Subsets of Real Numbers
Lori Jordan
Kate Dirga
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Printed: December 30, 2014
AUTHORS
Lori Jordan
Kate Dirga
www.ck12.org
C HAPTER
Chapter 1. Subsets of Real Numbers
1
Subsets of Real Numbers
Here you’ll learn how to identify the subsets of real numbers and place a real number into one of these subsets.
The number of survey participants who declined to respond can be represented by the decimal 0.14141414... How
would you write this decimal as a fraction?
By being able to write a repeating decimal as fraction, we know it is a rational number.
Watch This
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/57616
James Sousa: Identifying Sets of Real Numbers
Guidance
There are several types of real numbers. You are probably familiar with fractions, decimals, integers, whole numbers
and even square roots. All of these types of numbers are real numbers. There are two main types of numbers: real
and complex. We will address complex (imaginary) numbers in the Quadratic Functions chapter.
TABLE 1.1:
Real Numbers
Rational Numbers
Irrational Numbers
Integers
Whole Numbers
Natural Numbers
Any number that can be plotted on
a number line. Symbol: R
Any number that can be written as
a fraction, including repeating decimals. Symbol: Q
Real numbers that are not rational.
When written as a decimal, these
numbers do not end nor repeat.
All positive and negative “counting” numbers and zero. Symbol: Z
All positive “counting” numbers
and zero.
All positive “counting” numbers.
Symbol: N
Examples: 8, 4.67, − 13 , π
Examples: − 59 , 18 , 1.3, 16
4
√ √
3
Example: e, π, − 2, 5
Example: -4, 6, 23, -10
Example: 0, 1, 2, 3, ...
Example: 1, 2, 3, ...
A counting number is any number that can be counted on your fingers.
The real numbers can be grouped together as follows:
1
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Example A
What is the most specific subset of the real numbers that -7 is a part of?
Solution: -7 is an integer.
Example B
List all the subsets that 1.3 lies in.
Solution: 1.3 is a terminating decimal. Therefore, it is considered a rational number. It would also be a real number.
3
As a fraction, we would write 1 10
because the 3 is in the tenths position after the decimal.
Example C
True or False:
8
3
is a rational number.
Solution: Yes, by definition, because it is written as a fraction.
Intro Problem Revisit How do we write 0.14141414.... as a fraction? Let’s devise a step-by-step process.
Step 1: Set your repeating decimal equal to x.
x = 0.14141414
Step 2: Find the repeating digit(s).
In this case 14 is repeating.
Step 3: Move the repeating digits to the left of the decimal point and leave the remaining digits to the right.
14.14141414
Step 4: Multiply x by the same factor you multiplied your original repeating decimal to get your new repeating
decimal.
2
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Chapter 1. Subsets of Real Numbers
14.14141414 = 100(0.14141414)
So,
100x = 14.14141414
Step 5: Solve your system of linear equations for x.
(100x = 14.14141414) − (x = 0.14141414)
yields:
99x = 14
, so x =
14
99
What about 0.327272727... ? The 0.3 does not repeat. So, rewrite this as 0.727272727... − 0.4 Therefore, the
fraction will be:
72
4
−
99 10
2
8
−
11 5
40 22
−
55 55
18
55
Guided Practice
1. What type of real number is
√
5?
2. List all the subsets that -8 is a part of.
√
3. True or False: − 9 is an irrational number.
Answers
1.
√
5 is an irrational number because, when converted to a decimal, it does not end nor does it repeat.
2. -8 is a negative integer. Therefore, it is also a rational number and a real number.
√
3. − 9 = −3, which is an integer. The statement is false.
Vocabulary
Subset
A set of numbers that is contained in a larger group of numbers.
3
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Real Numbers
Any number that can be plotted on a number line.
Rational Numbers
Any number that can be written as a fraction, including repeating decimals.
Irrational Numbers
Real numbers that are not rational. When written as a decimal, these numbers do not end nor repeat.
Integers
All positive and negative “counting” numbers and zero.
Whole Numbers
All positive “counting” numbers and zero.
Natural Numbers or Counting Numbers
Numbers than can be counted on your fingers; 1, 2, 3, 4, ...
Terminating Decimal
When a decimal number ends.
Repeating Decimal
When a decimal number repeats itself in a pattern. 1.666..., 0.98989898... are examples of repeating decimals.
Explore More
What is the most specific subset of real numbers that the following numbers belong in?
1.
2.
3.
4.
5.
6.
5.67
√
− 6
9
5
0
-75
√
16
List ALL the subsets that the following numbers are a part of.
7. 4
8. 69
9. π
Determine if the following statements are true or false.
10.
11.
12.
13.
14.
15.
4
Integers are rational numbers.
Every whole number is a real number.
Integers are irrational numbers.
A natural number is a rational number.
An irrational number is a real number.
Zero is a natural number.
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Chapter 1. Subsets of Real Numbers
Rewrite the following repeating decimals as fractions.
16.
17.
18.
19.
20.
0.4646464646...
0.81212121212...
0.35050505050...
2.485485485485485...
1.25141414141414...
5