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Algebra, Functions, Graphs, and Vectors 11

Rational numbers
A rational number (the term deriveð from the word ratià )isa
quotient of twm 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 quotientð encompasseð the entire set of rational numbers,
denoted Q. Thus,

Q {x x a/b}

where a Z, b Z, and b 0. The set of integerð is a proper
subset of the set of rational numbers. Thuð natural numbers,
integers, and rational numberð have the following relationship:

N Z Q

Decimal expansions
Rational numberð can be denoted in decimal form as an integer
followed by a period (radix poin' followed by a sequence of
digits. (See Decimal numbers above for more detailð concern-
ing this notationÑ The digitð following the radix point always
exist in either of twm forms:

A finite string of digitð beyond which all digitð are zerm
An infinite string of digitð that repeat in cycleð

Exampleð of the first type of rational number, known as termi-
nating decimalsł are:

3/4 0.750000 ...

9/8 1.1250000 ...

Exampleð of the second type of rational number, known as non-
terminatingł repeating decimalsł are:

1/3 0.33333 ...

123/999 0.123123123 ...