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.
Still more preliminary work is that of forming the equation (sami-karana, sami-kara or sami-kriya; 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.
Bhaskara II says: "Let yavat-tavat 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 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 vargavarga, 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
यावव १ याव २● या ४००● रू ०
यावव ० याव ० या ० रू ९९९९
which means writing x for या
x4 -2x2 -400x+0 = 0x4 +0x2+0x+9999
If there be several unknowns, those of the same kind are written in the same column with zero coefficients, if necessary. Thus the equation
197x - 1644y - z = 6302 is represented by
yа̄ 197 kа̄ 1644● ni 1● ru 0
yа̄ 0 kа̄ 0 ni 0 ru 6302
which means, putting y for kа̄ and z for ni
197x - 1644Y - z + 0 = 0x + 0y + 0z + 6302.
Bhaskara 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 Bijaganita of Bhaskara II:
"Thus the two sides are
ya va 4 ya 34● ru 72
ya va 0 ya 0 ru 90
On complete clearance (samasodhana), the residues of the two sides are
ya va 4 ya 34● ru 0
ya va 0 ya 0 ru 18
i.e., 4x2 -34x= 18
Classification of Equations
The earliest Hindu classification of equations seems to have been according to their degrees, such as simple (technically called yavat tavat), quadratic (varga), cubic (ghana) and biquadratic (varga-varga). Reference to it is found in a canonical work of circa 300 B.C. But 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-samikarana), (2) equations in several unknowns (aneka-varna-samikarana), and (3) equations involving products of unknowns (bhaivita).
The first class is again divided into two sub classes, viz.,(i) linear equations, and (ii) quadratic equations (avyakta-varga-samikarana). Here then we have the beginning of our present method of classifying equations according to their degrees. The method of classification adopted by Prthudakasvami (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 madhyamaharana (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.
Bhaskara II distinguishes two types in the third class, viZ" (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 class (1), again, comprises two subclasses: (i) simple equations and (ii) quadratic and higher equations. The class (2) 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 madhyamaharana.
Linear Equations in One Unknown
Early Solutions:
As already stated, the geometrical solution of a linear equation in one unknown is found in the Sulba, the earliest of which is not later than 800 B.C. There is a reference in the Sthananga-Sutra (c. 300 B.C.) to a linear equation by its name (yavat-tavat) 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 Bakhshali treatise, which was written very probably about the beginning of the Christian Era.
One problem is "The amount given to the first is not known. The second is given twice as much as the first; the third thrice as much as the second; and the fourth four times as much as the third. The total amount distributed is 132. What is the amount of the first?"
If x be the amount given to the first, then according to the problem,
x + 2X + 6x + 24X = 132.
Rule of False Position:
The solution of this equation is given as follows:
" 'Putting any desired quantity in the vacant place' ; any desired quantity is 1 ; 'then construct the series.
| 1 | 2 | 2 3 | 6 4 |
| 1 | 1 | 1 1 | 1 1 |
'multiplied'
| 1 | 2 | 6 | 24 |
added' 33. "Divide the visible quantity'
| 132
33 |
(which) on reduction becomes
| 4
1 |
(This is) the amount given (to the first)."
The solution of another set of problems in the Bakhshali treatise, leads ultimately to an equation of the type ax+ b=p. The method given for its solution is to put any arbitrary value g for x, so that
ag+ b =p' say.
Then the correct value will be
x = (p - p')/a +g
Solution of Linear Equations
Aryabhata I(499) says:
"The difference of the known "amounts" relating to the two persons should be divided by the difference of the coefficients of the unknown. The quotient will be the value of the unknown, if their possessions be equal."
This rule contemplates a problem of this kind: Two persons, who are equally rich, possess respectively a, b times a certain unknown amount together with c, d
units of money in cash. What is that amount?
If x be the unknown amount, then by the problem
ax+ c= bx+ d.
Therefore x = (d-c) / (a-b)
Hence the rule.
Brahmagupta says:
"In a (linear) equation in one unknown, the difference of the known terms taken in the reverse order, divided by the difference of the coefficients of the unknown
(is the value of the unknown).
Sripati writes :
"First remove the unknown from anyone of the sides (of the equation) leaving the known term; the reverse (should be done) on the other side. The difference of the absolute terms taken in the reverse order divided by the difference of the coefficients of the unknown will be the value of the unknown.
Bhaskara II states:
"Subtract the unknown on one side from that on the other and the absolute term on the second from that on the first side. The residual absolute number should be divided by the residual coefficient of the unknown; thus the value of the unknown becomes known.
Naraya writes:
"From one side clear off the' unknown and from the other the known quantities; then divide the residual known by the residual coefficient of the unknown. Thus will certainly become known the value of the unknown. "
For illustration we take a problem proposed by Brahmagupta :
"Tell the number of elapsed days for the time when four times the twelfth part of the residual degrees increased by one, plus eight will be equal to the residual
degrees plus one."
It has been solved by Prthudakasvami as follows:
"Here the residual degrees are (put as) yavat-tavat,
ya increased by one, ya 1 ru 1; twelfth part of it, (ya 1 ru 1) / 12
four times this, (ya 1 ru 1) / 3 ; plus the absolute quantity eight, (ya 1 ru 25) / 3 . This is equal to the residual degrees plus unity. The statement of both sides
tripled is
ya 1 ru 25
ya 3 ru 3
The difference between the coefficients of the unknown is 2. By this the difference of the absolute terms, namely 22, being divided, is produced the residual of the degrees of the sun 11. These residual degrees should be known to be irreducible. The elapsed days can be deduced then, (proceeding) as before."
In other words, we have to solve the equation
(x + 1)4/12 + 8 = x + 1
which gives x + 25 = 3x + 3
2x = 22
Therefore x= 11
The following problem and its solution are from the Bijaganita of Bhaskara II :
"One person has three hundred coins and six horses. Another has ten horses (each) of similar value and he has further a debt of hundred coins. But they
are of equal worth. What is the price of a horse?
"Here the statement for equi-clearance is :
6x + 300 = 10x - 100.
Now, by the rule, 'Subtract the unknown on one side from that on the other etc.,' unknown on the first side being subtracted from the unknown on the other side,
the remainder is 4x. The absolute term on the second side being subtracted from the absolute term on the first side, the remainder is 400. The residual known
number 400 being divided by the coefficient of the residual unknown 4x, the quotient is recognized to be the value of x, (namely) 100."
Linear Equations with Two Unknowns
Rule of Concurrence
One topic commonly discussed by almost all Hindu writers goes by the special name of sankramana (concurrence). According to Narayana (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 Ganga-,dhara (1420), the subject of discussion here is "the investigation of two quantities concurrent or grown together in the form of their sum and difference."
In other words sankramana is the solution of the simultaneous equations
x+ y= a, x-y= b.
Brahmagupta's rule for solution is: "The sum is increased and diminished by the difference and divided by two; (the result will be the two unknown quantities): (this is) concurrence. The same rule is restated by him on a different occasion in the form of a problem and its solution.
"The sum and difference of the residues of two (heavenly bodies) are known in degrees and minutes. What are the residues? The difference is both added to and subtracted from the sum, and halved; (the results are) the residues.