Page 73 - ArithBook5thEd ~ BCC

absolute value greater than or equal to its denominator. It follows that a positive improper fraction
represents a number that is greater than (or equal to) one whole.

Example 77. The figure below represents three fractions with denominator 9: the first one (2/9) is
proper, representing a number less than 1. The other two are improper. The one in the middle (9/9)
represents the whole number 1. The last (10/9) represents a number that is greater than 1.

2 9 =1 10 =1 + 1
9 9 9 9

9/9 = 1 illustrates the general fact that the whole number 1 can be represented in infinitely many
ways as a fraction – just take any fraction whose numerator and denominator are equal. For example,

2 9 159
1= = = .
2 9 159

Intuitively, if an object is divided into equal parts, and all the parts are taken,then, in fact, the whole
(1 whole) has been taken.

For any whole number a (except 0),

a
=1.
a

Any whole number can be represented as a fraction. For example, to represent the whole number
3, we can think of three whole rectangles, each “divided” into 1 “part,” and write

3
3= .
1

More generally,

For any whole number a (including 0),

a
a = .
1

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