Download The Real Number System

The Real Number System
Objectives
Standard 8.2
The student will describe orally and
in writing the relationships
between the subsets of the real
number system.
Real Numbers
Real Numbers – a combination of all the number systems.
• A real number is EITHER rational, or irrational. It
cannot be both.
• Real numbers include natural numbers, whole
numbers, integers, rational numbers, and irrational
numbers.
ALL NUMBERS ARE REAL NUMBERS.
Venn Diagram of Real Number System
Never Write In Red Ink
Natural Numbers
Natural numbers – the numbers used for counting
how many items you have.
• Zero is NOT included in the set of natural numbers.
• Decimals and Negative numbers are NOT included in
the set of natural numbers.
Example of Natural Numbers: {1, 2, 3, 4, 5, 6,…..}
Are these natural numbers?
162
Yes
0.35
No, it’s a decimal
-3
No, it’s a negative
number.
4
7
No, it’s a fraction, that
simplifies to .571429
20
5
Yes. Although it’s a
fraction, it simplifies
to 4.
Which of the following is the set of
natural numbers?
A.
B.
C.
D.
… -3, -2, -1, 0, 1, 2, 3,…
0, 1, 2, 3,…
1, 2, 3, 4,….
Any number that can be written in the
form of a/b.
Whole Numbers
Whole numbers – include natural numbers and zero.
Decimals and Negative numbers are NOT included in the
set of whole numbers.
Example of Whole Numbers: {0, 1, 2, 3, 4, 5, 6,…}
Are these whole numbers?
0
Yes
0.45
No, it’s a decimal
-3
No, it’s a negative
number.
1
2
No, it’s a fraction, that
simplifies to .5
36
4
Yes. Although it’s a
fraction, it simplifies
to 9.
Which of the following is the set of
whole numbers?
A.
B.
C.
D.
… -3, -2, -1, 0, 1, 2, 3,…
0, 1, 2, 3,…
1, 2, 3, 4,….
Any number that can be written in the
form of a/b.
Integers
Integers – include natural numbers, the opposite of the
natural numbers (negative numbers), and zero.
• All whole numbers are integers.
• All natural numbers are integers.
• Decimals are NOT included in the set of integers.
Example of Integers: {…, -2, -1, 0, 1, 2,….}
Are these integers?
-11.46
No, it’s a decimal.
-4
Yes
0
Yes
1
2
No, it’s a fraction, that
simplifies to .50
2, 356
Yes.
Which statement is true?
A.
B.
C.
D.
All integers are natural numbers.
All integers are whole numbers.
All whole numbers are natural numbers.
All whole numbers are integers.
Explain why the other answer choices are not
true?
Which of the following numbers is a
natural number, a whole number, and an
integer?
A.
B.
C.
D.
3
0
0.3
-8
Rational Numbers
Rational Numbers – a number that can be expressed as
the ratio of two integers. This ratio is sometimes called a
fraction.
• The set of rational numbers includes integers, whole
numbers and natural numbers.
• Decimals are rational numbers if the decimal repeats
such as with 0.1212… or if the decimal
stops/terminates such as with 432.8.
• Square roots are rational numbers if they are perfect
squares.
Are these rational numbers?
0.27
0.010110111…
Yes, it’s a repeating decimal.
3
Yes, it’s a whole number and
can be written as 3/1.
121
1

4
Yes, it’s a perfect square
and simplifies to 11.
Yes, it simplifies to -0.25
which is a terminating
decimal.
No it does not repeat.

No, the decimal form
of pi is 3.1415926….
3
No, the decimal form is
1.7320508…Also it’s not
a perfect square.
Which of the following numbers is
NOT a rational number?
A.
B.
C.
D.
0.2
4
5.7
3.121221222…
Irrational Numbers
Irrational Numbers – a number that cannot be expressed
as the ratio of two integers. “opposite of rational”
• Decimals that are irrational never repeat and never
end or terminate.
• The square root of any number that is not a perfect
square is irrational.
Are these irrational numbers?
0.13
0.010110111…
No, it repeats.
5
Yes, it does not repeat and
does not end.
4
No
49
No, it’s a perfect square.
Yes, pi is always
irrational.
6
1

2
No
Yes, the decimal form is
2.449484…Also it’s not a
perfect square.
Which of the following numbers is
an irrational number?
A. 0.81
B. 9.02
C. 5
D. 2
Which statement is true?
A. All integers are rational numbers.
B. A number can be both rational and
irrational.
C. Every integer is a whole number.
D. All natural numbers are irrational.
What is the lowest subset to which
- 2 belongs? (Look at the Venn Diagram
for the subsets of real numbers)
A.
B.
C.
D.
Irrational Number
Rational Number
Whole Number
Integer
Although -2 is a rational number and an integer. It is an integer first
which makes it the lowest subset.
Which statement is true?
A.
B.
C.
D.
All real numbers are rational.
All irrational numbers are real.
All integers are whole numbers.
All rational numbers are irrational.
Other Types of Numbers
Prime Numbers – numbers that are only divisible by 1 and
itself. Ex: {2, 3, 5, 7, 11, 13, 17, 19,…}
Composite Numbers – numbers that are not prime numbers.
{Ex: 4, 6, 8, 9, 10, 12, 14, 15,…}
Even Numbers – numbers that can be divided evenly by 2.
{ Ex: 2, 4, 6, 8, 10, 12, 14, 16…}
Odd Numbers – numbers that cannot be divided evenly by 2.
{ Ex: 1, 3, 5, 7, 9, 11, 13, 15, 17…}
Prime Numbers
Prime vs. Composite Numbers
Prime vs. Composite Numbers
A = {2,3,5,7, 11,19, 23, 29}
Which statement is true?
A.
B.
C.
D.
All numbers in A are odd.
All numbers in A are prime.
All numbers in A are even.
All numbers in A are composites.
Which number is composite and
odd?
A.
B.
C.
D.
15
19
17
8
Prime vs. Composite Numbers
On the number grid below, please highlight
each prime number…
Prime Numbers Highlighted…