Properties of Rational Numbers
Jen Kershaw
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Printed: November 10, 2014
AUTHOR
Jen Kershaw
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C HAPTER
Chapter 1. Properties of Rational Numbers
1
Properties of Rational
Numbers
Here you’ll learn to identify a rational number as the ratio of two integers.
Have you ever compared numbers?
Molly has spent the day skiing. She did 12 runs on the diamond trails and was very pleased with her speed and
ability. She felt that out of the 12 runs, that 9 of them were particularly good.
She wrote
9
12 .
Besides being a fraction, this is another type of number too.
Do you know what it is?
This Concept will help you to understand rational numbers.
Guidance
Some numbers are considered rational numbers. A rational number is a number that can be written as a ratio.
What is a ratio?
A ratio is a comparison of two numbers. For example, you might discover that the ratio of boys to girls in your
class one day was 12 to 13. That same ratio could be also be expressed using a colon, 12 : 13, or as a fraction, 12
13 .
In fact, any number that can be written as the ratio of two integers is classified as a rational number. Let’s take a
closer look at how to identify rational numbers now.
How can we determine if an integer is a rational number?
That is a good question. Let’s look at a value and see if we can write it as a ratio.
10
This number can be written as a ratio. Each whole number can be written over 1. That means that it can be written
in the form of a ratio. Notice that the fraction bar is a way to tell if the integer can be written as a ratio.
10 is a rational number.
− 23
This fraction is a rational number. Notice that it is written as a ratio already. We are comparing the numerator and
the denominator. Yes, it is negative. But that is okay, because we can have negative fractions. We call them rational
numbers.
− 23 is a rational number.
What about a decimal?
.687
This decimal can be written as a rational number over 1000. This is a rational number too.
.687 is a rational number.
Are there any others?
Yes. Terminating decimals and repeating decimals are also rational numbers.
• Terminating decimals, which are decimals with a set number of digits, are always rational. For example,
1
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0.007 is a terminating decimal, so it is rational.
• Repeating decimals, which are decimals in which one or more digits repeat, are always rational. For example,
0.3̄ is a repeating decimal in which the digit 3 repeats forever, so it is rational.
Are there any numbers that are not rational?
Yes. Some decimals don’t terminate and they don’t repeat. They just go on and on and on forever. These are
a special group of numbers called irrational numbers . They are not rational numbers. You will learn more
about them in another Concept.
Determine whether each is a rational number.
Example A
−4
Solution: Yes
Example B
1
3
Solution: Yes
Example C
.89765....
Solution: No, it does not terminate or repeat.
Here is the original problem once again.
Molly has spent the day skiing. She did 12 runs on the diamond trails and was very pleased with her speed and
ability. She felt that out of the 12 runs, that 9 of them were particularly good.
She wrote
9
12 .
Besides being a fraction, this is another type of number too.
Do you know what it is?
Molly’s comparison is a rational number. It is a ratio that is written in fraction form.
Other types of rational numbers are negative numbers, fractions, terminating and repeating decimals.
Vocabulary
Rational Number
any number positive or negative that can be written as a ratio.
Ratio
a comparison between two quantities. Can be written using the word “to”, using a colon, or using a fraction
bar
Terminating Decimal
a decimal that has a definite ending
2
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Chapter 1. Properties of Rational Numbers
Repeating Decimal
a decimal where some of the digits repeat themselves.
Irrational Number
a decimal that does not terminate or repeat but continues indefinitely.
Guided Practice
Here is one for you to try on your own.
Show that the following number is rational by writing it as a ratio in fraction form.
.85
Answer
We can say that this is eighty -five hundredths.
Next, we convert it to a fraction.
85
100
This is a rational number.
Video Review
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/57616
This is a James Sousa video on identifying sets of real numbers including rational numbers.
Explore More
Directions: Rewrite each number as the ratio of two integers to prove that each number is rational.
1.−11
2. 3 16
3. 9
4. .08
5. −.34
6..678
7.
4
5
8.−19
9. 25
10. .17
3
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11. .2347
12. −17
13. 347
14. 87
15. −97
4