Lesson 19

COMMON FRACTIONS
PROPER FRACTIONS
MIXED NUMBERS
IMPROPER FRACTIONS

We are about to introduce a new kind of number, called a fraction. First, though, by a natural number we have meant a collection of indivisible ones.

6 people. 23 chairs. 10,000 names. If you try to take half of any 1, it will not be that same thing any more

A length, on the other hand, can be divided not only in half

-- it could be divided into any parts: thirds, fourths, fifths, sixths, millionths.

Therefore if we now think of 1 as the measure of a length,

then that gives rise to a new and different kind of number: a fraction. It is a number we need for measuring rather than counting. We are then able to say that a certain length, for example, is 5¼ cm. And so what we call a proper fraction is a number less than 1.  

Fractions refer to what is continuous rather than what is discrete. See The Evolution of the Real Numbers, Lesson 3.

We will see that everything we have learned about ratios will apply to fractions, because each numerator has a ratio to its denominator. And the fraction itself has that same ratio to 1.

a decimal fraction, or simply a decimal.

terms of the fraction.  We will see that they correspond to the terms of a ratio.

of 1, as we will see presently.

"one-half" because of the ratio of 1 to 2.  1 is one half of 2.

And that proper fraction itself, we will see, is one half of 1.

Answer.   Because of the ratio of 3 to 4.  3 is three fourths of 4.

Because the English names of the proper fractions are the same as the names of the parts, the important distinction between a fraction and a ratio has become blurred. "One half" is the name of a part, or ratio: which is a relationship between numbers. Numbers have ratios to one another: 6 is one half of 12, and that statement is not a measurement, we are not invoking any fraction. "One-half," on the other hand, written hyphenated, is the name that has been given to the fraction -- the measure -- that is one half of 1. That may sound like double-talk, but how else are to explain that fact? Obviously, one must first understand what it means to say that there is a number that is one half of 1. That is, one must first understand the ratio -- the part -- whose name is one half.

(To write "6 is ½ of 12," as lamentably some do, is not simply lazy. It shows a profound disrespect for the distinction between a fraction and a ratio.)

Spanish is much more scrupulous in distinguishing the names of the fractions from the names of the ratios. This fraction , for example, is called un dozavo, while the ratio of 1 to 12 is la duodécima parte.

We will respect the separation of ratio and number by writing the name of the number hyphenated -- three-fourths -- but the name of the ratio unhyphenated: three fourths. That separation gives life to the question, Can "the ratio of two lengths" (whatever that means) always be named? Is there a number? For, numbers have names. 5¼, 9.6, , . If we merely postulate that corresponding to every ratio, or every point on a line, there always is a number, then we will miss one of the most intriguing and disturbing questions in mathematics.

The interested student is referred to The Evolution of the Real Numbers.

For more on the tension between a fraction and a ratio, see Section 2.

Example 4.   Read these numbers:

	"Five-eighths"
	"Sixteen-thirds"	The improper fractions (see below) are named like the proper fractions.
	"Nine-halves"	(Not "Nine 2's.")
	"27 over 39"	With large numbers, it is not necessary to say "27 thirty-ninths."
	"56 thousandths"	When the denominator is a power of 10, however, we always say the decimal name.  (Lesson 2)
	"5 and 79 hundredths"	This is a mixed number; see below.

Ratio to 1

A fraction is a number; and as any number we know it relative to 1.

What is our understanding of "2"?  It is twice as much as 1.  What is "3"? It is three times 1.

Every number has a ratio to 1.  It is according to that ratio that we know each number.

What ratio, then, has ½ to 1?

½ is one half of 1.

½ has the same ratio to 1 as the numerator has to the denominator.

1 is one half of 2.  And the fraction ½ is one half of 1.

Since the numerator is to the denominator in the same ratio as two natural numbers, fractions are also called rational numbers.

This is the same ratio that 2 has to 3.

into three equal pieces.  Notice that to cut that unit into thirds, we cut it only twice.  "One third, two thirds."

Example 6.   This number  ¼  is called "one-quarter" or "one-fourth," because the numerator is one quarter of the denominator -- and ¼ itself is one quarter of 1.

To place ¼ on the number line, cut the line one less than the name of the part.  To divide 1 into fourths, cut the line three times.

Example 7.   What number is at the arrow?

Answer.  The unit has been cut five times -- into six equal pieces.

The proper fractions are the parts of 1.

The and in "2 and one-third" means plus.

into thirds.  We cut the line twice.

Problem.   Answer with a mixed number or with a whole number and a remainder, whichever makes sense.

a)  How many basketball teams -- 5 on a team -- can you make from 23
a)  students?

b)  You are going on a trip of 23 miles, and you have gone a fifth of the
b)  distance.  How far have you gone?

Answers.  

We can recognize an improper fraction when the numerator is greater than or equal to the denominator.

improper fraction.

We will now see explicitly why we use the division bar to signify a fraction.

"5 goes into 43 eight (8) times (40) with 3 left over."

The student should not have to write the division box. (Lesson 12.)

"9 goes into 32 three (3) times (27) with remainder 5."

The remainder is what we must add to 27 to get 32. (Lesson 10.)

"4 goes into 28 seven (7) times exactly."

"5 times 2 is 10, plus 3 is 13; over 5."

"8 times 3 is 24, plus 5 is 29; over 8."

To summarize:  Fractions that are less than 1 are called proper fractions, while fractions greater than or equal to 1 are improper. Improper fractions are equivalent to mixed numbers or whole numbers.

Please "turn" the page and do some Problems.

or

Continue on to Section 2.

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