Fundamentals of Mathematics, 2008a

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CHAPTER 4.
INTRODUCTION TO FRACTIONS AND MULTIPLICATION
AND DIVISION OF FRACTIONS
The closed circle at 0 indicates that 0 is included, while the open circle at 1 indicates that 1 is not included.
Some examples of positive proper fractions are
1
2 , 3 5 , 20 106
27
, and
255
Note that 1 < 2, 3 < 5, 20 < 27, and 106 < 225.
4.3.3 Positive Improper Fractions
Positive Improper Fractions
Fractions in which the whole number in the numerator is greater than or equal to the whole number in the
denominator are called positive improper fractions. On the number line, improper fractions lie to the
right of (and including) 1. Positive improper fractions are always greater than or equal to 1.
Some examples of positive improper fractions are
3
2 , 8 5 , 4 105
4
, and
16
Note that 3 ≥ 2, 8 ≥ 5, 4 ≥ 4, and 105 ≥ 16.
4.3.4 Positive Mixed Numbers
Positive Mixed Numbers
A number of the form
nonzero whole number + proper fraction
is called a positive mixed number. For example, 2 3 5
is a mixed number. On the number line, mixed
numbers are located in the interval to the right of (and including) 1. Mixed numbers are always greater than
or equal to 1.
4.3.5 Relating Positive Improper Fractions and Positive Mixed Numbers
A relationship between improper fractions and mixed numbers is suggested by two facts. The rst is that
improper fractions and mixed numbers are located in the same interval on the number line. The second fact,
that mixed numbers are the sum of a natural number and a fraction, can be seen by making the following
observations.
Divide a whole quantity into 3 equal parts.
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