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results when you divide 1 by 7 and 1 by 17:
1
=
7 ¼ 0
:
142857142857142857142857
1
=
17 ¼ 0
:
05882352941176470588235294117647
Whether you encounter a single repeating number or, as in the case of 1/17,
groups of 16 repeating numbers, you are still working with a number that you
can represent explicitly. You can show that at a given point, the number begins to
repeat itself. You need not show the repeating decimals more than once. For
example, you can also represent the numbers this way:
1
=
7 ¼ 0
:
142857142857
1
=
17 ¼ 0
:
0588235294117647
Irrational Numbers
When you perform some divisions, the number you end up with is neither exact
nor repeating. Consider, for example, the ratio of the length of the circumference
of a circle to the length of its diameter. The ancient Greek letter mathematicians
use to designate this ratio is
(pi). When you try to find a terminating or
repeating rational number to designate
, you fail. The quotient you generate is
not exact, and it does not repeat itself at periodic intervals. Here is a sampling of
the number that results:
:
3
141592653589793238462643383279502884197169399375 . . .
At this point in history,
has been calculated to billions of digits. No repeating
set of numbers has turned up. Such a number is known as an irrational number.
The mathematical symbol for an irrational number is H. Here is how to math-
ematically represent the set of rational numbers:
H ¼f x j x 2 R x
2 Qg
Again, the vertical bar means ''such that'' and the
means ''is a member of.'' The
R designates the set of real numbers (discussed below), and the
[
2 means ''is not a
member of.'' The definition reads, ''An irrational number (H) is any number x,
such that x is an element in the set of real numbers and is not in the set of rational
numbers.''
 
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