Number and operations

Summary: Numerous civilizations throughout history developed unique number systems and number operation methods, some principles of which survive into the twenty-first century.

The properties of numbers and operations are among the first concepts that most people learn about mathematics and they were also among the earliest type of mathematical knowledge developed historically. Number and operations are pervasive in school curricula. There are many types of numbers (for example, integers, irrational numbers, and imaginary numbers), each with their own properties. Learning how to work with different types of numbers is basic to the work of learning mathematics. The term “operations” refers to the practice of applying some rule on a set of numbers; the four basic operations are addition, subtraction, multiplication, and division. However, there are a wide variety of other mathematical operations or operation-like procedures on many types of mathematical objects, such as modular arithmetic, that may be explored at many levels.

In the twenty-first century, students in the earliest grades start to investigate whole numbers and common fractions along with addition and subtraction. In the later primary grades, they may study base-10 decimals, a broader range of fractions, negative numbers, and equivalent forms for fractions, decimals, and percentages. Operations extend to include addition and subtraction of common fractions or decimals; multiplication and division of whole numbers; and relationships between operations. Concepts like ratios and proportions, integers, factorization, prime numbers, and some alternative methods of notation for very large numbers begin to be introduced in middle school. Students learn more arithmetic procedures with fractions, decimals, or integers, as well as to simplify computations using addition and multiplication properties. They also investigate squares and square roots. In high school, students may study very large and small numbers; properties of numbers and various number systems; vectors and matrices with real number properties; and number theory. Operations begin to include addition and multiplication of vectors and matrices, as well as permutations and combinations. These concepts continue to be extended into college with new systems of numbers and operations or operation-like procedures.

Early Number Systems

The first type of numbers people generally learn about are called the “natural” or “counting numbers”: 1, 2, 3, 4, 5, 6… The historical record shows that the use of counting numbers is an ancient practice. As with measurement, body parts may have been used, and archaeologists have found other evidence, such as notched bones, that support the idea of tallying or counting. The Egyptians used a base-10 system, written either with hieroglyphs or hieratic (cursive) script, and using special symbols for powers of 10 (10, 100, 1000, and so on). The ancient Egyptians were also aware of fractions, which were primarily written as unit fractions of the form 1/n, such as 1/2 or 1/4, although some Egyptian texts contain fractions of the form 2/n or 3/n, and other quantities were expressed as combinations of unit fractions.

The Babylonians used a numerical system with a base of 60, a practice that survives into the twenty-first century in the convention of dividing a circle into 360 degrees and in units of time, such as 60 seconds in a minute and 60 minutes in an hour. They used only two symbols, one signifying 1 and the other 10, to write all the values 1–60, and used the place system so that the meaning of a symbol depended on its place within a number—a major advance that was crucial to the development of modern mathematics. In a decimal or base-10 system,

111 = (1 × 10²) + (1 × 10¹) + (1 × 10⁰).

The same digits in the base-60 system would mean

111 = (1 × 60²) + (1 × 60¹) + (1 × 60⁰).

or the same quantity as 3661 in a base-10 system.

The ancient Mayan developed a number system with a base of 20 and a place system, using dots (with a value of 1) and bars (with a value of 5) to write the numbers 1–19, with powers of 20 indicated vertically. The Mayans also understood the concept of zero as a placeholder and had a special symbol for it, which they used in their calendar system.

An acrophonic number system was used in Greece by the first millennium b.c.e. Acrophonic means that numbers are signified by the first letter of the word used for that number, with symbols for 1, 5, 10, 100, and so on. As with the more familiar Roman numerals, this system was an additive system (rather than place), so the value of a number was found by adding up the value of all the symbols that comprised it. A competing system also used in Greece was one in which each letter of the alphabet was assigned a numeric value reflecting its order in the alphabet. In this system, the first 10 letters (alpha through iota) correspond to the numbers 1–10, then the next letter (kappa) stands for 20, the next (lambda) for 30, until rho, which signifies 100.The next letter (sigma) signifies 200, and so on. This system was also an additive system, so that 12 was written as iota beta or 10+2 and 211 as epsilon iota alpha. Numbers 1000–9000 were written by adding a superscript or subscription to the letters alpha through theta, while larger numbers were written with the symbol M (meaning “myriad”) for 10,000, with multiples indicated by writing other numbers above the M.

Roman Numerals

The familiar system of Roman numerals was developed from about the third century b.c.e. It was used throughout the Roman Empire and in Europe into the Middle Ages and was eventually replaced by the more efficient Hindu–Arabic number system. The Roman number system has the benefit of using only a few symbols, but does not include the concepts of zero or of place, so the value of a number is calculated by adding together all the values of its elements. The symbols used include M for 1000, D for 500, C for 100, L for 50, X for 10, V for 5, and I for 1, with the later refinement that a smaller number could be placed next to a larger number to indicate subtraction. Roman numerals translate to Hindu–Arabic numerals as the following:

LXXIII = 73

CDXXXII = 432

MCMLXXXV = 1985

MMX = 2010

Roman numerals are still in use in the twenty-first century to indicate succession (for example, King Richard III of England) and sometimes in film release dates. The inefficiency of the Roman system compared to the modern system of Hindu–Arabic numerals can be illustrated by trying to quickly determine which of the following three dates is most recent: MCMXCIX, MCMLXXXVII, and MMVII. Now try again with the same values in Hindu-Arabic numerals: 1999, 1987, and 2007.

Indian or Hindu Numerals

Indian or Hindu numerals and the concept of zero (written as a dot or small circle and referred to by the Sanskrit term sunya, which means “empty”) also appear to date to the third century b.c.e. Historians have cited Brahmi numerals, which share a name with a family of alphabets or scripts; which evolved into Gupta numerals, named for the fourth to sixth century c.e. Gupta dynasty; and then Nagari or Devanagari numerals, also named for alphabet systems, beginning in about the ninth century; and finally symbols that looked very much like the familiar numerals 0–9 somewhere around the fourteenth century. There are many origin theories for Hindu numerals, which fall into two general classes: they came from an alphabet (as did the Greek system) or they came from some other earlier number system (as did Roman numerals). Hindu number systems were predominantly base 10, and documents suggest that Indians were using a place value system by the sixth century c.e. Mathematician Pierre-Simon Laplace said, “The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India… Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions.”

Hindu systems of numerals appear to have made their way into Arabic and Islamic cultures in the latter half of the first millennium c.e. These Hindu numerals, along with a base-60 system using Arabic letters to represent numbers (common among astronomers) and a “finger arithmetic” system (widely used in business), coexisted for some time in the Arabic world. In the ninth century, mathematician Abu Ja’far Muhammad ibn Musa Al-Khwarizmi wrote On the Calculation with Hindu Numerals. His contemporary Abu Yusuf Ya’qub ibn Ishaq al-Kindi also wrote On the Use of the Indian Numerals (c. 830 c.e.). Several Arabic and Islamic scholars studied Hindu numerals in the tenth century. Abu Ali al-Husain ibn Abdallah ibn Sina (also known as Avicenna) was purportedly taught by Egyptians, and Abu’l Hasan Ahmad ibn Ibrahim Al-Uqlidisi is credited with helping to modify Hindu numerals to replace the traditional “finger arithmetic.” Mathematician Abu Arrayhan Muhammad ibn Ahmad al-Biruni visited India in the eleventh century c.e., though even before his first travels he had examined Arabic translations of Indian mathematics texts.

Hindu–Arabic Number System

The Hindu–Arabic number system was adopted in Europe a few centuries later, replacing Roman numerals, as Europeans became familiar with Arabic manuscripts. The first known example of Hindu numerals in a European document are in the tenth-century Codex Vigilanus, but the beginnings of widespread use appear to date closer to the fifteenth century. The symbols used in this system are similar to those used in Europe in the twenty-first-century (0–9), while a different set of symbols is used with the same number system in the Middle East and in parts of India (thus many Arabic speakers do not use what in the United States are commonly called “Arabic numerals”). The Moroccan mathematician Abu Bakr Al-Hassar is credited with developing the modern method of notating fractions (two numbers separated by a horizontal bar) in the twelfth century. Liber Abaci, written by Italian mathematician Leonardo Fibonacci in the early thirteenth century, was also influential in spreading the use of Hindu–Arabic numerals (and the place system) throughout Europe.

Any number may be used as a base when numbers are written using the place system. For instance, the binary (base 2) and hexadecimal (base 16) systems are used in work with computers. In the binary system, there are two digits (0 and 1), and each successive place is a greater power of 2. In the hexadecimal system, there are 16 digits (letters are used to express the extra digits required, so A = 10, B = 11, C = 12, D = 13, E = 14, and f = 15). If necessary to avoid confusion, the base of the number system may be included as a subscript; for example, 17₁₀ would be 17 in the base-10 system.

Modular arithmetic was developed by German mathematician Leonhard Euler in the eighteenth century and was advanced by others, including German astronomer and mathematician Carl Friedrich Gauss, whose 1801 Disquisitiones Arithmeticae established many of the rules of number theory. Modular arithmetic is sometimes called “clock arithmetic” because the concept is similar to that of a 12-hour clock. If it is currently 9 o’clock, 6 hours later it will be 3 o’clock, not 15 o’clock, because the clock starts over with 0 as soon as it reaches 12.

Aids to Computation

Systems to aid computation are almost as old as number systems themselves. For instance, Egyptian scribes used tables to help them perform arithmetic with fractions, and the abacus or counting frame was used in several ancient cultures, including those of Mesopotamia, Egypt, Persia, and Rome. However, the abacus is most strongly identified today with Asia, in particular China, where it was used at least as early as the second century b.c.e. A Chinese abacus consists of a number of rods divided by a beam into two regions or decks: the upper deck of each rod has two beads, and the lower deck has five. Mathematical operations are carried out by sliding the beads toward or away from the deck, and expert abacus operators can rapidly solve problems involving not only the four basic functions (addition, subtraction, multiplication, and division) but also square and cube roots. The simplicity and efficiency of the abacus encouraged its spread to other Asian countries, including India, Japan, and Korea. Use of the modern Japanese abacus, which uses one bead in the upper deck and four in the lower, is still taught in primary schools in Japan today as it is believed to aid students in forming a mental representation of numbers.

Arabic mathematicians developed a system of lattice multiplication, which involves using a lattice or grid of boxes divided into diagonal halves. To perform lattice multiplication, the two numbers to be multiplied are written across the top and the side of the grid, the digits are multiplied separately and then added along the diagonals to produce the result. This system was introduced to Europe by Fibonacci in 1202. It was improved by Scottish mathematician John Napier in the early seventeenth century through a type of abacus referred to as “Napier’s bones,” which consists of a tray and a set of 10 rods, one for each digit 0–9. Each rod is divided into nine squares, with each but the top divided by a diagonal line. Each square contains the product of its own digit multiplied by each other digit; for instance, the rod for 5 contains the values 5, 1/0, 1/5, 2/0, and so on (the / indicating the diagonal of the square). Napier’s bones are used to multiply, divide, and extract square roots. For example, to multiply, the rods for one number are placed in the tray, and the values from the rows comprising the digits of the second number are read off, adding together the pairs of values on the diagonals.

Logarithms are another important aid to calculation. A logarithm is an exponent such that when the base of a number system is raised to that power, the result will be the number. For instance in base 10, the logarithm of 100 is 2 because 10²=100. In the system of natural logs, the base is e (sometimes called “Euler’s number” after the Swiss eighteenth-century mathematician Leonhard Euler), the irrational constant 2.718281.… The natural log of 100 is 4.6051 because e^4.6051≈100.

One common use of logarithms before the advent of electronic calculators and computers was to simplify multiplication, division, and the calculation of powers and roots. As such, logarithms played an important role in the development of astronomy and other mathematically based sciences.

Napier is usually credited as the inventor of the logarithm due to his 1614 publication Mirifici Logarithmorum Canonis Descriptio, which included tables of natural logarithms and explanations of their use. Important tables of base 10 logarithms were published in 1617 and 1624 by English mathematician Henry Briggs.

Multiplication using logarithms rests on the following rule. For any base b

For instance, if the base is 10, c is 108, and d is 379:

Conducting multiplication in this way requires only looking up the two logarithms in the table, adding them, and looking up the antilogarithm (the base 10 raised to a power) in another table, which for large numbers is much quicker than doing the multiplication by hand. The slide rule, also developed in the seventeenth century, made the process even quicker and remained in common use well into the twentieth century.

Bibliography

Dantzig, Tobias. Number, the Language of Science: A Critical Survey for the Cultured Non-Mathematician. New York: Doubleday, 1956.

Eymard, Pierre, and Jean-Pierre Lafon. The Number Pi. Translated by Stephen S. Wilson. Providence, RI: American Mathematical Society, 2004.

Flannery, David. The Square Root of 2: A Dialogue Concerning a Number and a Sequence. New York: Copernicus, 2006.

Hodger, Andrew. One to Nine: The Inner Life of Numbers. New York: W. W. Norton, 2008.

Kaplan, Robert. The Nothing That Is: A Natural History of Zero. New York: Oxford University Press, 2000.