Numbers (numerals)

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A number originally was a count or a measurement. Mathematicians have extended this concept to include abstractions such as negative numbers and the square root of $-1$ . In common usage, number symbols are often used as labels (highway numbers) or to indicate order (serial numbers).

Examples

The most familiar numbers are the counting numbers or natural numbers. Some writers include 0, thus: {0, 1, 2, ...}. Others do not: {1, 2, 3, ...}. In the base ten number system, now in almost universal use worldwide, the symbols for natural numbers are written using ten digits, 0 through 9. The symbol for the set of all natural numbers is $\N.$

If the negative whole numbers are combined with the positive whole numbers and zero, one obtains the integers $\mathbb{Z}$ (German Zahl, plural Zahlen).

Negative numbers are used to indicate an opposite. If a positive number is used to indicate distance to the right of some fixed point, a negative number indicates distance to the left. If a positive number indicates a bank deposit, a negative number indicates a withdrawal.

Rational numbers are made up of all numbers that can be expressed as a fraction, with integer numerator and non-zero natural number denominator. The fraction m/n represents the quantity arrived at when a whole is divided into n equal parts, and m of those equal parts are chosen. If m is greater than n, the fraction is greater than one. Fractions can be positive, negative, or zero. The set of all fractions includes the integers, since every integer can be written as a fraction with denominator 1. The symbol for the rational numbers is a bold face $\mathbb{Q}$ (for quotient).

The real numbers are made up of all numbers that can be expressed as a decimal. These are the measuring numbers, and in the base ten number system are written as a string of digits, with a period (US) or a comma (Europe) to the right of the ones place. The symbol for the real numbers is $\mathbb{R}.$

All measurements are necessarily approximations; the accuracy of the approximation depends on the accuracy of the measuring device. Therefore all measurements are properly represented by decimals that end, the last decimal place indicating the accuracy of the measurement. For example, 1.23 inches indicates a measurement accurate to the nearest hundredth of an inch. However, mathematically, when a rational number is expressed as a decimal, it may never end. Thus 1/3 becomes 0.3333... (unending threes). Mathematicians, therefore, consider both decimals that end and decimals that go on forever. The latter represent an infinite series. Some real numbers can be written as fractions, 0.3333... for example. Others cannot, 0.1010010001... for example. A decimal that can be written as a fraction is called rational, a decimal that cannot be written as a fraction is called irrational. A decimal is rational when it either ends or repeats forever.

There is a technical sense in which the real numbers are the ideal set of numbers. They are the only complete ordered field.

Moving to a greater level of abstraction, and away from counting and measuring, the real numbers can be extended to the complex numbers $\Complex.$ This set of numbers arose, historically, from consideration of the question of whether or not there was any sense in which negative numbers can have a square root. A new number was invented, the square root of negative one, denoted by i, a symbol assigned to this new number by Leonhard Euler. The complex numbers consist of all numbers of the form a + bi, where a and b are real numbers. If b is zero, then a + bi is real. If a is zero, then a + bi is called imaginary. The complex numbers are an algebraically closed field, meaning that every polynomial with complex coefficients can be factored into linear factors with complex coefficients.

While the natural numbers and the real numbers suffice for most everyday purposes, mathematicians have invented many other sets of numbers with specialized uses. Some are subsets of the complex numbers. For example the roots of polynomials with rational coefficients are called the algebraic numbers. Real numbers that are not algebraic are called transcendental numbers. The Gaussian integers are complex numbers a + bi where a and b are integers. Sets of numbers that are not subsets of the complex numbers include the quaternions $\mathbb{H}$ , invented by Sir William Rowan Hamilton, in which multiplication is not commutative, and the octonions, in which multiplication is not associative.

Further generalizations

The arithmetical operations of numbers, such as addition, subtraction, multiplication and division, are generalized in the branch of mathematics called abstract algebra, the study of abstract number systems such as groups, rings and fields.

Elements of function fields of finite characteristic behave in some ways like numbers and are often regarded as a kind of number by number theorists.

Extensions

Superreal, hyperreal and surreal numbers extend the real numbers by adding infinitesimal and infinitely large numbers. While real numbers may have infinitely long expansions to the right of the decimal point, one can also try to allow for infinitely long expansions to the left. The number system which results depends on what base is used for the digits: any base is possible, but a system with the best mathematical properties is obtained when the base is a prime number p. This leads to the p-adic numbers. For dealing with infinite collections, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former give the ordering of the collection, the latter its size. (For the finite case, the ordinal and cardinal numbers are equivalent; but they differ in the infinite case.)

Numerals

Numbers should be distinguished from numerals, the symbols used to represent numbers. The number five can be represented by both the base ten numeral 5 and by the Roman numeral V. Notations used to represent numbers are discussed in the article numeral systems. An important development in the history of numerals was the development of a positional system, like modern decimals, which can represent very large numbers. The Roman numerals require extra symbols for larger numbers.

References

Bibliography

Erich Friedman, What's special about this number?
Steven Galovich, Introduction to Mathematical Structures, Harcourt Brace Javanovich, 1989, ISBN 0154534683.
Paul Halmos, Naive Set Theory, Springer, 1974, ISBN 0387900926.
Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972.
Whitehead and Russell, Principia Mathematia to *56, Cambridge University Press, 1910.
What's a Number? at cut-the-knot

External links

Mesopotamian and Germanic numbers

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