Lesson 19

COMMON FRACTIONS
PROPER FRACTIONS
MIXED NUMBERS
IMPROPER FRACTIONS

What you see above is called a number line. On it will be the numbers we need for measuring rather than counting. But measuring is very different from counting. We count with the natural numbers -- 1 person, 2 persons, 3 persons -- and each 1 is indivisible. You cannot take half of 1 person. But when we measure a length, for example, it will not always be a whole number. A length could be divided into any parts -- halves, thirds, fourths, fifths, millionths In order to measure, then, we must have numbers that are parts of number 1. That requires a new and very different kind of number. What we call a proper fraction is a number that is less than 1.

Number 1 is now continuous, not discrete.

to .3, which is a decimal fraction, or simply a decimal.

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

equal parts, and that we are counting 3 of them.

Number 1, in other words, has been divided into tenths.  At this point, the student should be clear about the language of division into equal parts, and why we use ordinal numbers.  See Lesson 2, the topic Division into equal parts.

Throughout we will speak of  "number 1."  We mean not only the idea of 1, but whatever the unit of measure might be.  1 centimeter, 1 inch, 1 pound, 1 hour.  For, fractions are numbers we require for measuring.

In everyday speech, a fraction means a part of a whole, as in the phrase "a fraction of the students." In other words, it means a ratio! Moreover, students are discrete not continuous, and in mathematics a fraction is a number we need to measure things that are continuous. A mathematcal fraction is not simply a part of any whole. It is a part of the unit of measure, which is 1. As for a pie divided into equal parts, it can be and should be, described verbally -- not with fractions. We do not measure pies. We will go into this more below.

Example 2.   If number 1 is divided into 5 equal parts, and we count 4 of them, what fraction is that?  Also, into which parts has number 1 been divided?

Number 1 has been divided into fifths.

Note:  To divide number 1 into fifths, we cut the line four times. We cut the line one less than the name of the part.

We write the name of a fraction hyphenated, but the name of a part, or of a ratio, unhyphenated. We explain why below.

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

Example 5.   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, a relationship, 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 is one half of 1.

That ratio defines the meaning of a fraction and its place on the number line. It is equivalent to the answer to Question 2.

This is the same ratio that 2 has to 3.

into three equal parts.  To cut that unit into thirds, we cut it only twice. "One third, two thirds."

Example 7.   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 8.   What number is at the arrow?

Answer.  Number 1 has been cut five times -- into six equal parts. That

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.

In fact, when the numerator is equal to the denominator,

then the fraction is equal to 1.

We say that those fractions also are improper.

See the next Lesson, Example 4.

Problem.   Which of these fractions are less than 1, equal to 1, or greater than 1?

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!

*

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

When we change an improper fraction to a mixed number, we say that we are extracting, or taking out, the whole number..

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

We have extracted the whole number 8.

Compare Lesson 10, Division.

To extract whole numbers, 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."

For an explanation of why we do that, see Lesson 20, Question 3.

"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.

In English, the names of the proper fractions are the same as the names of the parts, and therefore a fraction and a ratio have become confused. "One quarter" is the name of a part, or a ratio, which is a relationship between numbers. 5  people are one quarter of 20 -- and that statement is not a measurement, we are not invoking any fraction. The fraction we call "one-quarter," on the other hand, is one quarter of a continuous 1. That may sound like double-talk, but how else are we to explain the meaning of the fraction ¼  and its place on the number line? Obviously, one must first understand the ratio -- the part -- whose name is one quarter

(To write "5 is ¼ of 20," or "She ate ¼ of the pie," is not just stylistically crude. It shows a profound disregard for the distinction between a fraction and the name of a ratio or part.)

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 problematic questions in mathematics.

The interested reader is referred to The mathematical existence of numbers.

For more on the tension between a fraction and a ratio, see Section 3.

Please "turn" the page and do some Problems.

or

Continue on to Section 2:  The Relative Sizes of Fractions

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