The Evolution of the

R E A L  N U M B E R S

7

# RATIONAL NUMBERS

THE RATIONAL NUMBERS are the numbers of ordinary arithmetic. They are the numbers with whose names we count and measure. They are the whole numbers, the fractions, the mixed numbers and decimals.

 Why are they called rational?  We have seen that every fraction ab has

the same ratio to 1 as the numerator has to the denominator:

 ab : 1  =  a : b.

A number that has the same ratio to 1 as two natural numbers -- whose relationship to 1 we can always name -- we say is rational.

Now we can write any number of arithmetic as a fraction and thus

 show that ratio to 1.  A whole number, such as 6, we can write as 61 ;  we

can write any mixed number as a fraction; and we can write any decimal as a fraction.

The rational numbers are simply the numbers of arithmetic.

(In algebra, those numbers of arithmetic are extended to their negative images. See Topic 2 of Precalculus.)

Problem 1.   Which of these numbers are rational?

 1    5 38 6¼ .005    9.2    1.6340812437

All of them!

Problem 2.   Write each of the following as a fraction.

 5 = 51 6¼ = 25 4 .35 = 35 100 9.2 = 9210 1.732 = 17321000

Problem 3.   To what does the word "rational" refer?

The ratio of two natural numbers.

The number line

We need the rational numbers for measuring.  Therefore we can think of a rational number as naming a distance from 0 along the number line.

But will the rational numbers account for every distance from 0? Will every length be a rational number of units?  To pursue that question, we have the following theorem:

Theorem.   Any two rational numbers have the same ratio as two natural numbers.

That is true because:

Fractions with the same denominator have the same ratio
as their numerators.

And we can always express two fractions with the same denominator.

 Example 1. 25 : 35 =  2 : 3

 25 is two thirds of 35 .

We could prove that by multiplying both fractions by their common denominator. (Lesson 3.)

 Example 2. 23 : 56

We can make the denominators the same.

 23 = 46

Therefore,

 23 : 56 = 46 : 56 = 4 : 5
 Example 3. 23 : 58

In this example, we can choose a common denominator, 3 × 8 = 24.  We can then obtain the numerators by cross-multiplying:

We can always express the ratio of two fractions by cross-multiplying.  Cross-multiplying gives the numerators of the common denominator.

 Example 4. 45 : 79 = 36 : 35
 Example 5. = 10 : 3
 Example 6.   Explicitly, what ratio has 12 to  1 34 ?

Explicitly means to verbally name that ratio.

 Answer. 12 : 1 34 = 12 : 74 =  4 : 14  =  2 : 7
 Explicitly, then, 12 is two sevenths of 1 34 .

Example 7.   .3 is to 1.24 in the same ratio as which two natural numbers?

Answer.   We can "clear of decimals" by multiplying both numbers by the same power of 10; in this case, 100:

 .3 : 1.24 = 30 : 124 = 15 : 62, upon dividing by 2.

We have now established the theorem:

Any two rational numbers have the same ratio
as two natural numbers
.

Example 8.   A photograph measures 2½ inches by 3½ inches. You want to enlarge it so that the shorter side will be 10 inches. How long will the larger side be?

Solution.  Proportionally,

 2½ inches : 3½ inches = 52 : 72 = 5 : 7.

So the question is:

5 : 7 = 10 inches : ? inches

Now, 5 has been multiplied by 2.  Therefore, 7 will also be multiplied by 2. (Lesson 3.)  The longer side will be 14 inches.

Problem 4.   Show that these rational numbers have the same ratio as two natural numbers.

 a) 59 : 79 = 5 : 7.  The denominators are the same.
 b) 15 3 : 16 3 = 15 : 16 c) 12 : 34 = 24 : 34 = 2 : 3
 d) 25 : 37 = 14 : 15.  Cross-multiply. e) 12 : 13 = 3 : 2
 f) 38 : 7 10 = 30 : 56 = 15 : 28 g) 49 : 23 = 12 : 18 = 2 : 3
 h) 2 : 12 = 4 : 1 i) 56 : 7 = 5 : 42 j) 23 : 1 = 2 : 3 k) 1 : 12 = 2 : 1 l) 85 : 1 = 8 : 5 m) 1 : 85 = 5 : 8
 n) 1 : 3 12 = 1 : 72 = 2 : 7 o) 6 78 : 5 = 55 8 : 5 = 55 : 40 = 11 : 8
 p) 2 34 : 3 12 = 11 4 : 72 = 22 : 28 = 11 : 14

Problem 5.   Explicitly, what ratio has

 a) 12 to  2? 12 : 2 = 1 : 4. 12 is one fourth of 2.
 b) 43 to 29 ? 43 : 29 = 36 : 6 = 6 : 1. 43 is six times 29 .
 c)   1 14 to 12 ? 1 14 : 12 = 54 : 12 = 54 : 24 = 5 : 2. 1 14 is two and a half times 12 .

Problem 6.   Show that these rational numbers have the same ratio as two natural numbers.

 a) .2 : .3  = 2 : 3 b) .2 : .03  = 20 : 3 c) 2 : .03  = 200 : 3
 d) .025 : 1  = 25 : 1000 = 1 : 40 e) .025 : .01  = 25 : 10 = 5 : 2

f)   6.1 : 6.01  = 610 : 601

 Problem 7.   A loaf of bread weighs 1 13 pounds, and you want to cut off

half a pound; where will you cut the loaf?

 (Hint:  What ratio has half a pound to 1 13 pounds?)
 12 : 1 13 = 12 : 43 = 3 : 8.   Cut three eighths of the loaf.

Problem 8.

a)   Corresponding to every rational number, is there a distance from 0
a)   on the number line?   Yes.

b)   Corresponding to every distance from 0, is there a rational number?
a)    Hmmm. Is there?

Next Topic:  Measurement:  Geometry and arithmetic

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