  ४<sup>'''.'''</sup>००    रू  ०


which means writing x for या
यावव ०  याव ०    या  ०        रू  ९९९९


x<sup>4</sup> -2x<sup>2</sup> -400x+0 = 0x<sup>4</sup> +0x<sup>2</sup>+0x+9999
== Operations with Algebraic Expressions ==
Bhāskara II gives the operations using algebraic terms as follows:


If there be several unknowns, those of the same kind are written in the same column with zero coefficients, if necessary. Thus the equation
स्याद्रूपवर्णाभिहतौ तु वर्णो द्वित्र्यादिकानां समजातिकानाम् ॥


197x - 1644y - z = 6302 is represented by
वधे तु तद्वर्गघनादयः स्युस्तद्भावितं चासमजातिघाते।


yа̄ 197 kа̄ 1644<sup>●</sup> ni 1<sup>●</sup> ru 0
भागादिकं रूपवदेव शेषं व्यक्ते यदुक्तं गणिते तदत्र ॥<ref>Bījagaṇita, ch. Avyaktādi-guṇana, vs.6,7, p.8</ref>


yа̄ 0 kа̄ 0 ni 0 ru 6302
“The product of a numerical constant and  an unknown quantity is an unknown quantity.  Products of two or three like terms are their squares or cubes (respectively). Product of unlike terms is bhāvita. Fractions etc. are as in the case of knowns. The other (processes) are same as explained in arithmetic."


which means, putting y for kа̄ and z for ni
=== Addition and Subtraction of Algebraic Expressions ===
Bhāskara II gives the rule for addition and subtraction of unknown quantities as follows:


197x - 1644Y - z + 0 = 0x + 0y + 0z + 6302.
योगोऽन्तरं तेषु समानजात्योर्विभिन्नजात्योश्च पृथक् स्थितिश्च।<ref>Bījagaṇita ch. Avyakta-saṅkalana-vyavakalana, vs.6, p.7</ref>


Bhāskara II says:
“Addition and subtraction are performed amongst like terms. The unlike terms are to be kept separately."


"Then the unknown on one side of it (the equation) should be subtracted from the unknown on the other side; so also the square and other powers of the unknown;
'''Explanation:'''


the known quantities on the other side should be subtracted from the known quantities of another side."
Addition and subtraction can be performed with like terms and unlike terms are to be kept separately. Same letter variables raised to the same powers are treated as like terms. E.g., या ४, या ५, या ६ are like terms. याव ७, याव ८, याव ९ are also like terms. का ३, का ७, का १५  are also like terms. Presently we say 4x, 5x, 6x are like terms. Similarly 7x<sup>2</sup>, 8x<sup>2</sup>, 9x<sup>2</sup>  are like terms. and 3y, 7y, 15y are also like terms. When we have like terms, the sum and difference can be simplified. E.g. 4x + 6x can be simplified as 10x.  9x<sup>2</sup> - 7x<sup>2</sup>  can be simplified as 2x<sup>2</sup>.
 
Unlike terms are those terms having different variables or variables with different powers. E.g. या ३, याव ३, याघ ४, का ५, काव, याकाभा . Presently, these are represented as 3x, 3x<sup>2</sup>, 4x<sup>3</sup>, 5y, y<sup>2</sup>, xy.
 
=== Multiplication of Algebraic Expressions ===
Bījagaṇita gives a rule for multiplication -
 
गुण्यः पृथग्गुणकखण्डसमो निवेश्यस्तैः खण्डकैः क्रमहतः सहितो यथोक्त्या।
 
अव्यक्तवर्गकरणीगणनास चिन्त्यो व्यक्तोक्तखण्डगुणनाविधिरेवमत्र॥<ref>Bījagaṇita ch. Avyaktādi-guṇana, vs.8, p.8</ref>
 
“Place the multiplicand at as many places as the terms of the multiplier. Multiply with the terms of the multiplier in order separately and add the results as  directed in the problem. This is applicable in the case of squares of unknown numbers and surds also. The method of partial products stated in the case of arithmetic numbers is applicable here also.”
 
'''Explanation'''
{| class="wikitable"
|+
!Ancient Indian notation
!Modern notation
|-
|If या २ रू ४ and या ३ रू ५ are multiplicand and multiplier respectively,
we can get their product as mentioned below
|If 2x + 4 and 3x + 5 are multiplicand and multiplier respectively,
we can get their product as mentioned below
|-
|The multiplier contains two terms, i.e., या ३ and रू ५
|The multiplier contains two terms, i.e., 3x and 5
|-
|Multiply the multiplicand with the two terms of the multiplier separately as mentioned below.
(या २ रू ४) X या ३ = याव ६ या १२


The following illustration is from the Bījagaṇita of Bhāskara II:
(या २ रू ४) X रू ५ = या १०  रू  २०
|Multiply the multiplicand with the two terms of the multiplier separately as mentioned below.
(2x + 4) X 3x = 6x<sup>2</sup> + 12x


"Thus the two sides are
(2x + 4) X 5 = 10x + 20
|-
|Add the results.