Equations - Revision history

Mani: /* Quadratic Equations */

2022-01-25T12:01:15Z

Quadratic Equations

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	h = ½(d-√d<sup>2</sup> – c<sup>2</sup>)		h = ½(d-√d<sup>2</sup> – c<sup>2</sup>)

	'''Sridhara's Rule.''' ~~Sddhara~~ (c. 750) expressly indicates his method of solving the quadratic equation.		'''Sridhara's Rule.''' Sridhara (c. 750) expressly indicates his method of solving the quadratic equation.

	His treatise on algebra is now lost. But the relevant portion of it is preserved in quotations by Bhaskara II and others. Sridhara's method is: .		His treatise on algebra is now lost. But the relevant portion of it is preserved in quotations by Bhaskara II and others. Sridhara's method is: .

Mani: /* Second Type */

2022-01-25T11:45:03Z

Second Type

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	Where Σx stands for x<sub>1</sub> + x<sub>2</sub> +....+x<sub>n</sub>		Where Σx stands for x<sub>1</sub> + x<sub>2</sub> +....+x<sub>n</sub>

	But it will not be proper to say that equations of this type have been treated in the Bakhshali treatise. l They have however, been solved by Aryabhata (499) and ~~Mahivira~~ (850). The former 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.		But it will not be proper to say that equations of this type have been treated in the Bakhshali treatise. l They have however, been solved by Aryabhata (499) and Mahāvīra (850). The former 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.

	Σx = Σ<sup>n</sup><sub>r=1</sub> a<sub>r / (n-1)</sub>		Σx = Σ<sup>n</sup><sub>r=1</sub> a<sub>r / (n-1)</sub>

Mani: /* Second Type */

2022-01-25T11:41:47Z

Second Type

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	Where Σx stands for x<sub>1</sub> + x<sub>2</sub> +....+x<sub>n</sub>		Where Σx stands for x<sub>1</sub> + x<sub>2</sub> +....+x<sub>n</sub>

	But it will not be proper to say that equations of this. type have been treated in the ~~Bakhshili~~ treatise. l They have however, been solved by Aryabhata (499) and Mahivira (850). The former 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.		But it will not be proper to say that equations of this type have been treated in the Bakhshali treatise. l They have however, been solved by Aryabhata (499) and Mahivira (850). The former 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.

	Σx = Σ<sup>n</sup><sub>r=1</sub> a<sub>r / (n-1)</sub>		Σx = Σ<sup>n</sup><sub>r=1</sub> a<sub>r / (n-1)</sub>

Mani: /* Solution of Linear Equations */

2022-01-25T08:29:08Z

Solution of Linear Equations

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	Therefore x= 11		Therefore x= 11

	The following problem and its solution are from the ~~Bijaganita~~ of ~~Bhaskara~~ II :		The following problem and its solution are from the Bījaganita of Bhāskara 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		"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

Mani: changed from z to 2

2022-01-25T05:38:21Z

changed from z to 2

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	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 Chaturveda Prthudaka Swami (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.		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 Chaturveda Prthudaka Swami (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.''		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 (2) 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 ==		== Linear Equations in One Unknown ==

Mani: Name edited

2022-01-25T04:35:54Z

Name edited

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	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-varg''a). 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 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-varg''a). 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.		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 Chaturveda Prthudaka Swami (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.''		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.''

Mani: ā instead of a

2022-01-20T12:18:06Z

ā instead of a

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	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.		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.		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 Notations ==		== Algebraic Notations ==

Ramamurthys: /* Simultaneous Quadratic Equations */

2022-01-06T13:16:36Z

Simultaneous Quadratic Equations

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	y=½(a -√(a<sup>2</sup> - 4b ))		y=½(a -√(a<sup>2</sup> - 4b ))

			Narayana says: "The square-root of the square of the sum minus four times the product is the difference."

			For (3) Mahivlra gives the rule: "Add to and subtract twice the area (of a rectangle) from the square of the diagonal and extract the square roots. By ''sankramana'' between the greater and lesser of these (roots), the side and upright are found.

			x =½(√(c + 2b) + √(c - 2b))

			y =½(√(c + 2b) - √(c - 2b))

			For equations (4) Aryabhata I writes: "From the square of the sum (of two quantities) subtract the sum of their squares. Half of the remainder is their product."

			The remaining operations will be similar to those for the equations (2); so that

			x =½(a + √(2c –a<sup>2</sup>))

			y =½(a - √(2c –a<sup>2</sup>))

			Brahmagupta says: "Subtract the square of the sum from twice the sum of the squares; the square-root of the remainder being added to and subtracted from the sum and halved, gives the desired residues."

			Narayana has given two other forms of simultancous quadratic equations, namely,


			x - y = d......(5)

			x² - y² =m; xy = b ......(6)

			For the solution of (5) he gives the rule: "The square-root of twice the sum of the squares decreased by the square of the difference is equal to the

			sum."

			x + y = √(2c- d²)

			Therefore

			x = ½(√(2c- d²) + d)

			y = ½(√(2c- d²) - d)

			x = ~(V 2c - d2 + d), _'Y = -Hv 2.C - d2 - d).

			For (6) Narayana writes: "-

			"Suppose the square of the product as the product (of two quantities) and the difference of the squares as their difference. From them by ''sankrama'' will be obtained the (square) quantities. Their square-roots severally will give the quantities (required)."

			We have

			x² - y² =m

			x² y² = b<sup>2</sup>

			These are of the form (1). Therefore

			x² = ½(√(m<sup>2</sup> + 4b<sup>2</sup>) + m )

			y² = ½(√(m<sup>2</sup> + 4b<sup>2</sup>) - m )

			hence we get the values of x and y.

Ramamurthys: /* Equations Of Higher Degrees */

2022-01-06T08:29:02Z

Equations Of Higher Degrees

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	Here on adding four hundred x plus unity to the first side, the root can be extracted, but on adding the same to the other side, there will be no root of it. Thus the work (of reduction) does not proceed. Hence here , ingenuity (is called for). Here adding to both the sides four times the square of x, four hundred x and unity and then extracting roots, we get		Here on adding four hundred x plus unity to the first side, the root can be extracted, but on adding the same to the other side, there will be no root of it. Thus the work (of reduction) does not proceed. Hence here , ingenuity (is called for). Here adding to both the sides four times the square of x, four hundred x and unity and then extracting roots, we get

	~~x 2~~ + ox + I = ~~ox2~~ + ~~2.X~~ + 100.		x² + 0x+ 1 = 0x² + 2x + 100.

	Again, forming equation with these and proceeding as		Again, forming equation with these and proceeding as before, the value of x is obtained as 11 In similar instances the value of the unknown must be determined

	~~before,~~ the ~~value~~ of ~~x is obtained as 1 I~~. ~~In similar~~		by the ingenuity of the mathematician."

	~~instances the value~~ of the ~~unknown must be determined~~		== Simultaneous Quadratic Equations ==
			'''Common Forms''' Various problems involving simultaneous quadratic equations of the following forms have been treated by Hindu writers:

	by the ~~ingenuity o~~~ the ~~mathematician~~.		x - y = d ; xy = b ......(1)

			x + y = a ; xy = b ......(2

			x² + y² = c ; xy = b ......(3)

			x² + y² = c ; x + y = a ......(4)

			For the solution of (1) Aryabhata I (499) states the following rule:

			"The square-root of four times the product (of two quantities) added with the square of their difference, being added and diminished by their difference and halved gives the two multiplicands."

			i.e., x =½(√(d² + 4b + d)) y=½(√(d² + 4b - d))

			Brahmagupta (628) says: "The square-root of the sum of the square of the difference of the residues and two squared times the product of the residues, being added and subtracted by the difference of the residues, and halved (gives) the desired residues severally ."

			Narayana (1357) writes: "The square-root of the square of the difference of two quantities plus four times their product is their sum."

			"The square of the difference of the quantities together with twice their product is equal to the sum of their squares. The square-root of this result plus twice the product is the sum."

			For the solution of (2) the following rule is given by Mahavira(850) : "Subtract four times the area (of a rectangle) from the square of the semi-perimeter; then by ''sankramana'' between the square-root of that (remainder) and the semi-perimeter, the base and the upright are obtained."

			x =½(a +√(a<sup>2</sup> - 4b ))

			y=½(a -√(a<sup>2</sup> - 4b ))

Ramamurthys: /* Equations Of Higher Degrees */

2022-01-06T07:48:44Z

Equations Of Higher Degrees

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	== Equations Of Higher Degrees ==		== Equations Of Higher Degrees ==
	'''Cubic and Biquadratic.''' The Hindus did not achieve much in the solution of the cubic and biquadratic equations. Bhaskara II (1150) attempted the application of the method of the ''madhyamaharana'' (elimination of the middle) to those equations also so as to reduce them by means of advantageous transformations and introduction of auxiliary quantities to simple and quadratic equations respectively. He thus anticipated one of the modern methods of solving the biquadratic. "If, however," observes Bhaskara II, "due to the presence of the cube, biquadrate, etc., the work of reduction cannot proceed any further, after the performance of such operations, for want of a root of the unknown side (of an equation), then the value of the unknown must be obtained by the ingenuity (of the mathematician)." He has given two examples, one of the cubic and the other of the biquadratic, in which such reduction is possible		'''Cubic and Biquadratic.''' The Hindus did not achieve much in the solution of the cubic and biquadratic equations. Bhaskara II (1150) attempted the application of the method of the ''madhyamaharana'' (elimination of the middle) to those equations also so as to reduce them by means of advantageous transformations and introduction of auxiliary quantities to simple and quadratic equations respectively. He thus anticipated one of the modern methods of solving the biquadratic. "If, however," observes Bhaskara II, "due to the presence of the cube, biquadrate, etc., the work of reduction cannot proceed any further, after the performance of such operations, for want of a root of the unknown side (of an equation), then the value of the unknown must be obtained by the ingenuity (of the mathematician)." He has given two examples, one of the cubic and the other of the biquadratic, in which such reduction is possible.

			Example 1. "What is that number, which being multiplied by twelve and increased by the cube of the number, is equal to six times the square of the number added with thirty-five.

			Solution. "Here the number is x. This multiplied by twelve and increased by the cube of the number becomes x³ + 12x. It is equal to 6x² + 35. On making clearance, there appears on the first side x³ - 6x²+ 12x; on .the other side 35 . Adding negative eight to both the sides and extracting cube-roots, we get x - 2. = 0x + 3. And from this equation the number is found to be 5.

			Example 2. "What is that number which being multiplied by 200 and added to the square of the number, and then multiplied by 2 and subtracted from the fourth power of the number will become one myriad less unity? Tell that number .

			Solution. "Here the number is x; multiplied by 200 it becomes 200x; added to the square of the number, becomes x² + 200x; this being multiplied by two, 2x² + 400x; by this being diminished the fourth power of the number, namely, this x<sup>4</sup> becomes x<sup>4</sup>- 2x² - 400x. This is equal to a myriad less unity. Equiclearance having been made, the two sides will, be

			x<sup>4</sup>- 2x² - 400x. = 0x<sup>4</sup> + 0x² + 0x + 9999.

			Here on adding four hundred x plus unity to the first side, the root can be extracted, but on adding the same to the other side, there will be no root of it. Thus the work (of reduction) does not proceed. Hence here , ingenuity (is called for). Here adding to both the sides four times the square of x, four hundred x and unity and then extracting roots, we get

			x 2 + ox + I = ox2 + 2.X + 100.

			Again, forming equation with these and proceeding as

			before, the value of x is obtained as 1 I. In similar

			instances the value of the unknown must be determined

			by the ingenuity o~ the mathematician.