The fifth in a series. See the November 20 entry, “What is algebra?” for the first, the December 13 entry “When did algebra begin?” for the second, the December 19 entry “The golden age of Arabic mathematics” for the third, and the December 25 entry “al-Khwārizmī” for the fourth.
Modern algebra is generally acknowledged to have begun with the appearance around 830 CE of al-Khwārizmī’s book al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqābala. What better source can there be to find the answer to that perennial student question, “What is this stuff good for?” In the introduction to his seminal work, al-Khwārizmī stated that its purpose was to explain:
… what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.
It doesn’t get much more practical and useful than that! Either in 830 or today! Throughout history, the nations that led the world in mathematics led the world in commerce, industry, and science. In the 9th century, Baghdad was the commercial, industrial, and scientific center of world. In the 13th century, the leadership role crossed the Mediterranean to Italy, then over the ensuing centuries continued gradually westwards through Europe, crossing the Atlantic to the East coast of the US in the middle third of the 20th century, arriving in California in the 1980s, and likely to cross the Pacific (back) to China within the next couple of decades.
Al-Khwārizmī’s strong emphasis on practical applications typified Arabic texts of the time, every bit as much as the intense focus on applications of mathematics and science you find in today’s Silicon Valley.
The book was divided into three parts. The first part was devoted to algebra, giving the rules together with 39 worked problems, all abstract. Then came a short section on the rule of three and mensuration. Two mensuration problems dealing with surveying were solved with algebra. Finally, al-Khwārizmī presented a long section on inheritance problems solved by algebra.
The term al-jabr (“restoration” or “completion”) in al-Khwārizmī’s title refers to a procedure whose modern counterpart is eliminating negative terms from a (linear or quadratic) equation by adding an appropriate qantity to both sides of the equation. For example, using one of al-Khwārizmī’s own examples (but expressed using modern symbolic notation), al-jabr transforms
x2 = 40x – 4x2
5x2 = 40x.
The other key term in the title, al-muqābala (“confrontation”) refers to the process of eliminating identical quantities from the two sides of the equation. For example, (again in modern notation) one application of al-muqābala simplifies
50 + 3x + x2 = 29 + 10x
21 + 3x + x2 = 10x
and a second application simplifies that to
21 + x2 = 7x.
Procedurally (but not conceptually) these are the methods we use today to simplify and solve equations. Hence, a meaningful, modern English translation for Hisâb al-Jabr wa’l-Muqābala would be, simply, “Calculation with Algebra.”
The symbolic notation is not the only difference between medieval algebra and its present-day counterpart. The medieval mathematicians did not acknowledge negative numbers. For instance, they viewed “ten and a thing” (10 + x) as a composite expression (it entails two types of number: “simple numbers” and “roots”), but they did not see “ten less a thing” (10 – x) as composite. Rather, they thought of it as a single quantity, a “diminished” 10, or a 10 with a “defect” of x. The 10 retained its identity, even though x had been taken away from it. When an x was added to both sides of an equation, the diminished 10, (10 – x), was restored to its rightful value. Hence the terminology.
The first degree unknown, our x, was usually called shay’ (“thing”), but occasionally jidhr (“origin” or “base”, also “root” of a tree, giving rise to our present-day expression “root of an equation”). The second power, our x2, was called māl (a sum of money/property/ wealth). Units were generally counted in dirhams, a denomination of silver coin, occasionally simply “in number”. For example, al-Khwārizmī’s (rhetorical) equation “a hundred ten and two māls less twenty-two things equals fifty-four dirhams” corresponds to our symbolic equation 110 + 2x2 – 22x = 54.
Arabic authors typically explained the methods of algebra in two stages. First they provided an explanation of the names of the powers, described six simplified forms of equations and their solutions, and gave rules for operating on polynomials and roots. They then followed this introduction by a collection of solved problems which illustrated the methods.
Their solutions followed a standard template:
Stage 1: an unknown quantity was named (usually referred to as a “thing”), and an equation was set up.
Stage 2: the equation was simplifed to one of six canonical types.
Stage 3: the appropriate procedure was applied to arrive at the answer.
Because they allowed only positive coefficients, they had to consider six equation types, rather than the single template ax2 + bx + c = 0 we use today:
(1) māls equals roots (in modern terms, ax2 = bx),
(2) māls equals numbers (ax2 = c),
(3) roots equals numbers (bx = c),
(4) māls and roots equals numbers (ax2 + bx = c),
(5) māls and numbers equals roots (ax2 + c = bx),
(6) māls equals roots and numbers (ax2 = bx + c).
We see how al-Khwārizmī used the two simplification steps in Stage 2, al-jabr wa’l-muqābala, (“restoration and confrontation”) in his solution to a quadratic equation, which he described in these words:
If some one say: “You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times.” Computation: You say, ten less thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts.
The American scholar Jeffrey Oaks has translated this (fairly literally) as follows, adding headings to assist the reader:
If [someone] said, ten: you divided it into two parts. You multiplied one of the parts by itself, which is the same as eighty-one times the other.
Setting up and simplifying the equation
The rule for this is that you say ten less a thing by itself is a hundred and a mal less twenty things [which] equal eighty-one things. Restore the hundred and a mal by the twenty things and add them to the eighty-one [things]. This yields: a hundred and a mal equal a hundred roots and a root.
Solving the simplified equation
So halve the roots, which yields fifty and a half, and multiply it by itself, which yields two thousand five hundred fifth and a fourth. Subtract from it the hundred, leaving two thousand four hundred fifty and a fourth. Take its [square] root, which is forty-nine and a half. Subtract it from half the roots, which is fifty and a half. There remains one, which is one of the two parts.
Using modern notation, and substituting the letter x for “thing”, al-Khwārizmī was solving the equation
(10 − x)2 = 81x
which can be written in the equivalent form
x2 + 100 = 101x
Al-Khwārizmī did not state the equation
(10 − x)2 = 81x
Rather, he set up the equation
100 + x2 – 20x = 81x.
Nothing like the equation (10 − x)2 = 81x was ever stated in medieval algebra; the left side of such an expression entails what was then an unrealized operation. Medieval algebraists worked out all operations before stating equations, so al-Khwārizmī did not begin with (10 − x)2 = 81x, as we would, rather he first worked out the multiplication.
Having demonstrated methods for solving linear and quadratic equations, al-Khwārizmī proceeded to examine how to manipulate algebraic expressions. For example he showed how to multiply out specific numerical instances
(a + bx) (c + dx)
expressing everything in words, not symbols.
He ended the first section of the book by presenting the solutions to 39 problems.
In the following section, al-Khwārizmī presented solutions to some mensuration problems, including rules for finding the area of figures such as the circle and for finding the volume of solids such as the sphere, cone, and pyramid.
The final part of the book dealt with the complicated Islamic rules for inheritance, which involved the solution of linear equations.
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COMING UP: In the final article in this series I’ll summarize some of the amazing developments in algebra that were made in the Arabic period subsequent to al-Khwārizmī.