by DK
After the death of the Prophet Mohammed in 632, Islam rapidly became a major political as well as religious power in the Middle East and beyond, spreading from Arabia across Persia and into Asia as far as the Indian subcontinent. The new religion had a high regard for philosophy and scientific enquiry, and the “House of Wisdom,” a center of learning and research established in Baghdad, attracted scholars from all over the expanding Islamic Empire.
This thirst for knowledge prompted the study of ancient texts, especially those of the great Greek philosophers and mathematicians. Islamic scholars not only preserved and translated the ancient Greek texts, but provided commentaries on them and developed their own original concepts. Open to new ideas, they also adopted many of the Indian innovations, in particular their numeral system. The Islamic world, like India, entered a “Golden Age” of learning that lasted until the 1300s, and produced a succession of influential mathematicians—such as al-Khwarizmi, a key figure in the development of algebra (the word “algebra” derives from the Arabic term for rejoining), and other scholars whose contributions to the binomial theorem and the treatment of quadratic and cubic equations were groundbreaking.
From East to West
In Europe, mathematical study was under the control of the Church, and was confined to a few early translations of some of Euclid’s work. Progress was hindered by the continued use of the cumbersome Roman system of numerals, necessitating the use of the abacus for calculation. However, from the 12th century onward, during the Crusades, contact with the Islamic world increased, and some recognized the wealth of scientific knowledge Islamic scholars had amassed. Christian scholars now gained access to Greek and Indian philosophical and mathematical texts, and to the work of the Islamic scholars. Al-Khwarizmi’s treatise on algebra was translated into Latin in the 12th century by Robert of Chester, and soon after, complete translations of Euclid’s Elements and other important texts began to appear in Europe.
Mathematical renaissance
City-states in Italy were quick to trade with the Islamic Empire, and it was an Italian, Leonardo of Pisa, nicknamed Fibonacci, who spearheaded the revival of mathematics in the West. He adopted the Hindu-Arabic numeral system, and the use of symbols in algebra, and contributed many original ideas, including the Fibonacci arithmetical sequence.
With the growth in trade in the later Middle Ages, mathematics—especially the fields of arithmetic and algebra—became increasingly important. Advances in astronomy also demanded sophisticated calculations. Mathematical education was now taken more seriously. With the invention of the movable-type printing press in the 1400s, books of all sorts, including the Treviso Arithmetic, became widely available, spreading the newfound knowledge across Europe. These books inspired a “scientific revolution” that would accompany the cultural rebirth known as the Renaissance.
IN CONTEXT
KEY FIGURE
Brahmagupta (c. 598–668 CE)
FIELD
Number theory
BEFORE
c. 700 BCE On a clay tablet, a Babylonian scribe indicates a placeholder zero with three hooks; it is later written as two slanted wedge marks.
36 BCE A shell-shaped zero is recorded on a Mayan stela (stone slab) in Central America.
c. 300 CE Parts of the Indian Bakshali text reveal many circular placeholder zeros.
AFTER
1202 In his book Liber Abaci, Leonardo of Pisa (Fibonacci) introduces zero to Europeans.
17th century Zero is finally established as a number and is in widespread use.
A number that represents the absence of something is a difficult concept, which may be why zero took so long to become widely accepted. Several ancient civilizations, including the Babylonians and the Sumerians, could claim to have invented zero, but its use as a number was pioneered in the 7th century CE, by Brahmagupta, an Indian mathematician.
The development of zero
Any system for recording numbers eventually reaches a point at which it becomes positional; that is to say, digits are ordered according to their value to cope with increasingly large numbers. All place value (positional) systems require a way of denoting “there is nothing here.” The Babylonians (1894–539 BCE), for example, who at first used context to differentiate between, say, 35 and 305, eventually used a double wedge mark rather like inverted commas to indicate the empty value. In this way, zero entered the world as a form of punctuation.
The problem for historians has been finding evidence for early civilizations using zero and recognizing it as such, which has been made more difficult by the fact that zero fell in and out of use over time. In about 300 BCE, for example, the Greeks were starting to develop a more sophisticated form of mathematics based on geometry, with quantities being represented by the lengths of lines. There was no need for zero, or indeed negative numbers (numbers less than 0), as the Greeks did not have a positional number system (lengths cannot be nonexistent or negative).
As the Greeks developed the use of mathematics in astronomy, they began to use an “O” to represent zero, although it is not clear why. In his astronomical manual Almagest, written in the 2nd century CE, the Greco-Roman scholar Ptolemy used a circular symbol positionally between digits and at the end of a number, but did not consider it a number in its own right.
In Central America, during the 1st millenium CE, the Mayans used a place value system, which included zero as a numeral, denoted by a shell shape. It was one of three symbols used by the Mayans for arithmetic; the other two were a dot representing 1 and a bar for 5. While the Mayans could calculate up to hundreds of millions, their geographical isolation meant that their mathematics never spread to other cultures.
In India, mathematics advanced rapidly in the early centuries of the 1st millennium CE. By the 3rd and 4th centuries, a place value system had long been in use, and by the 7th century—the time of Brahmagupta—the use of a circular symbol as a placeholder was already well established there.
An abax, a table or board covered in sand, was used by the Greeks to count. Some scholars have suggested that “O” was used because it was the shape left when a counter was removed.
BRAHMAGUPTA
Born in 598 CE, astronomer and mathematician Brahmagupta lived in Bhillamala, northwest India—a center of learning in those fields. He became head of the leading astronomical observatory at Ujjain, and incorporated new work on number theory and algebra into his studies on astronomy.
Brahmagupta’s use of the decimal number system and the algorithms he devised spread throughout the world and informed the work of later mathematicians. His rules for calculating with positive and negative numbers, which he called “fortunes” and “debts,” are still cited today. Brahmagupta died in 668, only a few years after completing his second book.
Key works
628 Brahmasphutasiddhanta (The Correctly Established Doctrine of Brahma)
665 Khandakhadyaka (Morsel of Food)
The Nadi Yali yantra is part of an 18th-century observatory in Ujjain, India. A center of mathematics and astronomy since Brahmagupta worked there in the 7th century, it lies on the intersection of a former zero meridian of longitude and the Tropic of Cancer.
Zero as a number
Brahmagupta established rules for calculating with zero. He began by defining it as the result of subtracting a number from itself— for example, 3 - 3 = 0. That established zero as a number in its own right as opposed to simply a figurative notation or placeholder. He then explored the effect of calculating with zero. Brahmagupta showed that if he added zero to a negative number, the result was equal to that negative number. Similarly, adding zero to a positive number produced the same positive number. Brahmagupta also described subtracting zero from both a negative and a positive number, and noted again that it left the numbers unchanged.
Brahmagupta went on to describe the effect of subtracting numbers from zero. He calculated that a positive number subtracted from zero becomes a negative number and a negative number subtracted from zero becomes a
positive number. This calculation brought negative numbers into the same number system as positive numbers. Like zero, negative numbers were an abstract concept rather than positive values such as lengths or quantities.
First-century Indian numerals did not use zero. By the 9th century, Brahmagupta’s zero (highlighted in pink) was widely used in India, from where it spread via the Arab world to Europe. There, it met some initial opposition from Christian religious leaders, who found the concept of zero satanic because they associated nothingness with the devil.
Black holes are where God divided by zero.
Steven Wright
American comedian
Multiplying and dividing
Brahmagupta went on to examine zero in relation to multiplication and described how the product of multiplying any number with zero is zero, including zero multiplied by zero. The next step was to explain division by zero, which was more problematic. Recording the result of dividing a number, n, by zero as n⁄0, Brahmagupta suggested that a number is unchanged when it is divided by zero. However, this was later found to be impossible, as is demonstrated by multiplying any number by zero (division being defined as finding the missing number in a multiplication). The result cannot be the original number, as any number multiplied by zero equals zero.
Mathematicians now describe division by zero as “undefined.” Some have suggested that the required answer to n⁄0 is “infinity,” but infinity is not a number and cannot be used in calculations. Dividing zero itself by zero has proved even trickier. The result could be zero, if zero divided by any number is thought to be zero. It could also be 1, as any number divided by itself is 1.
The spread of Islam through parts of India in the 8th century led to Indian mathematicians sharing their knowledge, including the concept of zero, with scholars in the Arab world. In the 9th century, the Islamic mathematician al-Khwarizmi wrote a treatise on Hindu–Arabic numbers, which described the place value system including zero. Yet 300 years later when Leonardo of Pisa (better known as Fibonacci) introduced Hindu–Arabic numerals to Europe, he was still wary of zero and treated it as an operator like + and ˗ rather than a number. Even in the 1500s, Italian polymath Gerolamo Cardano solved quadratic and cubic equations without zero. Europeans finally accepted zero in the 1600s, when English mathematician John Wallis incorporated zero in his number line.
Zero is the most magical number we know. It is the number we’re striving toward every day.
Bill Gates
A vital concept
Mathematics without zero would mean many of the articles in this book could not have been written: there would be no negative numbers, no coordinate systems, no binary systems (and hence no computers), no decimals, and no calculus, because it would not be possible to describe infinitesimally small quantities. Advances in engineering would have been severely restricted. Zero is perhaps the most important number of all.
The Treviso Arithmetic
This grid method of multiplication from the Treviso Arithmetic multiplies the number 56,289 by 1,234. Zero is used as a placeholder in the calculation and in the final solution—70,072,626. The book also illustrated other methods of multiplication.
The figure zero first became known in Italy from the Arte dell’ Abbaco (Art of Calculation, also known as The Treviso Arithmetic), published anonymously in 1478 and the first printed mathematics textbook in Europe. It was revolutionary because it was written in everyday Venetian for merchants and anyone else who wanted to solve calculation problems. It outlined the Hindu–Arabic decimal place value system and described how the number system worked. The unknown author makes 0 the 10th number and calls it a “cipher” or “nulla”—something that has no value unless it is written to the right of other numbers to increase their value.
In the Treviso description, zero is just a placeholder number, which itself was still a new notion. The idea of zero as a number was not accepted for centuries. It was also of little interest to the readers of the Arte dell’ Abbaco, most of whom wanted to learn how to use numbers in practical business calculations in everyday trading.
See also: Positional numbers • Negative numbers • Binary numbers • The law of large numbers • The complex plane
IN CONTEXT
KEY FIGURE
Al-Khwarizmi (c. 780–c. 850)
FIELD
Algebra
BEFORE
1650 BCE The Egyptian Rhind papyrus includes solutions to linear equations.
300 BCE Euclid’s Elements lays the foundations of geometry.
3rd century CE Greek mathematician Diophantus uses symbols to represent unknown quantities.
7th century CE Brahmagupta solves the quadratic equation.
AFTER
1202 Leonardo of Pisa’s Liber Abaci uses the Hindu-Arabic number system.
1591 François Viète introduces symbolic algebra, in which letters are used to abbreviate terms in equations.
The origins of algebra— a mathematical method for calculating unknown quantities—can be traced back to ancient Babylonians and Egyptians, as equations on cuneiform tablets and papyri reveal. Algebra evolved from the need to solve practical problems, often of a geometrical nature, requiring the determination of a length, area, or volume. Mathematicians gradually developed rules to handle a wider range of general problems. To work out lengths and areas, equations involving variables (unknown quantities) and squared terms were devised. Using tables, the Babylonians could also calculate volumes, such as the space within a grain store.
A search for new methods
Over the centuries, as mathematics developed, problems became longer and more complex, and scholars sought new ways to shorten and simplify them. Although early Greek mathematics was largely geometry-based, Diophantus developed new algebraic methods in the 3rd century CE, and was the first to use symbols for unknown quantities. However, it would be more than a thousand years before standard algebraic notation was accepted.
After the fall of the Roman Empire, mathematics in the Mediterranean area declined, but the spread of Islam from the 7th century had a revolutionary impact on algebra. In 762 CE, Caliph al-Mansur established a capital in Baghdad, which swiftly became a major center of culture, learning, and commerce. Its status was enhanced by the acquisition and translation of manuscripts from earlier cultures, including works by the Greek mathematicians Euclid, Apollonius, and Diophantus, as well as Indian scholars such as Brahmagupta. They were housed in a great library, the House of Wisdom, which became a center for research and the dissemination of knowledge.
The early algebraists
Scholars at the House of Wisdom produced their own research, and in 830, Muhammad Ibn Musa al-Khwarizmi presented his work to the library—The Compendious Book on Calculation by Completion and Balancing. It revolutionized ways of calculating algebraic problems, introducing principles that are the foundation of modern algebra. As in earlier periods, the types of problems discussed were largely geometrical. The study of geometry was important in the Islamic world, partly because the human form was forbidden in religious art and architecture, so many Islamic designs were based on geometric patterns.
Al-Khwarizmi introduced some fundamental algebraic operations, which he described as reduction, rejoining, and balancing. The process of reduction (simplifying an equation) could be done by rejoining (al-jabr)—moving subtracted terms to the other side of an equation—and then balancing the two sides of the equation. The word “algebra” comes from al-jabr.
Al-Khwarizmi was not working in a total vacuum, as he had the translated works of earlier Greek and Indian mathematicians at his disposal. He introduced the Indian decimal place-value system to the Islamic world, which later led to the adoption of the Hindu-Arabic numeral system widely used today.
Al-Khwarizmi began by studying linear equations, so-called because they create a straight line when plotted on a graph. Linear equations involve only one variable, which is expressed only to the power of 1, rather than squared or to any higher po
wer.
Quadratic equations
Al-Khwarizmi did not employ symbols; he wrote his equations in words, supported by diagrams. For example, he wrote out the equation (x⁄3 + 1)(x⁄4 + 1) = 20 as: “A quantity: I multiplied a third of it and a dirham by a fourth of it and a dirham; it becomes twenty,” a dirham being a single coin, used by al-Khwarizmi to signify a single unit. According to al-Khwarizmi, by using his completion and balancing methods, all quadratic equations—those in which the highest power of x is x2 —can be simplified to one of six basic forms. In modern notation, these would be: ax2 = bx; ax2 = c; ax2 + bx = c; ax2 + c = bx; ax2 = bx + c; and b2 = c. In these six types, the letters a, b, and c all represent known numbers, and x represents the unknown quantity.
Al-Khwarizmi approached more complex problems too, producing a geometrical method for solving quadratic equations that used the technique known as “completing the square” . He went on to search for a general solution to cubic equations—in which the highest power of x is x3—but was unable to find one. However, his pursuit of this goal showed how mathematics had progressed since the time of the ancient Greeks. For centuries, algebra had just been a tool to solve geometric problems, but now became a discipline in its own right, where calculating increasingly difficult equations was the end goal.
The principal object of Algebra… is to determine the value of quantities which were before unknown… by considering attentively the conditions given… expressed in known numbers.
