The Math Book
Page 12
In mathematics, Khayyam is best remembered for his work on cubic equations, but he also produced an important commentary on Euclid’s fifth postulate, known as the parallel postulate. As an astronomer, he helped to construct a highly accurate calendar that was used until the 1900s. Ironically, Khayyam is now best known for a work of poetry for which he may not have been the sole author—the Rubaiyat, which was translated into English by Edward Fitzgerald in 1859.
Key works
c. 1070 Treatise on Demonstration of Problems of Algebra
1077 Commentaries on the difficult postulates of Euclid’s book
A parabola (pink) for the equation x2 = 6y intersects the circle (blue) (x˗2)2 + y2 = 4. A line from G, the point of intersection, to H on the x axis, gives the value for x (3.14) in the cubic equation x3 + 36x = 144.
Khayyam’s contribution
While Archimedes, working in the 3rd century BCE, may well have examined the intersection of conic sections in a bid to solve cubic equations, what marks Khayyam out is his systematic approach. This enabled him to produce a general theory. He extended his mix of geometry and algebra to solve cubic equations using circles, hyperbolas, and ellipses, but never explained how he constructed them, simply saying he “used instruments.”
Khayyam was among the first to realize that a cubic equation could have more than one root, and therefore more than one solution. As can be shown on a modern graph that plots a cubic equation as a curve snaking above and below the x axis, a cubic equation has up to three roots. Khayyam suspected two, but would not have considered negative values. He did not like having to use geometry as well as algebra to find a solution, and hoped that his geometrical efforts would one day be replaced by arithmetic.
Khayyam anticipated the work of 16th-century Italian mathematicians, who solved cubic equations without direct recourse to geometry. Scipione del Ferro produced the first algebraic solution to cubic equations, discovered in his notebook after his death. He and successors Niccolò Tartaglia, Lodovico Ferrari, and Gerolamo Cardano all worked on algebraic formulae to solve cubic equations. Cardano published Ferro’s solution in his book Ars Magna in 1545. Their solutions were algebraic but differed from those of today, partly because zero and negative numbers were little used at the time.
I have shown how to find the sides of the square-square, quatro-cube, cubo-cube… to any length, which has not been [done] before now.
Omar Khayyam
Toward modern algebra
Mathematicians who continued the quest for cubic equation solutions included Rafael Bombelli. He was among the first to state that a cubic root could be a complex number, that is, a number that makes use of an “imaginary” unit derived from the square root of a negative number, something not possible with “real” numbers. In the late 1500s, Frenchman François Viète created more modern algebraic notation, using substitution and simplifying to reach his solutions. By 1637, René Descartes had published a solution to the quartic equation (involving x4), reducing it to a cubic equation and then to two quadratic equations to solve it. Today, a cubic equation can be written in the form ax3 + bx2 + cx + d = 0, provided a itself is not 0. Where the coefficients (a, b, and c, which multiply the variable x) are real numbers, rather than complex numbers, the equation will have at least one real root and up to three roots in total.
Khayyam’s method is still taught today. His painstaking work advanced early algebra, while later mathematicians have continued to refine its expression and scope.
Algebras are geometric facts which are proved by propositions.
Omar Khayyam
A passion for geometric forms is evident in Islamic architecture, seen here in the tile patterns, curved arches, and domes of the Masjid-i Kabud, the “Blue Mosque,” in Tabriz, Iran.
The length of the year
In 1074, the ruling sultan of Persia, Jalal al-Din Malik Shah I, commissioned Omar Khayyam to reform the lunar calendar used since the 7th century, replacing it with a solar calendar. A new observatory was built in the capital Isfahan, and Khayyam assembled a team of eight astronomers to assist him with the work.
The year—computed to a highly accurate 365.24 days—began at the vernal equinox in March, when the center of the visible Sun is directly above the equator. Each month was worked out by the passage of the sun into the corresponding zodiac region, which required both computations and actual observations. Because solar transit times could vary by 24 hours, months were between 29 and 32 days long, but their length could differ from year to year. The new Jalali calendar, named after the sultan, was adopted on March 15, 1079 and was only modified in 1925.
See also: Quadratic equations • Euclid’s Elements • Conic sections • Imaginary and complex numbers • The complex plane
IN CONTEXT
KEY FIGURE
Leonardo of Pisa, also known as Fibonacci (1170–c. 1250)
FIELD
Number theory
BEFORE
200 BCE The number sequence later known as the Fibonacci sequence is cited by the Indian mathematician Pingala in relation to Sanskrit poetic meters.
700 CE The Indian poet and mathematician Virahanka writes about the sequence.
AFTER
17th century In Germany, Johannes Kepler notices that the ratio of successive terms in the sequence converges.
1891 Édouard Lucas coins the name Fibonacci sequence in Théorie des Nombres (Number Theory).
One sequence of numbers occurs time and again in the natural world. In this sequence, every number is the sum of the previous two (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on). Originally referred to by the Indian scholar Pingala in around 200 BCE, it was later called the Fibonacci sequence after Leonardo Pisano (Leonardo of Pisa), an Italian mathematician known as Fibonacci. Fibonacci explored the sequence in his 1202 book Liber Abaci (The Book of Calculation). The sequence has important forecasting applications in nature, geometry, and business.
A problem with rabbits
One of the problems Fibonacci raised in Liber Abaci concerned the growth of rabbit populations. Starting with a single pair of rabbits, he asked his readers to work out how many pairs there would be in each successive month. Fibonacci made several assumptions: no rabbit ever died; rabbit pairs mated every month, but only after they were two months old, the age of maturity; and each pair produced one male and one female offspring every month. For the first two months, he said, there would only be the original pair: by the end of three months, there would be a total of two pairs; and at the end of four months, there would be three pairs, as only the original pair was old enough to breed.
Thereafter, the population grows more quickly. In the fifth month, both the original pair and their first offspring produce baby rabbits, although the second pair of offspring is still too young. This results in a total of five pairs of rabbits. The process continues in successive months, resulting in a number sequence in which each number is the sum of the previous two: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on – a sequence that became known as the Fibonacci sequence. As with many mathematical problems, it is based on a hypothetical situation: Fibonacci’s assumptions about how the rabbits behave are unrealistic.
FIBONACCI
Born Leonardo Pisano, probably in Pisa, Italy, in 1170, Fibonacci did not become known as Fibonacci (“son of Bonacci”) until long after his death. Leonardo traveled widely with his diplomat father and studied at a school of accounting in Bugia, North Africa. There he came across the Hindu–Arabic symbols used to represent the numbers 1 to 9. Impressed by these numerals’ simplicity compared with the lengthy Roman numerals used in Europe, he discussed them in Liber Abaci (The Book of Calculation), which he wrote in 1202.
Leonardo also traveled to Egypt, Syria, Greece, Sicily, and Provence, exploring different number systems. His work was widely read and came to the attention of the Holy Roman Emperor, Frederick II. Fibonacci died c. 1240–50.
Key works
1202 Liber Abaci (The Book of Calculation)
&n
bsp; 1220 Practica Geometriae (Practical Geometry)
1225 Liber Quadratorum (The Book of Squares)
Generations of bees
An example of the Fibonacci sequence cropping up in nature concerns bees in a beehive. A male bee, or drone, develops from the unfertilized egg of a queen bee. Since the egg is unfertilized, the drone has only one parent, its “mother.” Drones have different roles in the beehive, one of which is to mate with the queen and fertilize her eggs. Fertilized eggs develop into female bees, which can either be queens or workers. This means that one generation back the drone has only one ancestor, its mother; two generations back it has two ancestors, or “grandparents”—the mother and father of its mother; and three generations back, it has three “great grandparents”—its grandmother’s two parents and its grandfather’s mother. Further back, there are five members of the previous generation, eight of the one before that, and so on. The pattern is clear: the number of members in each generation of ancestors forms the Fibonacci sequence. The sum of the number of parents of a male and a female from the same generation of bees is three. Their parents total five grandparents, whose own parents add up to eight great-grandparents. When the pattern is traced back to earlier generations, the Fibonacci sequence continues, with 13, 21, 34, 55 ancestors, and so on.
Each month, some rabbits mature and others breed. In the first six months, the number of pairs has increased in the sequence 1, 1, 2, 3, 5, and 8. Future generations over the next four months can be forecast to contain 13, 21, 34, and 55 pairs of rabbits.
The Fibonacci sequence turns out to be the key to understanding how nature designs.
Guy Murchie
American writer
Plant life
The Fibonacci sequence can also be seen in the arrangement of leaves and seeds in some plants. Pine cones and pineapples, for example, display Fibonacci numbers in the spiral formation of their exterior scales. Many flowers have three, five, or eight petals—numbers that belong to the Fibonacci sequence. Ragwort flowers have 13 petals, chicory often has 21, and different types of daisy have 34 or 55. However, many other flowers have four or six petals, so while numbers from the sequence are common, other patterns are also found.
Each Fibonacci number is the sum of the previous two, so the first two have to be stated before the third can be calculated. The Fibonacci sequence can be defined by a recurrence relation—an equation that defines a number in a sequence in terms of its previous numbers. The first Fibonacci number is written as f1, the second as f2, and so on. The equation is fn = f(n-1) + f(n-2), where n is greater than 1. If you are trying to find the fifth Fibonacci number (f5), for example, you must add together f4 and f3.
[If] a spider climbs so many feet up a wall each day and slips back a fixed number each night, how many days does it take him to climb the wall?
Fibonacci
Fibonacci ratios
Calculating the ratios of successive terms in the Fibonacci sequence is particularly interesting. Dividing each number by the previous number in the sequence produces the following: 1⁄1 = 1, 2⁄1 = 2, 3⁄2 = 1.5, 5⁄3 = 1.666…, 8⁄5 = 1.6, 13⁄8 = 1.625, 21⁄13 = 1.61538…, 34⁄21 = 1.61904… By continuing this process indefinitely, it can be shown that the numbers approach 1.618, approximately. This is referred to as the golden ratio or the golden mean. The same number is also significant in a curve called the golden spiral, which gets wider by a factor of 1.618 for every quarter turn it makes. This spiral crops up commonly in nature: for example, the seeds of pine cones, sunflowers, and coneflowers tend to grow in golden spirals.
The scales of a pine cone, viewed from above, can be seen to run in two sets of spirals. Both sets run from the outside to the center: one clockwise, and the other counterclockwise. The numbers of spirals in each set are 13 (clockwise) and 8 (counterclockwise)—two Fibonacci numbers.
Arts and analysis
The Fibonacci sequence can also be found in poetry, art, and music. A pleasing rhythm in poetry, for example, is produced when successive lines have 1, 1, 2, 3, 5, and 8 syllables, and there is a long tradition of 6-line, 20-syllable poetry structured in this way. Around 200 BCE, Pingala was aware of this pattern in Sanskrit poetry, and the Roman poet Virgil used it in the 1st century BCE.
The sequence has also been used in music. French composer Claude Debussy (1862–1918) employed Fibonacci numbers in several compositions. In the dramatic climax of his Cloches à travers les feuilles (Bells Through the Leaves), the ratio of total bars in the piece to climax bars is approximately 1.618.
Although it is often associated with the arts, the Fibonacci sequence has also proved a useful tool in finance. Today, ratios derived from the sequence are used as an analytical tool to forecast the point at which stock market prices will stop rising or falling.
A piano keyboard scale from C to C spans 13 keys, eight white and five black. The black keys are in groups of two and three. These numbers all form part of the Fibonacci sequence.
Practical solutions
A page from the original manuscript of Liber Abaci shows the Fibonacci sequence listed on the right.
Fibonacci’s work was intended to have a useful purpose. In Liber Abaci (1202), for example, he described how to solve many of the problems encountered in commerce, including calculating profit margins and converting currencies. In Practica Geometriae (1220), he solved problems associated with surveying, such as finding the height of a tall object using similar triangles (triangles that have identical angles, but different sizes). In his Liber Quadratorum (1225), he tackled several topics in number theory, including finding Pythagorean triples—groups of three integers that represent the lengths of the sides of right-angled triangles. In these triangles, the square of the length of the longest side (the hypotenuse) equals the sum of the squares of the lengths of the two shorter sides. Fibonacci found that, starting with 5, every second number in his sequence (13, 34, 89, 233, 610, and so on) is the length of the hypotenuse of a right-angled triangle when the lengths of the two shorter sides are integers.
See also: Positional numbers • Pythagoras • Trigonometry • Algebra • The golden ratio • Pascal’s triangle • Benford’s law
IN CONTEXT
KEY FIGURE
Sissa ben Dahir (3rd or 4th century CE)
FIELD
Number theory
BEFORE
c. 300 BCE Euclid introduces the concept of a power to describe squares
c. 250 BCE Archimedes uses the law of exponents, which states that multiplying exponents can be achieved by adding the powers.
AFTER
1798 British economist Thomas Malthus predicts that the human population will grow exponentially while the food supply will increase more slowly, causing a catastrophe.
1965 American co-founder of Intel Gordon Moore observes how the number of transistors on a microchip doubles roughly every 18 months.
The first written record of the wheat on a chessboard problem was made in 1256 by Muslim historian Ibn Khallikan, though it is probably a retelling of an earlier version that arose in India in the 5th century. According to the story, the inventor of chess, Sissa ben Dahir, was summoned to an audience with his ruler, King Sharim. The king was so delighted with the game of chess that he offered to grant Sissa any reward that he wanted. Sissa asked for some grains and explained the quantity he desired using the squares on the 8 × 8 chessboard. One grain of wheat (or rice, in some versions of the story) was to be placed on the bottom left square of the chessboard. Moving right, the number of grains would then be doubled, so the second square had two grains, the third had four, and so on, moving left to right along each row to the 64th square at the top right.
Puzzled by what seemed to be a paltry reward, the king ordered that the grains be counted out. The 8th square had 128 grains, the 24th had more than 8 million, and the 32nd, the last square on the chessboard’s first half, had over 2 billion. By then, the king’s granary was running low, and he realized that the next square alone, number 33, woul
d need 4 billion grains, or one large field’s worth. His advisers calculated that the final square would need 9.2 million trillion grains, and the total number of grains on the chessboard would be 18,446,744,073,709,551,615 (264 – 1). The story has two alternative endings: in one, the king made Sissa his chief adviser; in the other, Sissa was executed for making the king look foolish.
Sissa’s concept is an example of what is known as a geometric series, in which every successive term is the previous one multiplied by two: 1 + 2 + 4 + 8 + 16, and so on. From 2 onward, these numbers are all powers of 2: 1 + 2 + 22 + 23 + 24, and so on. The superscript number, the exponent, shows how many times the other number, in this case 2, is multiplied by itself. The last term in the series, 263, is 2 multiplied by itself 63 times.
Bacteria dividing is an example of exponential growth; when a single cell divides, it creates two cells that divide to make four, and so on. This allows bacteria to spread very quickly.
Power of exponents
The growth of the values in this series is described as exponential. Exponents can be viewed as instructions for how many times 1 should be multiplied by a given number. For example, 23 means that 1 will be multiplied by 2 three times: 1 × 2 × 2 × 2 = 8, while 21 means that 1 will be multiplied by 2 just once: 1 × 2 = 2. The first square of the chessboard contains 1 grain, so 1 is the first term of this series. The number 1 can be written as 20, because it is equivalent to 1 multiplied by 2 zero times, leaving 1 unaffected. In fact, any number to the power of 0 equals 1 for this reason.