Page 71 - ArithBook5thEd ~ BCC

Chapter 3

Fractions and Mixed Numbers

Fractions are expressions such as

3 a
or more generally ,
19 b

where a and b are whole numbers, and b ̸=0. For now, we assume that a and b are both positive (we
will introduce negative fractions in due course.) The numbers a and b are called terms. The term on
top is called the numerator, and the term on the bottom is called the denominator. The horizontal
line in the middle is called the fraction bar. Sometimes to save space, we write fractions in one line,
using a slash instead of the fraction bar, putting the numerator on the left and the denominator on the
right:
a
= a/b.
b
When reading a fraction out loud, we attach the suffix th or ths to the denominator, saying “abths”
for a/b.

Example 74. The fractions 1 and 6 are spoken “one eighth,” and “six thirteenths,” respectively.
8 13
If the denominator is 2 or 3, we say “halves” or “thirds” (not twoths or threeths!), respectively. If the
denominator is 4, we sometimes say “fourths,” and sometimes, ”quarters.”

Example 75. 5/2is spoken “five halves.” 4/3 is spoken “four thirds.” 3/4 is spoken “three fourths,”
or “three quarters.”

Sometimes, instead of using the th suffix, we just say “a over b.” Thus 4/9can be spoken as “four
ninths,” or just ”four over nine.”

3.1 What positive fractions mean

Positive fractions represent parts of a whole in a precise way. In the fraction a/b,the denominator
b represents the number of equal parts into which the whole has been divided, and the numerator
represents how many of those parts are being taken into account. For example, if a rectangle is divided
up into 3 equal parts, and 2 of those parts are shaded, then the shaded portion represents 2 (two-thirds)
3
of the whole rectangle.

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