number

A complex number can be represented graphically as a line whose end-point coordinates equal the real and imaginary parts of the complex number. This type of diagram is called an Argand diagram after the French mathematician Jean Argand (1768–1822) who devised it.

Coordinates are numbers that define the position of points in a plane or in space. In the Cartesian coordinate system, a point in a plane is charted based upon its location along intersecting horizontal and vertical axes. In the polar coordinate system, a point in a plane is defined by its distance from a fixed point and direction from a fixed line.

The simplest of all magic squares (figure 1) is formed by the 9 digits, with 5 in the centre and the even numbers at the corners, so that the sum of any row or column is 15. In figure 2 the numbers 1 to 5 are arranged in any order in the first row; the second commences with the fourth number from the first row and proceeds in the same relative order. The third row starts with the fourth number from the second row, and so on. Figure 3 consists of the numbers 0 to 4 multiplied by 5, and each row starts with the third number from the row above. Adding together the corresponding numbers from figures 2 and 3 produces the magic square in figure 4.

Symbol used in counting or measuring. In mathematics, there are various kinds of number. The everyday number system is the decimal (‘proceeding by tens’) system, using the base ten.

Real numbers include all rational numbers (integers, or whole numbers, and fractions) and irrational numbers (those not expressible as fractions). Complex numbers include the real and imaginary numbers (real-number multiples of the square root of −1). The binary number system, used in computers, has two as its base. The natural numbers, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, give a counting system that, in the decimal number system, continues 10, 11, 12, 13, and so on. These are whole numbers (integers), with fractions represented as, for example, 1/4, 1/2, 3/4, or as decimal fractions (0.25, 0.5, 0.75). They are also rational numbers.

Irrational numbers cannot be represented in this way and require symbols, such as √2, π, and e. They can be expressed numerically only as the (inexact) approximations 1.414, 3.142, and 2.718 (to three places of decimals), respectively. The symbols π and e are also examples of transcendental numbers, because they (unlike √2) cannot be derived by solving a polynomial equation (an equation with one variable quantity) with rational coefficients (multiplying factors). Complex numbers, which include the real numbers as well as imaginary numbers, take the general form a + bi, where i = √−1 (that is, i² = −1), and a is the real part and bi the imaginary part.

Evolution of number systems

Defining types of numbers Concepts such as negative number, rational number, and irrational number can be rigorously and precisely defined in terms of the natural numbers. There remains then the problem of defining the natural numbers. A modern approach defines the natural numbers in terms of sets. Zero is defined to be the empty set: 0 = ø (i.e. the set with no elements). Then 1 is defined to be the union of 0 and the set that consists of 0 (which is a set with 1 element, zero). Now we can define 2 as the union of 1 and the set containing 1 (which is a set containing 2 elements, zero and one), and so on.

An alternative procedure for constructing a number system is to define the real numbers in terms of their algebraic and analytical properties.

number

In grammar, the feature of a noun, verb, or pronoun that denotes whether it is singular or plural. The number of a noun will determine the number of its verb; singular nouns and pronouns usually take a singular verb (‘The cat runs’), and plural nouns and pronouns usually take a plural verb (‘the cats run’).

In practice there is little difference between singular and plural in English verbs . In the past tense they are identical. In the present tense, the only difference is that the third person singular usually has an s on the end, whereas the plural and the first and second person singular do not: I come, he comes, they come. Exceptions to this are the modal verb or auxiliary verb, such as: can, may, will, which do not have an s in the third person singular (as in ‘she can’); and the verb to be, which has distinctive forms in the first and third person singular in both the present and the past tenses .

Some singular nouns which refer to groups of people or animals can take a plural verb. See collective noun.

It can often happen that another noun or pronoun comes between the verb and its grammatical subject. If this is different in number from the subject, it can affect the number of the verb; for example ‘The contents of the book submitted to our legal department has raised a few eyebrows’. The subject of the verb here is contents, not book or department, so the verb should be plural (have), not singular (has). Nouns and pronouns linked by ‘and’ can take a plural or a singular verb.

Measurements of quantity, distance, or similar, that contain a plural noun can be regarded as a single unit, and therefore take a singular verb: ‘Fifty pounds is too much to pay’; ‘Twenty miles is a long way to walk’.

There are some nouns ending in -s that take a singular verb, for example ‘the news is good’. They include the names of certain games (‘darts is my favourite game’) and the names of certain diseases (‘mumps is unpleasant for adult males to catch’). Some nouns ending in -s take a singular verb when they denote an area of study (‘acoustics is a tricky subject’) but a plural verb when the noun has a different meaning (‘the acoustics of the hall are very good’) .