Numbers
A number is an abstract expression of a quantity. Mathematicians define numbers in terms of sets containing sets. All the known numbers can be built up from a starting point of zero. Numerals are the written symbols that are agreedon to represent numbers.
Natural and Whole Numbers
The natural numbers, also called whole numbers or counting numbers, are built up from a starting point of 0 or 1, depending on which text you consult. The set of natural numbers is denoted N. If we include 0, we have this:
 N = {0, 1, 2, 3, . . ., n, . . .}
In some instances, 0 is not included, so:
 N = {1, 2, 3, 4, . . ., n, . . .}
Natural numbers can be expressed as points along a geometric ray or halfline, where quantity is directly proportional to displacement (Fig. 14).
Integers
The set of natural numbers, including zero, can be duplicated and inverted to form an identical, mirrorimage set:
 –N = {0, –1, –2, –3, . . ., –n, . . .}
The union of this set with the set of natural numbers produces the set of integers, commonly denoted Z:
 Z = N –N
 = {. . ., –n, . . ., –2, –1, 0, 1, 2, . . ., n, . . .}
Integers can be expressed as individual points spaced at equal intervals along a line, where quantity is directly proportional to displacement (Fig. 15). In the illustration, integers correspond to points where hash marks cross the line. The set of natural numbers is a proper subset of the set of integers:
 N Z
For any number a, if a is an element of N, then a is an element of Z. The converse of this is not true. There are elements of Z (namely, the negative integers) that are not elements of N.
Rational Numbers
A rational number (the term derives from the word ratio) is a quotient of two integers, where the denominator is positive. The standard form for a rational number r is:
 r = a/b
Any such quotient is a rational number. The set of all possible such quotients encompasses the entire set of rational numbers, denoted Q. Thus:
 Q = {x  x = a/b}
where a Z, b Z, and b > 0. (Here, the vertical line means ''such that.'') The set of integers is a proper subset of the set of rational numbers. Thus, the natural numbers, the integers, and the rational numbers have the following relationship:
 N Z Q
Decimal Expansions
Rational numbers can be denoted in decimal form as an integer followed by a period (radix point) followed by a sequence of digits. The digits following the radix point always exist in either of two forms:
 a finite string of digits beyond which all digits are zero
 an infinite string of digits that repeat in cycles
Examples of the first type, known as terminating decimals, are:
 3/4 = 0:750000 . . .
 –9/8 = –1:1250000 . . .
Examples of the second type, known as nonterminating, repeating decimals, are:
 1/3 = 0:33333 . . .
 –123/999 = –0:123123123 . . .
Irrational Numbers
An irrational number is a number that cannot be expressed as the ratio of any two integers. (This is where the term ''irrational'' comes from; it means ''existing as no ratio.'') Examples of irrational numbers include:
 the length of the diagonal of a square in a flat plane that is 1 unit long on each edge; this is 2^{1/2}, also known as the square root of 2
 the circumferencetodiameter ratio of a circle as determined in a flat plane, conventionally named by the lowercase Greek letter pi (π)
All irrational numbers share one quirk: they cannot be expressed precisely using a radix point. When an attempt is made to express such a number in this form, the result is a nonterminating, nonrepeating decimal. No matter how many digits are specified to the right of the radix point, the expression is only an approximation of the actual value of the number. The best we can do is say things like this:
 2^{1/2} = 1:41421356 . . .
 π = 3:14159 . . .
Sometimes, ''squiggly equals signs'' are used to indicate that values are approximate:
 2^{1/2} ≈ 1.41421356
 π ≈ 3.14159
The set of irrational numbers can be denoted S. This set is entirely disjoint from the set of rational numbers, even though, in a sense, the two sets are intertwined:
 S Q =

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