6
RATIONAL NUMBERS
| WE HAVE SEEN THAT EVERY FRACTION | a b |
has the same ratio to 1 as the |
numerator has to the denominator:
| a b |
: 1 = a : b. |
Any number that is to 1 in the same ratio as two natural numbers, is called a rational number. A rational number, then, is any number that we could write as a fraction.
We need the rational numbers for measuring. We therefore conceive of rational numbers as being continuous.
| Any whole number, such a 6 -- which we could write as | 6 1 |
-- is |
rational; any mixed number, such as 3½, is rational; and any decimal, such as 6.732, is rational. The rational numbers are simply the numbers of arithmetic.
(In algebra, these numbers of arithmetic are extended to their negative images.)
Problem 1. Which of these numbers are rational?
| 1 5 | 3 8 |
6¼ | .005 9.2 1.6340812437 |
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All of them!
Problem 2. Write each of the following as a fraction.
| 5 = | 5 1 |
6¼ = | 25 4 |
.35 = | 35 100 |
9.2 = | 92 10 |
1.732 = | 1732 1000 |
Problem 3. To what does the word "rational" refer?
The ratio of two natural numbers.
The number line
We can think of a rational number as a distance from 0 along the number line. For, we use the rational numbers for measuring.
But will the rational numbers account for every distance from 0? Is every length 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.
For,
Fractions with the same denominator are in the same ratio
as their numerators.
| Example 1. | 2 5 |
: | 3 5 |
= 2 : 3 |
| 2 5 |
is two thirds of | 3 5 |
. |
We could prove that by multiplying both fractions by their common denominator. (Topic 2.)
| Example 2. | 2 3 |
: | 5 6 |
Here, we can make the denominators the same.
| 2 3 |
= | 4 6 |
Therefore,
| 2 3 |
: | 5 6 |
= | 4 6 |
: | 5 6 |
= | 4 : 5 |
| Example 3. | 2 3 |
: | 5 8 |
In this example, we can choose a common denominator, 3· 8 = 24. We can then obtain the numerators by cross-multiplying:
Therefore we can always express the ratio of two fractions by cross-multiplying. Cross-multiplying gives the numerators of the common denominator.
| Example 4. | 4 5 |
: | 7 9 |
= | 36 : 35 |
| Example 5. |  |
= | 10 : 3 |
| Example 6. Explicitly, what ratio has | 1 2 |
to 1 | 3 4 |
? |
Explicitly means to verbally name that ratio.
| Answer. | 1 2 |
: 1 | 3 4 |
= | 1 2 |
: | 7 4 |
= 4 : 14 = 2 : 7 |
| Explicitly, then, | 1 2 |
is two sevenths of 1 | 3 4 |
. |
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.
Problem 4. Show that these rational numbers have the same ratio as two natural numbers.
| a) | 5 9 |
: | 7 9 |
= | 5 : 7. The denominators are the same. |
| b) | 15 3 |
: | 16 3 |
= | 15 : 16 | c) | 1 2 |
: | 3 4 |
= | 2 4 |
: | 3 4 |
= 2 : 3 |
| d) | 2 5 |
: | 3 7 |
= | 14 : 15. Cross-multiply. | e) | 1 2 |
: | 1 3 |
= | 3 : 2 |
| f) | 3 8 |
: | 7 10 |
= | 30 : 56 = 15 : 28 | g) | 4 9 |
: | 2 3 |
= | 12 : 18 = 2 : 3 |
| h) | 2 | : | 1 2 |
= | 4 : 1 | i) | 5 6 |
: | 7 | = | 5 : 42 | j) | 2 3 |
: | 1 | = | 2 : 3 | ||
| k) | 1 | : | 1 2 |
= | 2 : 1 | l) | 8 5 |
: | 1 | = | 8 : 5 | m) | 1 | : | 8 5 |
= | 5 : 8 |
| n) | 1 : 3 | 1 2 |
= | 1 : | 7 2 |
= 2 : 7 | o) | 6 | 7 8 |
: 5 | = | 55 8 |
: 5 = 55 : 40 = 11 : 8 |
| p) | 2 | 3 4 |
: 3 | 1 2 |
= | 11 4 |
: | 7 2 |
= 22 : 28 = 11 : 14 |
Problem 5. Explicitly, what ratio has
| a) | 1 2 |
to 2? | 1 2 |
: 2 = 1 : 4. | 1 2 |
is one fourth of 2. |
| b) | 4 3 |
to | 2 9 |
? | 4 3 |
: | 2 9 |
= 36 : 6 = 6 : 1. | 4 3 |
is six times | 2 9 |
. |
| c) 1 | 1 4 |
to | 1 2 |
? | 1 | 1 4 |
: | 1 2 |
= | 5 4 |
: | 1 2 |
= | 5 4 |
: | 2 4 |
= 5 : 2. | |||
| 1 | 1 4 |
is two and a half times | 1 2 |
. | ||||||||||||||||
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 | 1 3 |
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 | 1 3 |
pounds?) |
| 1 2 |
: 1 | 1 3 |
= | 1 2 |
: | 4 3 |
= | 3 : 8. Cut three eighths of the loaf. |
Problem 8.
a) Corresponding to every rational number, is there a point on the
a) number line?
Yes.
b) Corresponding to every point on the number line, is there a
a) rational number?
Hmmm. Is there?
Next Topic: Measurement: Geometry and arithmetic
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