THE RATIONAL NUMBERS are the numbers of ordinary arithmetic. They are the whole numbers, the fractions, the mixed numbers, and decimals. Those are the numbers with whose names we count and measure.
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 ; 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?
To see the answer, pass your mouse over the colored area.
All of them!
Problem 2. Write each of the following as a fraction.
Problem 3. To what does the word "rational" refer?
The ratio of two natural numbers.
The number line
We need the numbers of arithmetic for measuring. Therefore we can think of them as naming a distance from 0 along the number line.
But will those 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
And we can always express two fractions with the same denominator.
We could prove that by multiplying both fractions by their common denominator. (Lesson 3.)
We can make the denominators the same.
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.
Explicitly means to verbally name that ratio.
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:
We have now established the theorem:
Any two rational numbers have the same ratio
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?
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.
Problem 5. Explicitly, what ratio has
Problem 6. Show that these rational numbers have the same ratio as two natural numbers.
f) 6.1 : 6.01 = 610 : 601
want to cut off half a pound; where will you cut the loaf?
a) Corresponding to every rational number, is there a distance from 0
b) Corresponding to every distance from 0, is there a rational number?
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Copyright © 2016 Lawrence Spector
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