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The Compendious Book on Calculation by Completion and Balancing contains methods for finding positive roots of linear and quadratic algebraic equations with positive coefficients. Select from the following menu to show roots of the six equation types considered by al-Khwārizmī:

Out[]=

​
equation a 2 x bx b>0&&a>0&&x b a

Use the following sliders to explore the solution of a quadratic equation of type 4 as demonstrated graphically by making implicit use of the algebraic identity

4

2

b

+4ab+

2

a



2

(a+2b)

:

Out[]=

​
a b

Computational Explanation

Al-Khwārizmī's work The Compendious Book on Calculation by Completion and Balancing, dating from around 820 CE, consists of several chapters. The first and best-known chapter provides a complete treatment for determining the positive roots of linear and quadratic polynomial equations. The second chapter is devoted to area and volume computation and contains a number of approximations to π. The remainder of the book focuses on the complex Islamic rules of inheritance, the determination of which requires the solution of linear equations.

Linear and Quadratic Equations

The Compendious Book provides a complete methodology for determining the positive roots of linear and quadratic polynomial equations. The book classifies quadratic equations into six basic types and gives methods to solve them. In the work, the equations were verbally described in terms of "squares"

(

2

x

)

, "roots" (

x

) and "numbers" (positive integers explicitly written out as words).

As described in the terminology of al-Khwārizmī (cf. van der Waeden 1985, p. 5), the equation types and their modern algebraic notations are given by:

1

.

Squares equal roots (

a

2

x

bx

)

2

.

Squares equal number (

a

2

x

c)

3

.

Roots equal number (

bxc

)

4

.

Squares and roots equal number (

a

2

x

+bxc

)

5

.

Squares and number equal roots (

a

2

x

+cbx

)

6

.

Roots and number equal squares (

bx+ca

2

x

)

Here, the constants a, b and c are all considered to be positive, and only positive solutions in x are allowed.

These equations can be solved directly in the Wolfram Language as follows:

In[]:=

eqns={a

2

x

==bx,a

2

x

==c,bxc,a

2

x

+bxc,a

2

x

+cbx,bx+ca

2

x

};

In[]:=

TextGrid[Table[{eqn,Reduce[eqn,x,PositiveReals]},{eqn,eqns}],DividersAll]//TraditionalForm

Out[]//TraditionalForm=

a 2 x bx	b>0∧a>0∧x b a
a 2 x c	c>0∧a>0∧x c a
bxc	c>0∧b>0∧x c b
a 2 x +bxc	c>0∧b>0∧a>0∧x 1 2 4ac+ 2 b 2 a - b 2a
a 2 x +cbx	c>0∧b>0∧0<a< 2 b 4c ∧x b 2a - 1 2 2 b -4ac 2 a ∨x 1 2 2 b -4ac 2 a + b 2a ∨a 2 b 4c ∧x b 2a - 1 2 2 b -4ac 2 a
bx+ca 2 x	c>0∧b>0∧a>0∧x 1 2 4ac+ 2 b 2 a + b 2a

Al-Khwārizmī solved these equations using operations known as al-ğabr ("forcing" or "restoring") and al-muqabala ("balancing" or "corresponding"). Here, al-ğabr is the operation of moving a "deficient" (negative) quantity from one side of the equation to the other (where it becomes positive) and the al-muqabala operation subtracts the same positive quantity from both sides.

To see how these methods worked, consider his solution to an equation of type 4, as detailed by van der Waerden (1985, pp. 7–9). In al-Khwārizmī's own words:

Roots and Squares equal to numbers.
​For instance: one square and ten roots of the same amount to thirty-nine dirhems; that is to say, what must the square which, when increased by ten of its own roots, amounts to thirty-nine?
The solution is: you halve the number of roots, which in the present instance yields five. This you multiply by itself; the product is twenty-five. Add this to thirty-nine; the sum is sixty-four. Now take the root of this, which is eight, and subtract from it half of the number of the roots, which is four. The remainder is three. This is the root of the square you thought for; the square itself is nine.

Converted to modern notation, the problem asks for the positive root of:

2

x

+10x39

… which it determines using the following steps:

2

(x+5)

39+2564​​x+5=

64

8​​x8-53.

The solution

x3

agrees with the result obtained by direct computation in the Wolfram Language:

In[]:=

Reduce[

2

x

+10x39,x,PositiveReals]

Out[]=

x3

While Islamic mathematicians did not consider negative numbers, the full solution set of this equation is given by:

In[]:=

Reduce[

2

x

+10x39,x]

Out[]=

x-13||x3

(The fact that a quadratic polynomial has two roots is a special case of the fundamental theorem of algebra.)

Al-Khwārizmī also provided a graphical demonstration of the solution by drawing a square with side length equal to the desired root x and rectangles of length 10/4 constructed on each of the sides:

The combined area of the central square (shown in light blue below) and rectangles (shown in yellow) is then taken as 39. The square is completed by adding four light-red rectangles with a total area given by four times 10/4, namely

4

2

10

4

25

:

Out[]=

This gives the area of the large square as

39+2564

, meaning it has side length

64

8

. The side length x of the original square is then

8-210/48-53

.

As illustrated, this construction is a special case of the general algebraic identity

4

2

b

+4ab+

2

a



2

(a+2b)

… which can be demonstrated as follows:

In[]:=

Factor[4

2

b

+4ab+

2

a

]

Out[]=

2

(a+2b)

For this example, al-Khwārizmī concluded by giving a simpler geometric proof that constructed rectangles only on two adjacent sides of the rectangle:

Out[]=

Mensuration

The second chapter of The Compendious Book is concerned with mensuration and consists mainly of rules for finding areas and volumes. For example, the area of a circle is given as the product of half its diameter by half its circumference.

Three rules for finding a circle's circumference C in terms of its diameter d, with the constant of proportionality corresponding to the quantity today known as π, were also given:

C3+

1

7

d​​C

10

2

d

​​C

62832

20000

d.

The first of these is due to Archimedes and Heron of Alexandria, the second to Brahmagupta and the last to Indian astronomer Aryabhata (cf. van Waerden 1985, p. 6).

Numerically, the ratios have values:

In[]:=

N3+

1

7

,

10

,

62832

20000

,10

Out[]=

{3.142857143,3.162277660,3.141600000}

… and they differ from the true value of π by the following amounts:

In[]:=

N3+

1

7

,

10

,

62832

20000

-π

Out[]=

{0.00126449,0.020685,7.34641×

-6

10

}

The Arithmetic of Inheritance

The rules for distributing estates as laid out in the Koran are complicated and require the solution of linear equations. Al-Khwārizmī devoted fully half of The Compendious Book to the algebra of Islamic inheritance, possibly as a result of the pronouncement attributed to the prophet Mohammed that the laws of inheritance comprise one-half of all useful knowledge (Gandz 1938; van der Waerden 1985, p. 6; King 1994, p. 83).

Additional Reading

• Berggren, J. L. "Al-Khwārizmī's Compendium on Calculation by Completion and Reduction." In The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (Ed. V. J. Katz). Princeton, NJ: Princeton University Press, pp. 542–548, 2007.
• Gandz, S. "The Algebra of Inheritance." Osiris, Vol. 5, pp. 319–391, 1938.
• Hughes, B. B. (Ed.). Robert of Chester's Latin Translation of Al-Khwarizmi's Al-Jabr: A New Critical Edition. Wiesbaden, Germany: F. Steiner Verlag, 1989.
• Katz, V. J. and Parshall, K. H. Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century. Princeton, NJ: Princeton University Press, 2020.
• King, D. A. "Mathematics Applied to Aspects of Religious Ritual in Islam: The Arithmetic of Inheritance." Ch. 1.7, § 5 in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Vol. 1 (Ed. I. Grattan-Guinness). London: Routledge, p. 83, 1994.
• Merzbach, U. C. and Boyer, C. B. "Al-Kharizmi." In A History of Mathematics, 3rd ed. New York: Wiley, pp. 206–212, 2011.
• Rashed, R. The Development of Arabic Mathematics: Between Arithmetic and Algebra. Dordrecht, Netherlands: Kluwer, 1994.
• Rosen, T. (Ed. and trans.). The Algebra of Mohammed ben Musa. London: J. L. Cox, 1831.
• van der Waerden, B. L. "Three Muslim Authors: Al-Khwārizmī." Ch. 1, Part A in A History of Algebra: From al-Khwārizmī to Emmy Noether. Berlin: Springer-Verlag, pp. 3–15, 1985.

Additional Links

• The Algebra of Mohammed ben Musa: Edited and Translated by Fredric Rosen
• Columbia University of New York: Butler Library: QA32 .K48 2009g
• MacTutor History of Mathematics Site: Abu Ja'far Muhammad ibn Musa Al-Khwarizmi
• Mathematical Treasures - Al-Khwarizmi's Algebra
• Wikipedia: The Compendious Book on Calculation by Completion and Balancing
• Wikipedia: Muhammad ibn Musa al-Khwarizmi