Equations: Difference between revisions

We need to carry out certain preliminary operations on equations  before moving into actual solution.

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.

[[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 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>

[[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).

== Classification of Equations ==

In the canonical work of circa 300 B.C. found that Hindu classification of of equations  seems to have been according to their degrees, such as simple (technically called ''yāvat tāvat )'', quadratic (''varga''), cubic (''ghana'') and biquadratic (''varga-varg''a).

In the absence of further corroborative evidence, we cannot be sure of it. Brahmagupta (628) has classified equations as: (I) equations in one unknown (''eka-varna-samīkaraṇa''), (2) equations in several unknowns (''aneka-varna-samīkaraṇa''), and (3) equations involving products of unknowns (bhaivita).

Equations in one unknown (''eka-varna-samīkaraṇa'') 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. He classified as : (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.

Bhāskara II distinguishes two types in the third class, namely " (i) equations in one unknown in its second and higher powers and (ii) equations having two or more unknowns in their second and higher powers.' According to Krsna (1580) equations are primarily of two classes: (1) equations in one unknown and (z) equations in two or more unknowns. The first classification again, comprises of  two subclasses: (i) simple equations and (ii) quadratic and higher equations. The second classification has three subclasses: (i) simultaneous linear equations, (ii) equations involving the second and higher powers of unknowns, and (iii) equations involving products of unknowns. He then observes that these five classes can be reduced to four by including the second subclasses of classes (1) and (2) into one class as ''madhyamāharaṇa.''

A Linear equation is an equation having only the first power of the variables, coefficients and constants. For example, the equation 4x + 7 = 8 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.

=== Early Solutions: ===

In ''śulba'' geometrical solution of a linear equation in one unknown is found , the earliest of which is not later than 800 B.C.

''Sthānāṅga-Sūtra''  (c. 300 B.C.) has a reference to a linear equation by its name (''yāvat -tāvat'' )  which is suggestive of the method of solution followed at that time.

Bakhshālī treatise has problems involving simple algebraic equations and solution method, probably written in the beginning of the Christian Era.

<math>x+2x+6x+24x=132</math>

added

 

1 + 2 + 6 + 24 = 33

 

. "Divide the visible quantity'

on reduction becomes

This is the amount given to the first."

[[Aryabhata|āryabhaṭa I]](499) says:

This rule considers a problem of this kind: Two persons, who are equally rich, possess respectively a, b times a certain unknown amount together with c, d

Let x be the unknown amount, with the given information

 

Narayana writes:

One topic commonly discussed by almost all Hindu writers goes by the special name of ''sankramana'' (concurrence). According to [[Development of Mathematics|Nārāyana(1350)]], it is also called ''sankrama'' and ''sankraama''. Brahmagupta (628) includes it in algebra while others consider it as falling within the scope of arithmetic. As explained by the commentator Gangadhara (1420), the subject of discussion here is "the investigation of two quantities concurrent or grown together in the form of their sum and difference."

Bakhshālī  treatise talks about the earliest Hindu solving of linear equations involving several unknowns.

 

One problem in it runs as follows:

Another problem is "Five persons possess a certain amount of riches each. The riches of the first and the second mixed together amount to 16; the riches of the second and the third taken together are known to be 17; the riches of the third and the fourth taken together are known to be 18; the riches of the fourth and the fifth mixed together are 19; and the riches of the first and the fifth together amount to 20. Tell me what is the amount of each.

The general format of the linear equations is mentioned below.

A system of linear equations of this type is solved in the Bakhshālī  treatise as shown below.

But it will not be proper to say that equations of this. type have been treated in the Bakhshili treatise. They have however, been solved by āryabhaṭa(499) and Mahivira (850). āryabhaṭa says: "The (given) sums of certain (unknown) numbers, leaving out one number in succession, are added together separately and divided by the number of terms less one; that (quotient) will be the value of the whole.

Nārāyana says: "The sum of the depleted amounts divided by the number of persons less one, is the total amount. On subtracting from it the stated amounts severally will be found the different amounts."

A more generalized system of linear equations will be

"Three merchants begged money mutually from one another. The first on begging 4 from the second and 5 from the third became twice as rich as the others.The second on having 4 from the first and 6 from the third became thrice as rich. The third man on begging 5 from the first and 6.from the second became five times as rich as the others. O mathematician, if you know the ''citra-kuṭṭaka-miśra'' tell me quickly what was the amount in the hand of each."

Pṛthūdakasvāmī  (860) has explained it thus: "In an example in which there are two or more unknown quantities, colours such as ''yāvat -tāvat'' , ,etc.should be assumed for their values. Upon them should be performed all operations conformably to the statement of the example and thus should be carefully framed two or more sides and also equations. Equi-clearance should be made first between two and two of them and so on to the last: from one side one unknown should be cleared, other unknowns reduced to a common denominator and also the absolute numbers should be cleared from the side opposite. The residue of other unknowns being divided by the residual coefficient of the first unknown will give the value of the first unknown. If there be obtained several such values, then with two and two of them, equations should be formed after reduction to common denominators. Proceeding in this way to the end find out the value of one unknown. If that value be in terms of another unknown then the coefficients of those two will be reciprocally the values of the two unknowns. If, however, there be present more unknowns in that value, the method of the pulveriser should be employed. Arbitrary values may then be assumed for some of the unknowns." The above rule accepts the indeterminate as well as the determinate equations. All the examples provided by Brahmagupta in illustration of the rule are of indeterminate character.  

'''Bhāskara's Rule'''. Bhāskara II has given the same rule as that of Brahmagupta for the solution of simultaneous linear equations involving several unknowns.

Here is the illustrations from his works.

Substituting in the remaining equation, we get w = 200. Therefore

In the early canonical works of the Jainas (500-300 B.C) we see the  geometrical solution of the simple quadratic equation

<math>4h^2 -4dh =- c^2</math> . Also in the ''Tattvathadhigama-Sūtra''  of Umasvati (c. 150 B. C.) as  <math>{\displaystyle h = {\frac {1}{2}} (d-\sqrt{d^2-c^2})}</math>.

'''śrīdhara's Rule.'''   śrīdhara (c. 750) clearly indicates his method of solving the quadratic equation.

To solve the equation

Multiply by 4a on both sides

<math>(2ax+b)^2 = 4ac + b^2</math>  

'''Sripati's Rules'''. Sripati (1039) indicates two methods of solving the quadratic. There is a lacuna in our manuscript in the rule describing the first method, but it can be easily recognized to be the same as that of śrīdhara .  

 

"Multiply by four times the coefficient of the square of the unknown and add the square of the coefficient of the unknown; then extract the square-root divided by twice the coefficient of the square of the unknown, is said to be the value of the unknown."  

 

"Or multiplying by the coefficient of the square of the unknown and adding the square of half the coefficient of the unknown, extract the square-root. Then proceeding as before, it is diminished by half the coefficient of the unknown and divided by the coefficient of the square of the unknown. This quotient is said to be the value of the unknown."