- The length of the diagonal of a square that measures 1 unit on each edge.
- The length of the diagonal of a cube that measures 1 unit on each edge.
- The ratio of a circle’s circumference to its diameter.
- The decimal number 0.01001000100001000001...
Whenever we try to express an irrational a number in decimal form, the result is an endless
nonrepeating decimal. (The last item in the above list has a pattern of sorts, but it is not
a repeating pattern like the decimal expansion of a rational.) No matter how many digits
we write down to the right of the decimal point, the expression is an approximation of the
actual value. A pattern can never be found that allows us to convert the expression to a ratio
of integers.
The set of irrationals can be denoted S. This set is disjoint from the set Q of rationals. No
irrational number is rational, and no rational number is irrational. In set notation,
S∩Q=∅Real numbers
The set of real numbers, denoted by R,is the union of the set Q of all rationals and the set S
of all irrationals:
R=Q∪S1–1/2(^12)
1–1/2
1–3/4
2
1–1/2 1–3/4
1–5/8
1–1/2 1–5/8
1–9/16
1–9/16 1–5/8
1–19/32
Figure 9-1 An interval on the rational-number line can be cut in
half over and over, and you can always find infinitely
many numbers in it.
The Number Hierarchy 125