by DK
G. H. Hardy
British mathematician
Modern numeration
The Hindu–Arabic decimal system used throughout the world today has its origins in India. In the 1st to 4th centuries CE, the use of nine symbols along with zero was developed to allow any number to be written efficiently, through the use of place value. The system was adopted and refined by Arab mathematicians in the 9th century. They introduced the decimal point, so that the system could also express fractions of whole numbers.
Three centuries later, Leonardo of Pisa (Fibonacci) popularized the use of Hindu–Arabic numerals in Europe through his book Liber Abaci (1202). Yet the debate about whether to use the new system rather than Roman numerals and traditional counting methods lasted for several hundred years, before its adoption paved the way for modern mathematical advances.
With the advent of electronic computers, other number bases became important—particularly binary, a number system with base 2. Unlike the base-10 system with its 10 symbols, binary has just two: 1 and 0. It is a positional system, but instead of multiplying by 10, each column is multiplied by 2, also expressed as 21, 22, 23 and upward. In binary, the number 111 means 1 × 22 + 1 × 21 + 1 × 20, that is 4 + 2 + 1, or 7 in our decimal number system.
In binary, as in all modern number systems whatever their base, the principles of place value are always the same. Place value—the Babylonian legacy—remains a powerful, easily understood, and efficient way to represent large numbers.
The fact that we work in 10s as opposed to any other number is purely a consequence of our anatomy. We use our ten fingers to count.
Marcus du Sautoy
British mathematician
Ebisu, the Japanese god of fishermen and one of the seven gods of fortune, uses a soroban to calculate his profits in The Red Snapper’s Dream by Utagawa Toyohiro.
Mayan numeral system
The Dresden Codex, the oldest surviving Mayan book, dating from the 13th or 14th century, illustrates Mayan number symbols and glyphs.
The Mayans, who lived in Central America from around 2000 BCE, used a base-20 (vigesimal) number system from around 1000 BCE to perform astronomical and calendar calculations. Like the Babylonians, they used a calendar of 360 days plus festivals, to make 365.24 days based on the solar year; their calendars helped them work out the growing cycles of crops.
The Mayan system employed symbols: a dot representing one and a bar representing five. By using combinations of dots over bars they could generate numerals up to 19. Numbers larger than 19 were written vertically, with the lowest numbers at the bottom, and there is evidence of Mayan calculations up to hundreds of millions. An inscription from 36 BCE shows that they used a shell-shaped symbol to denote zero, which was widely used by the 4th century.
The Mayans’ number system was in use in Central America until the Spanish conquests in the 16th century. Its influence, however, never spread further.
See also: The Rhind papyrus • The abacus • Negative numbers • Zero • The Fibonacci sequence • Decimals
IN CONTEXT
KEY CIVILIZATIONS
Egyptians (c. 2000 BCE), Babylonians (c. 1600 BCE)
FIELD
Algebra
BEFORE
c. 2000 BCE The Berlin papyrus records a quadratic equation solved in ancient Egypt.
AFTER
7th century CE The Indian mathematician Brahmagupta solves quadratic equations using only positive integers.
10th century CE Egyptian scholar Abu Kamil Shuja ibn Aslam uses negative and irrational numbers to solve quadratic equations.
1545 Italian mathematician Gerolamo Cardano publishes his Ars Magna, setting out the rules of algebra.
Quadratic equations are those involving unknown numbers to the power of 2 but not to a higher power; they contain x2 but not x3, x4, and so on. One of the main rudiments of mathematics is the ability to use equations to work out solutions to real-world problems. Where those problems involve areas or paths of curves such as parabolas, quadratic equations become very useful, and describe physical phenomena, such as the flight of a ball or a rocket.
Ancient roots
The history of quadratic equations extends across the world. It is likely that these equations first arose from the need to subdivide land for inheritance purposes, or to solve problems involving addition and multiplication.
One of the oldest surviving examples of a quadratic equation comes from the ancient Egyptian text known as the Berlin papyrus (c. 2000 BCE). The problem contains the following information: the area of a square of 100 cubits is equal to that of two smaller squares. The side of one of the smaller squares is equal to one half plus a quarter of the side of the other. In modern notation, this translates into two simultaneous equations: x2 + y2 = 100 and x = (1⁄2 + 1⁄4)y = 3⁄4 y. These can be simplified to the quadratic equation (3⁄4 y)2 + y2 = 100 to find the length of a side on each square.
The Egyptians used a method called “false position” to determine the solution. In this method, the mathematician selects a convenient number that is usually easy to calculate, then works out what the solution to the equation would be using that number. The result shows how to adjust the number to give the correct solution the equation. For example, in the Berlin papyrus problem, the simplest length to use for the larger of the two small squares is 4, because the problem deals with quarters. For the side of the smallest square, 3 is used because this length is 3⁄4 of the side of the other small square. Two squares created using these false position numbers would have areas of 16 and 9 respectively, which when added together give a total area of 25. This is only 1⁄4 of 100, so the areas must be quadrupled to match the Berlin papyrus equation. The lengths therefore must be doubled from the false positions of 4 and 3 to reach the solutions: 8 and 6.
Other early records of quadratic equations are found in Babylonian clay tablets, where the diagonal of a square is given to five decimal places. The Babylonian tablet YBC 7289 (c. 1800–1600 BCE) shows a method of working out the quadratic equation x2 = 2 by drawing rectangles and trimming them down into squares. In the 7th century CE, Indian mathematician Brahmagupta wrote a formula for solving quadratic equations that could be applied to equations in the form ax2 + bx = c. Mathematicians at the time did not use letters or symbols, so he wrote his solution in words, but it was similar to the modern formula shown above.
In the 8th century, Persian mathematician al-Khwarizmi employed a geometric solution for quadratic equations known as completing the square. Until the 10th century, geometric methods were were often used, as quadratic equations were used to solve real-world problems involving land rather than abstract algebraic challenges.
The Berlin papyrus was copied and published by German Egyptologist Hans Schack-Schackenburg in 1900. It contains two mathematical problems, one of which is a quadratic equation.
Negative solutions
Indian, Persian, and Arab scholars thus far had used only positive numbers. When solving the equation x2 + 10x = 39, they gave the solution as 3. However, this is one of two correct solutions to the problem; -13 is the other. If x is -13, x2 = 169 and 10x = -130. Adding a negative number gives the same result as subtracting its equivalent positive number, so 169 + -130 = 169 - 130 = 39.
In the 10th century, Egyptian scholar Abu Kamil Shuja ibn Aslam made use of negative numbers and algebraic irrational numbers (such as the square root of 2) as both solutions and coefficients (numbers multiplying an unknown quantity). By the 1500s, most mathematicians accepted negative solutions and were comfortable with surds (irrational roots – those that cannot be expressed exactly as a decimal). They had also started using numbers and symbols, rather than writing equations in words. Mathematicians now utilized the plus or minus symbol, ±, in solving quadratic equations. With the equation x2 = 2, the solution is not just x = but x = ±. The plus or minus symbol is included because two negative numbers multiplied together make a positive number. While × = 2, it is also true that - ×- = 2.
> In 1545, Italian scholar Gerolamo Cardano published his Ars Magna (The Great Art, or the Rules of Algebra) in which he explored the problem: “What pair of numbers have a sum of ten and product of 40?” He found that the problem led to a quadratic equation which, when he completed the square, gave . No numbers available to mathematicians at the time gave a negative number when multiplied by themselves, but Cardano suggested suspending belief and working with the square root of negative 15 to find the equation’s two solutions. Numbers such as would later be known as “imaginary” numbers.
The quadratic formula is a way to solve quadratic equations. By modern convention, quadratic equations include a number, a, multiplied by x2; a number, b, multiplied by x; and a number, c, on its own. The illustration above shows how the formula uses a, b, and c to find the value of x. Quadratic equations often equal 0, because this makes them easy to work out on a graph; the x solutions are the points where the curve crosses the x axis.
Politics is for the present, but an equation is for eternity.
Albert Einstein
Structure of equations
Modern quadratic equations usually look like ax2 + bx + c = 0. The letters a, b, and c represent known numbers, while x represents the unknown number. Equations contain variables (symbols for numbers that are unknown), coefficients, constants (those that do not multiply variables), and operators (symbols such as the plus and equals sign). Terms are the parts separated by operators; they can be a number or variable, or a combination of both. The modern quadratic equation has four terms: ax2, bx, c, and 0.
A graph of the quadratic function y = ax2 + bx + c creates a U-shaped curve called a parabola. This graph plots the points (in black) of the quadratic function where a = 1, b = 3, and c = 2. This expresses the quadratic equation x2 + 3x + 2 = 0. The solutions for x are where y = 0 and the curve crosses the x axis. These are -2 and -1.
Parabolas
A function is a group of terms that defines a relationship between variables (often x and y). The quadratic function is generally written as y = ax2 + bx + c, which, on a graph, produces a curve called a parabola. When real (not imaginary) solutions to ax2 + bx + c = 0 exist, they will be the roots—the points where the parabola crosses the x axis. Not all parabolas cut the x axis in two places. If the parabola touches the x axis only once, this means that there are coincident roots (the solutions are equal to each other). The simplest equation of this form is y = x2. If the parabola does not touch or cross the x axis, there are no real roots. Parabolas prove useful in the real world because of their reflective. properties. Satellite dishes are parabolic for this reason. Signals received by the dish will reflect off the parabola and be directed to one single point—the receiver.
Parabolic objects have special reflective properties. With a parabolic mirror, any ray of light parallel to its line of symmetry will reflect off the surface to the same fixed point (A).
Practical applications
Quadratic equations are used by military specialists to model the trajectory of projectiles fired by artillery—such as this MIM-104 Patriot surface-to-air missile, commonly used by the US Army.
Although they were initially used for working out geometric problems, today quadratic equations are important in many aspects of mathematics, science, and technology. Projectile flight, for example, can be modeled with quadratic equations. An object thrown up into the air will fall back down again as a result of gravity. The quadratic function can predict projectile motion—the height of the object over time. Quadratic equations are used to model the relationship between time, speed, and distance, and in calculations with parabolic objects such as lenses. They can also be used to forecast profits and loss in the world of business. Profit is based on total revenue minus production cost; companies create a quadratic equation known as the profit function with these variables to work out the optimal sale prices to maximize profits.
See also: Irrational numbers • Negative numbers • Diophantine equations • Zero • Algebra • The binomial theorem • Cubic equations • Imaginary and complex numbers
IN CONTEXT
KEY CIVILIZATION
Ancient Egyptians (c. 1650 BCE)
FIELD
Arithmetic
BEFORE
c. 2480 BCE Stone carvings record flood levels on the River Nile, measured in cubits—about 201⁄2 in (52 cm)—and palms—about 3 in (7.5 cm).
c. 1800 BCE The Moscow papyrus provides solutions to 25 mathematical problems, including the calculation of the surface area of a hemisphere and the volume of a pyramid.
AFTER
c. 1300 BCE The Berlin papyrus is produced. It shows that the ancient Egyptians used quadratic equations.
6th century BCE The Greek scientist Thales travels to Egypt and studies its mathematical theories.
The Rhind papyrus in the British Museum in London provides an intriguing account of mathematics in ancient Egypt. Named after Scottish antiquarian Alexander Henry Rhind, who purchased the papyrus in Egypt in 1858, it was copied from earlier documents by a scribe, Ahmose, more than 3,500 years ago. It measures 121⁄2 in (32 cm) by 781⁄2 in (200 cm) and includes 84 problems concerned with arithmetic, algebra, geometry, and measurement. The problems, recorded in this and other ancient Egyptian artifacts such as the earlier Moscow papyrus, illustrated techniques for working out areas, proportions, and volumes.
The Eye of Horus, an Egyptian god, was a symbol of power and protection. Parts of it were also used to denote fractions whose denominators were powers of 2. The eyeball, for example, represents 1⁄4, while the eyebrow is 1⁄8.
Representing concepts
The Egyptian number system was the first decimal system. It used strokes for single digits and a different symbol for each power of 10. The symbols were then repeated to create other numbers. A fraction was shown as a number with a dot above it. The Egyptian concept of a fraction was closest to a unit fraction—that is, 1⁄n, where n is a whole number. When a fraction was doubled, it had to be rewritten as one unit fraction added to another unit fraction; for example, 2⁄3 in modern notation would be 1⁄2 + 1⁄6 in Egyptian notation (not 1⁄3 + 1⁄3 because the Egyptians did not allow repeats of the same fraction).
The 84 problems in the Rhind papyrus illustrate the mathematical methods in common use in ancient Egypt. Problem 24, for example, asks what quantity, if added to its seventh part, becomes 19. This translates as x + x⁄7 = 19. The approach applied to problem 24 is known as “false position.” This technique—used well into the Middle Ages—is based on trial and improvement, choosing the simplest, or “false,” value for a variable and adjusting the value using a scaling factor (the required quantity divided by the result).
In the workings for problem 24, one-seventh is easiest to find for the number 7, so 7 is used first as a “false” value for the variable. The result of the calculation—7 plus 7⁄7 (or 1)—is 8, not 19, so a scaling factor is needed. To find how far the guess of 7 is from the required quantity, 19 is divided by 8 (the “false” answer). This produces a result of 2 + 1⁄4 + 1⁄8 (not 23⁄8, as Egyptian multiplication was based on doubling and halving fractions), which is the scaling factor that should be applied. So 7 (the original “false” value) is multiplied by 2 + 1⁄4 + 1⁄8 (the scaling factor) to give the quantity 16 + 1⁄2 + 1⁄8 (or 165⁄8).
Many problems in the papyrus deal with working out shares of commodities or land. Problem 41 asks for the volume of a cylindrical store with a diameter of 9 cubits and a height of 10 cubits. The method finds the area of a square whose side length is 8⁄9 of the diameter, then multiplies this by the height. The figure of 8⁄9 is used as an approximation for the proportion of the area of a square that would be taken up by a circle if it were drawn within the square. This method is used in problem 50 to find the area of a circle: subtract 1⁄9 from the diameter of the circle, and find the area of the square with the resulting side length.
Ancient Egyptians used vertical lines to denote the numbers 1 to 9. Powers of 10, particularly t
hose inscribed on stone, were depicted as hieroglyphs—picture symbols.
Level of accuracy
Since the Ancient Greeks, the area of a circle has been found by multiplying the square of its radius (r2) with the number pi (π), written as πr2. The ancient Egyptians had no concept of pi, but the calculations in the Rhind papyrus show that they were close to its value. Their circle area calculation—with the circle diameter as twice the radius (2r)—can be expressed as (8⁄9 × 2r)2, which, simplified, is 256⁄81 r2, giving an equivalent for pi of 256⁄81. As a decimal, this is about 0.6 percent greater than the true value of pi.
Instruction books
The Rhind papyrus scribe used the hieratic system of writing numerals. This cursive style was more compact and practical than drawing complex hieroglyphs.
The Rhind and Moscow papyri are the most complete mathematical documents to survive from the height of the ancient Egyptian civilization. They were painstakingly copied by scribes well versed in arithmetic, geometry, and mensuration (the study of measurements) and are likely to have been used for training of other scribes. Although they captured probably the most advanced mathematical knowledge of the time, they were not seen as works of scholarship. Instead, they were instruction manuals for use in trade, accounting, construction, and other activities that involved measurement and calculation.