The Math Book
Page 11
Leonhard Euler
Rational answers
Many of the equations that al-Khwarizmi was dealing with had solutions that could not be expressed rationally and completely using the Hindu-Arabic decimal system. Although numbers such as —the square root of 2—had been known since ancient Greek times and from even earlier Babylonian clay tablets, in 825 CE, al-Khwarizmi was the first to make the distinction between rational numbers—which can be made into fractions—and irrational numbers, which have an indefinite string of decimals with no recurring pattern. Al-Khwarizmi described rational numbers as “audible” and irrational numbers as “inaudible.”
Al-Khwarizmi’s work was developed further by Egyptian mathematician Abu Kamil Shuja ibn Aslam (c. 850–930 CE), whose Book of Algebra was designed to be an academic treatise for other mathematicians, rather than for educated people who had a more amateur interest. Abu Kamil embraced irrational numbers as possible solutions to quadratic equations, rather than rejecting them as awkward anomalies. In his Book of Rare Things in the Art of Calculation, Abu Kamil attempted to solve indeterminate equations (those with more than one solution). He further explored this topic in his Book of Birds, in which he posed a miscellany of bird-related algebra problems, including: “How many ways can one buy 100 birds in the market with 100 dirhams?”
Algebra is but written geometry and geometry is but figured algebra.
Sophie Germain
French mathematician
Geometric solutions
Up until the era of the Arab “algebraists”—from al-Khwarizmi in the 9th century to the death of the Moorish mathematician al-Qalasadi in 1486—the key developments within algebra were underpinned by geometrical representations. For example, al-Khwarizmi’s method of “completing the square” in order to solve quadratic equations relies on consideration of the properties of a real square; later scholars worked in a similar way. Mathematician and poet Omar Khayyam, for example, was interested in solving problems using the relatively new discipline of algebra, but employed both geometrical and algebraic methods. His Treatise on Demonstration of Problems of Algebra (1070) notably includes a fresh perspective on the difficulties within Euclid’s postulates, a set of geometric rules that are assumed to be true without requiring a proof. Picking up on earlier work by al-Karaji, Khayyam also develops ideas about binomial coefficients, which determine how many ways there are to select a number of items from a larger set. He solved cubic equations, too, inspired by al-Khwarizmi’s use of Euclid’s geometrical constructions for working out quadratic equations.
Al-Khwarizmi showed how to solve quadratic equations by a method known as “completing the square.” This example shows how to find x in the equation x2 + 10x = 39.
Polynomials
During the 10th and early 11th centuries, a more abstract theory of algebra was developed, which was not reliant on geometry—an important factor in establishing its academic status. Al-Karaji was instrumental in this development. He established a set of procedures for performing arithmetic on polynomials—expressions that contain a mixture of algebraic terms. He created rules for calculating with polynomials, in much the same way that there were rules for adding, subtracting, or multiplying numbers. This allowed mathematicians to work on increasingly complex algebraic expressions in a more uniform way, and reinforced algebra’s essential links with arithmetic.
Mathematical proof is a vital part of modern algebra and one of the tools of proof is called mathematical induction. Al-Karaji used a basic form of this principle, whereby he would show an algebraic statement to be true for the simplest case (say n = 1), then use that fact to show that it must also be true for n = 2 and so on, with the inevitable conclusion that the statement must hold true for all possible values of n.
One of al-Karaji’s successors was the 12th-century scholar Ibn Yahya al-Maghribi al-Samaw’al. He noted that the new way of thinking of algebra as a kind of arithmetic with generalized rules involved the algebraist “operating on the unknown using all the arithmetical tools, in the same way as the arithmetician operates on the known.” Al-Samaw’al continued al-Karaji’s work on polynomials, but also developed the laws of indices, which led to much later work on logarithms and exponentials, and was a significant step forward in mathematics.
An ounce of algebra is worth a ton of verbal argument.
John B. S. Haldane
British mathematical biologist
Islamic mathematicians gather in the library of a mosque in an illustration from a manuscript by the 12th-century poet and scholar Al-Hariri of Basra.
Plotting equations
Cubic equations had challenged mathematicians since the time of Diophantus of Alexandria. Al-Khwarizmi and Khayyam had made significant progress in understanding them—work further developed by Sharaf al-Din al-Tusi, a 12th-century scholar, probably born in Iran, whose mathematics appears to have been inspired by the work of earlier Greek scholars, especially Archimedes. Al-Tusi was more interested in determining types of cubic equation than al-Khwarizmi and Khayyam had been. He also developed an early understanding of graphical curves, articulating the significance of maximum and minimum values. His work strengthened the connection between algebraic equations and graphs—between mathematical symbols and visual representations.
As the sun eclipses the stars by its brilliancy, so the man of knowledge will eclipse the fame of others in assemblies of the people if he proposes algebraic problems, and still more if he solves them.
Brahmagupta
A new algebra
The discoveries and rules set down by medieval Arab scholars still form the basis of algebra today. The works of al-Khwarizmi and his successors were key to establishing algebra as a discipline in its own right. It was not until the 1500s, however, that mathematicians began to abbreviate equations by using letters to stand for known and unknown variables. French mathematician François Viète was key to this development. In his works, he pioneered the move away from the Arabic algebra of procedures toward what is known as symbolic algebra.
In his Introduction to the Analytic Arts (1591), Viète suggested that mathematicians should use letters to symbolize the variables in an equation: vowels to represent unknown quantities and consonants to represent the known. Although this convention was eventually replaced by René Descartes—in which letters at the beginning of the alphabet represent known numbers and letters at the end represent the unknown—Viète nonetheless was responsible for simplifying algebraic language far beyond what the Arab scholars had imagined. The innovation allowed mathematicians to write out increasingly complex and detailed abstract equations, without using geometry. Without symbolic algebra, it would be difficult to imagine how modern mathematics would have ever developed.
Islamic algebraists wrote equations as text with accompanying diagrams, as in the 14th-century Treatise on the Question of Arithmetic Code by Master Ala-El-Din Muhammed El Ferjumedhi.
AL-KHWARIZMI
Born in c. 780 CE near what is now Khiva, Uzbekistan, Muhammad Ibn Musa al-Khwarizmi moved to Baghdad, where he became a scholar at the House of Wisdom.
Al-Khwarizmi is regarded as the “father of algebra” for his systematic rules for solving linear and quadratic equations. These were outlined in his major work on calculation by “completion and balancing”—methods he devised that are still used today. Other achievements include his text on Hindu numerals, which, in its Latin translation, introduced Europe to Hindu-Arabic numerals. He wrote a book on geography, helped construct a world map, took part in a project to determine the circumference of Earth, developed the astrolabe (an earlier Greek tool for navigation), and compiled a set of astronomical tables. Al-Khwarizmi died around 850.
Key works
c. 820 On the Calculation with Hindu Numerals
c. 830 The Compendious Book on Calculation by Completion and Balancing
See also: Quadratic equations • The Rhind papyrus • Diophantine equations • Cubic equations • The algebraic resolution of equations �
� The fundamental theorem of algebra
IN CONTEXT
KEY FIGURE
Al-Karaji (c. 980–c. 1030)
FIELD
Number theory
BEFORE
c. 250 CE In Arithmetica, Diophantus lays down ideas about algebra later taken up by al-Karaji.
c. 825 CE The Persian astronomer and mathematician al-Khwarizmi develops algebra.
AFTER
1653 In Traité du triangle arithmétique (Treatise on the Arithmetical Triangle), Blaise Pascal reveals the triangular pattern of coefficients in the bionomial theorem in what is later called Pascal’s triangle.
1665 Isaac Newton develops the general binomial series from the binomial theorem, forming part of the basis for his work on calculus.
At the heart of many mathematical operations lies an important basic theorem—the binomial theorem. It provides a shorthand summary of what happens when you multiply out a binomial, which is a simple algebraic expression consisting of two known or unknown terms added together or subtracted. Without the binomial theorem, many mathematical operations would be almost impossible to achieve. The theorem shows that when binomials are multiplied out, the results follow a predictable pattern that can be written as an algebraic expression or displayed on a triangular grid (known as Pascal’s triangle after Blaise Pascal, who explored the pattern in the 1600s).
Making sense of binomials
The binomial pattern was first observed by mathematicians in ancient Greece and India, but the man credited with its discovery is the Persian mathematician al-Karaji, one of many scholars who flourished in Baghdad from the 8th to the 14th century. Al-Karaji explored the multiplication of algebraic terms. He defined single terms called monomials”—x, x2, x3, and so on—and showed how they can be multiplied or divided. He also looked at “polynomials” (expressions with multiple terms), such as 6y2 + x3 - x + 17. But it was his discovery of the formula for multiplying out binomials that had the most impact.
The binomial theorem concerns powers of binomials. For example, multiplying out the binomial (a + b)2 by converting it to (a + b) (a + b) and multiplying each term in the first parentheses by each term in the second parentheses results in (a + b)2 = a2 + 2ab + b2. The calculation for the power 2 is manageable, but for greater powers, the resulting expression becomes increasingly complicated. The binomial theorem simplifies the problem by unlocking the pattern in the coefficients—numbers, such as 2 in 2ab, by which the unknown terms are multiplied. As al-Karaji discovered, the coefficients can be laid out in a grid, with the columns showing the coefficients needed for multiplying out each power. The coefficients in a column are calculated by adding together pairs of numbers in the preceding column. To determine the powers in the expansion, you take the degree of the binomial as n. In (a + b)2, n = 2.
Al-Karaji created a table to work out the coefficients of binomial equations. The first five lines of it are shown here. The top line is for powers, with the coefficients for each power listed in the column below. The first and final numbers are always 1. Each other number is the sum of its adjacent number in the preceding column and the number above that adjacent number.
Algebra breaks free
Al-Karaji’s discovery of the binomial theorem helped to open the way for the full development of algebra, by allowing mathematicians to manipulate complicated algebraic expressions. The algebra developed by al-Khwarizmi 150 years or so previously had used a system of symbols to work out unknown quantities and was limited in scope. It was tied to the rules of geometry, and the solutions were geometric dimensions, such as angles and side lengths. Al-Karaji’s work showed how algebra could instead be based entirely on numbers, liberating it from geometry.
The binomial theorem and a Bach fugue are, in the long run, more important than all the battles of history.
James Hilton
British novelist
AL-KARAJI
Born around 980 CE, Abu Bakr ibn Muhammad ibn al-Husayn al-Karaji most likely got his name from the city of Karaj, near Tehran, but he lived most of his life in Baghdad, at the court of the caliph. It was here around 1015 that he probably wrote his three key mathematics texts. The work in which al-Karaji developed the binomial theorem is now lost, but later commentators preserved his ideas. Al-Karaji was also an engineer, and his book Extraction of Hidden Waters is the first known manual on hydrology.
Later in life, al-Karaji moved to “mountain countries” (possibly the Elburz mountains near Karaj), where he spent his time working on practical projects for drilling wells and building aqueducts. He died around 1030 CE.
Key works
Glorious on algebra
Wonderful on calculation
Sufficient on calculation
See also: Positional numbers • Diophantine equations • Zero • Algebra • Pascal’s triangle • Probability • Calculus • The fundamental theorem of algebra
IN CONTEXT
KEY FIGURE
Omar Khayyam (1048–1131)
FIELD
Algebra
BEFORE
3rd century BCE Archimedes solves cubic equations using the intersection of two conics.
7th century CE Chinese scholar Wang Xiaotong solves a range of cubic equations numerically.
AFTER
16th century Mathematicians in Italy create jealously guarded methods to solve cubic equations in the fastest time.
1799–1824 Italian scholar Paolo Ruffini and Norwegian mathematician Niels Henrik Abel show that no algebraic formulas exist for equations involving terms to the power of 5 and higher.
In the ancient world, scholars considered problems in a geometric way. Simple linear equations (which describe a line), such as 4x + 8 = 12, where x is to the power of 1, could be used to find a length, while a squared variable (x2) in a quadratic equation could represent an unknown area—a two-dimensional space. The next step up is the cubic equation, where the x3 term is an unknown volume—a three-dimensional space.
The Babylonians could solve quadratic equations in 2000 BCE, but it took another 3,000 years until Persian poet-scientist Omar Khayyam found an accurate method for solving cubic equations, using curves called conic sections—such as circles, ellipses, hyperbolas, or parabolas—formed by the intersection of a plane and a cone.
Problems with cubes
The ancient Greeks, who used geometry to work out complex problems, puzzled over cubes. A classic conundrum was how to produce a cube that was twice the volume of another cube. For example, if the sides of a cube are each equal to 1 in length, what length sides do you need for a cube twice the volume? In modern terms, if a cube with side length 1 has a volume of 13, what side length cubed (x3) produces twice that volume; that is, since 13 = 1, what is x if x3 = 2? The ancient Greeks used a ruler and compasses to attempt constructing a solution to this cubic equation but they never succeeded. Khayyam saw that such tools were not enough to solve all cubic equations, and set out his use of conic sections and other methods in his treatise on algebra.
Using modern conventions, cubic equations can be expressed simply, such as x3 + bx = c. Without the economy of modern notation, Khayyam expressed his equations in words, describing x3 as “cubes”, x2 as “squares,” x as “lengths,” and numbers as “amounts.” For example, he described x3 + 200x = 20x2 + 2,000 as a problem of finding a cube that “with two hundred times its side” is equal to “twenty squares of its side and two thousand.” For a simpler equation, such as x3 + 36x = 144, Khayyam’s method was to draw a geometric diagram. He found that he could break down the cubic equation into two simpler equations: one for a circle, and the other for a parabola. By working out the value of x for which both these simpler equations are true simultaneously, he could solve the original cubic equation. This is shown in the graph below. At the time, mathematicians did not have these graphical methods and Khayyam would have constructed the circle and parabola geometrically.
Khayyam had also explored the properties of conic sections, and had deduced that a s
olution to the cubic equation could be found by giving the circle in the diagram a diameter of 4. This measure was arrived at by dividing c by b, or 144⁄36 in the example below. The circle passed through the origin (0,0) and its center was on the x axis at (2,0). Using this diagram, Khayyam drew a perpendicular line from the point where the circle and parabola intersected down to the x axis. The point where the line crossed the x axis (where y = 0) gives the value for x in the cubic equation. In the case of x3 + 36x = 144, the answer is x = 3.14 (rounded to two decimal places).
Khayyam did not use coordinates and axes (which were invented about 600 years later). Instead, he would have drawn the shapes as accurately as possible and carefully measured the lengths on their diagrams. He would then have found an approximate numerical solution using trigonometric tables, which were common in astronomy. For Khayyam, the solution would always have been a positive number. There is an equally valid negative answer, as shown by the minus numbers in the graph below, but although the concept of negative numbers was recognized in Indian mathematics, it was not generally accepted until the 1600s.
OMAR KHAYYAM
Born in Nishapur, Persia (now Iran), in 1048, Omar Khayyam was educated in philosophy and the sciences. Although he won renown as an astronomer and mathematician, when his patron Sultan Malik Shah died in 1092, he was forced into hiding. Finally rehabilitated 20 years later, he lived quietly and died in 1131.