Equations: Difference between revisions

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.

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.

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.

What is an algebraic expression ? Let us try to understand this with an example.

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

Number of marbles of Geeta = Number of marbles of Mala = 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.

Number of marbles of Geeta = x+10.

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.

!Constituent of an algebraic expression

|या  ३

का  ४

नी  ८

|3x

या ३ का ४

3x + 4y

याकाभा  ३

|रू  ३

|3

रू ४

| -4

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.

|x + 1

|या १ रू १

|3x - 7

|या ३ रू ७^'''.'''

|2x – 8

|या २ रू ८^'''.'''

|15x² + 7x - 2

|याव १५  या ७ रू २^'''.'''

|1x⁴ + 6x³ + 25x² + 48x + 64

|यावव १ याघ ६ याव २५  या ४८ रू  ६४

|18x² + 12xy - 6xz -6x

|याव १८ याकाभा १२  यानीभा ६^'''.'''  या ६^'''.'''

We will see how algebraic expressions are written by ancient Indian mathematicians.  

Consider the equation 10 x - 8 = x² +1

0x² + 10 x - 8 = 1x² + 0x + 1

Observe the positions of x², x¹, x⁰ (constant term). Can you notice any 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.

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).

We shall see how this equation was written by Pṛthūdakasvāmin (864 CE) in his commentary on the Brāhma-sphuṭa-siddhānta. He writes the equation

 

10x - 8 = x² + 1 as follows:

|याव ०  या १०  रू ८'''^.'''

याव १  या ०    रू १

|Yāva  0 yā  10 rū  8'''^.'''  

Yāva 1  yā  0  rū 1

|0x² + 10 x - 8 = 1x² + 0x + 1

Another example of an equation from Bījagaṇita of Bhāskara II is:

It is well known that the addition and subtraction can be performed only amongst like terms and unlike terms are to be kept separately. Like terms are the terms that contain the same letter variables raised to the same powers. E.g., या ३, या ४, या ५  are like terms. याव २, याव ५, याव ७ are also like terms. का ३, का ७, का १५  are also like terms. Nowadays we say 3x, 4x, 5x are like terms. Similarly 2x², 5x², 7x²  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. 3x + 5x can be simplified as 8x.  10x² - 4x²  can be simplified as 6x².  

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 . In modern notation, these are represented as 3x, 3x², 4x³, 5y, y², xy.

|If या ३ रू ५ and या ४ रू ७ are multiplicand and multiplier respectively,  

their product can be obtained as follows:

|If 3x + 5 and 4x + 7 are multiplicand and multiplier respectively,

their product can be obtained as follows:

|The multiplier has two terms, i.e., या ४ and रू ७

|The multiplier has two terms, i.e., 4x and 7

|Place the multiplicand at two places. Multiply them with the terms of the multiplier separately as shown.

(या ३ रू ५) X या ४ = याव १२ या २०

(या ३ रू ५) X रू ७ = या २१ रू  ३५

|Place the multiplicand at two places. Multiply them with the terms of the multiplier separately as shown.

(3x + 5) X 4x = 12x² + 20x

(3x + 5) X 7 = 21x + 35

The multiplication result is: याव् १२ या ४१ रू ३५

The multiplication result is: 12x² + 41x + 35

If ax + b and cx + d  are multiplicand and multiplier respectively, their product can be obtained as follows:

The multiplier has two terms, i.e., cx and d.  Place the multiplicand at two places. Multiply them with the terms of the multiplier separately as shown.  

(ax +b) x cx = acx² + bcx

(ax + b) xd = adx + bd  

Add the results.  

The multiplication result is: acx² + (bc + ad)x + bd

== Algebraic Notations ==

* 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.

* A coefficient is placed next to the symbol. The constant term is denoted by the initial symbol ''rū'' of ''rūpa'' (form).

* A dot is placed above the negative integer

* The two sides of an equation are placed one below the other. Thus the equation X⁴ - 2X² - 400x = 9999; is written as

 

 

 

x⁴ -2x² -400x+0 = 0x⁴ +0x²+0x+9999

 

 

197x - 1644y - z = 6302 is represented by

 

 

 

 

197x - 1644Y - z + 0 = 0x + 0y + 0z + 6302.

 

Bhāskara II says:

 

"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;

 

the known quantities on the other side should be subtracted from the known quantities of another side."

The following illustration is from the Bījagaṇita of Bhāskara II:

 

 

 

 

 

 

 

 

The first class is again divided into two sub classes, viz.,(i) linear equations, and (ii) quadratic equations (''avyakta-varga-samīkaraṇa''). Here then we have the beginning of our present method of classifying equations according to their degrees. The method of classification adopted by Pṛthūdakasvāmī (860) is slightly different. His four classes are: (1) linear equations with one unknown, (2) linear equations with more unknowns, (3) equations with one, two or more unknowns in their second and higher powers, and (4) equations involving products of unknowns. As the method of solution of an equation of the third class is based upon the principle of the elimination of the middle term, that class is called by the name ''madhyamāharaṇa'' (from madhyama, "middle", aharana "elimination", hence meaning "elimination of the middle term"). For the other classes, the old names given by Brahmagupta have been retained. This method of classification has been followed by subsequent writers.

 

 

A Linear equation is an equation containing only the first power of the variables, coefficients and constants. For example, the equation 2x + 4 = 5 is a linear equation in one variable. This is called first-order equation since the power of the variable (x) is one. If the equation has highest power of x as two, i.e. x² , then it will  be a quadratic (second order) equation.

 

As already stated, the geometrical solution of a linear equation in one unknown is found in the ''śulba'' , the earliest of which is not later than 800 B.C. There is a reference in the ''Sthānāṅga-Sūtra''  (c. 300 B.C.) to a linear equation by its name (''yāvat -tāvat'' ) which is suggestive of the method of solution! followed at that time.We have, however, no further evidence about it. The earliest Hindu record of doubtless value of problems involving simple algebraic equations and of a method for their solution occurs in the Bakhshālī  treatise, which was written very probably about the beginning of the Christian Era.