Download Real Numbers

By Abdullah Alturki
Lecturer in Mathematics
Kingdom of Saudi Arabia
Prince Sattam Bin Abdulaziz University
College of Science and Humanities
in Hotat Bani Tamim
Department of Mathematics

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
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Natural Numbers
Whole Numbers
Integers
Rational Numbers
Irrational Numbers
Real Numbers
Examples of Real Numbers
Properties of Real Numbers
Examples for Properties of Real Numbers
 Natural
numbers are the set of counting
numbers which starts from 1.
 They
have only positive value numbers.
 Natural
N
numbers are denoted by N
={1,2,3,4,5,…}
 Whole
numbers are the set of numbers that
include 0 plus the set of natural numbers.
 They
have one number zero as nether positive
nor negative but all other numbers are positive.
 Whole
numbers are denoted by W
W
={0,1,2,3,4,5,…}
W
={0} U N

Integers are the set of whole numbers with negative
values.

Positive Integers Z+ = { 1 , 2 , 3 , 4 , 5 , … } = N

Negative Integers Z− = { -1 , -2 , -3 , -4 , -5 , … }

Integers Z = { … , -3 , -2 , -1 , 0 , 1 , 2 , 3 , … }
={0,±1,±2,±3,±4,±5,…}

They have a zero and the positive numbers with
negative numbers.

Z+ U { 0 } U Z− = Z

Rational numbers are any numbers that can be expressed in the
a
form of
(fractions) , where a and b are integers, and b ≠ 0.
b

They are written as an integer divided by another integer and the
denominator is not zero and both numbers do not have common
factors.

Rational numbers can be called fractions.

b
Any number in the form a such that a,b,c  Z , c  0 is rational
c
1
number . Example: 1
is rational number.
2

They can always be expressed by using terminating decimals
or repeating decimals.

Terminating decimals are decimals that contain a finite number of
1
3
1
digits. Examples : 0.5 =
, 1.5 =
, 0.125 =
2
2
8

Repeating decimals are decimals that contain infinite number of digits.
1
5
Examples : 0.3333…= 0.3 = 3 , 0.050505… = 0.05 = 99

Rational numbers are denoted by Q

m
Q={ n : mZ , nZ , n0 }

5
m
Every integer is a rational number. Example 5 = 1 in general m  Z , m = 1

Irrational numbers are any numbers that cannot be written as a simple fraction .

They are expressed as non-terminating(non-ending),non-repeating decimal.

A prime number (or a prime) is a natural number greater than 1
that can be divided, without a remainder, only by 1 and itself.

The roots of prime number are irrational =
If p is prime number , then 𝑝 is irrational number .

Irrational numbers are denoted by

Examples for Irrational numbers :
Pi = 𝜋  3.14… , e  2.71… , 2 , 3 , - 5 ,
1
2
,
2
5
, 1
𝜋
2
 The
real numbers include all the rational numbers
and all the irrational numbers .
 Real
numbers are
denoted byR
 R={
N
I
x  x is rational or
x is irrational }
WZQ R
R
 Which
of the following is a real number ?
A.
- 21
B.
0.123
C.
3
2
D.
E.
13
All of the answer choices are correct.
 Real
Numbers include:
A.
Natural Numbers and Whole numbers
B.
Integers
C.
Rational Numbers and Irrational Numbers
D.
All of the answer choices are correct.
 Which
classification describes the number
2
A.
Rational number
B.
Irrational number
C.
Real number
D.
B and C
 Which
A.
i
B.
3+i
C.
−1
D.
4
one of the following is a real number ?
 Which
A.
5
2
B.
-5
C.
D.
22
7
𝜋
one of the following is not a rational number?
 Which
one of the following is not an irrational
number?
B.
1
3
e
C.
7
A.
D.
3
2
 The
number 0.1212212221222212… is
A.
Natural Number
B.
Integer
C.
Rational Number
D.
Irrational Number

Z+ U { 0 } U Z− =
A.
Natural Numbers N
B.
Whole Numbers W
C.
Integers Z
D.
Irrational Number I
 Which
one of the following is a real number ?
A.
i ( Imaginary Number )
B.
3 + −1
C.
2 - −1
D.
( −1 )2
 Which
A.
i
B.
3+i
C.
−1
D.
𝑖2
one of the following is a real number ?
 Which
classification describes the number -2.3
A.
Positive number
B.
Whole number
C.
Irrational number
D.
Rational number
 Is
A.
B.
0.50 a rational number ?
Yes
No
 Is
A.
B.
( - 0.252525 ) a rational number?
Yes
No
 Is
A.
B.
0.123456789 a rational number?
Yes
No
20  a rational number ?
A. Yes
B. No
1
 Is ( - 2
) a rational number ?
5
A. Yes
B. No
8
 Is
a rational number ?
2
A. Yes
B. No
 Is
 Is
A.
B.
1
5
a rational number ?
Yes
No
 Is
3
5
a rational number ?
Yes
B. No
 Is 𝜋 a rational number ?
A. Yes
B. No
A.
Consider the following set of numbers.
9
1
{ 0 , 1 , -2 , , 16 , 0.1 , 4 , 0.5 , 11 }
3
2
List the numbers in the set that are :
 Natural Numbers
 Whole Numbers
 Integers
 Rational Numbers
 Irrational Numbers
 Real numbers
List the numbers in the set below that
belong to the set of rational numbers.
6
{ - , - 9 , - 0.25 , 𝜋 , e , 7 }
3
A. { 𝜋 , e , 7 }
6
B. { - , - 9 , - 0.25 }
3
6
C. { - , - 9 , - 0.25 , e }
3
6
D. { - , - 9 , - 0.25 , 𝜋 }
3
 Classifying
Real Numbers : Name the set(s) of
numbers to which the number belongs to .
0.5
 Real number and rational number .

3
 Real number and irrational number .
 3.14
 Real number and rational number .
 -7
 Real number, rational number and integer .

 Counter
Example: an example that proves a
statement false .
 True
or False . If false give a counter example .
 All
A.
B.
whole numbers are rational numbers .
True
False
 All
A.
B.
whole numbers are natural numbers .
True
False
1. Commutative Property of Addition
For any real numbers a and b .
a+b=b+a
2. Commutative Property of Multiplication
For any real numbers a and b .
a•b=b•a
3. Associative Property of Addition
For any real numbers a , b and c .
a+(b+c)=(a+b)+c
or
(a+b)+c=a+(b+c)
4. Associative Property of Multiplication
For any real numbers a , b and c .
a•(b•c)=(a•b)•c
or
(a•b)•c=a•(b•c)
5. Distributive Property
For any real numbers a , b and c .
a•(b+c)=a•b+a•c
or
(a+b)•c=a•c+b•c
6. Additive Identity Property
For any real numbers a .
a+0=a=0+a
7. Multiplicative Identity Property
For any real numbers a .
a•1=a=1•a
8. Additive Inverse Property
For any real numbers a .
a+(-a)=0=(-a)+a
9. Multiplicative Inverse Property
For any real numbers a .
a • ( a𝟏 ) = 1 = ( a𝟏 ) • a , a  0
10. Zero Property
For any real numbers a .
a•0=0=0•a
11. Closure Property of Addition
For any real numbers a and b .
a + b is real number .
12. Closure Property of Multiplication
For any real numbers a and b .
a • b is real number .
 To
combine real numbers involving
negatives, we use these properties.
x + 3 = 3 + x is an example of which property?
A.
associative property of addition
B.
additive identity
C.
commutative property of addition
D.
additive inverse
5( a + 1 ) = 5a + 5 is an example of which property?
A.
associative property of multiplication
B.
distributive property
C.
commutative property of multiplication
D.
multiplicative inverse property
a + ( b + 2 ) = a + ( 2 + b ) is an example of which property?
A.
associative property of addition
B.
distributive property
C.
additive identity
D.
commutative property of addition
( 3 b ) • ( 1 ) = 3 b is an example of which property?
A.
multiplicative identity property
B.
multiplicative inverse property
C.
commutative property of multiplication
D.
associative property of multiplication
( a b ) c = a ( b c ) is an example of which property?
A.
commutative property of multiplication
B.
associative property of multiplication
C.
distributive property
D.
multiplicative inverse property
( a + 2 ) ( 3 + a ) = ( a + 2 )•( 3 )+( a + 2 )•( a )
is an example of which property?
A.
associative property of addition
B.
commutative property of multiplication
C.
associative property of multiplication
D.
distributive property
(x + 3) + (-x + -3) = 0 is an example of which property?
A.
multiplicative identity property
B.
additive identity property
C.
multiplicative inverse property
D.
additive inverse property
1
3y . ( 3𝑦
) = 1 is an example of which property?
A.
multiplicative identity property
B.
multiplicative inverse property
C.
commutative property of multiplication
D.
associative property of multiplication
Determine which equation illustrates the commutative property.
A.
x+0=x
B.
x + (y + z ) = (x + y) + z
C.
x+y=y+x
D.
x(y + z) = xy + xz
Which property is illustrated by the equation
xa + xb = x(a + b)
A.
Associative
B.
Commutative
C.
Distributive
D.
Identity
 Decide
which property goes along with each
of the following examples
( 1 ) = 1438 is …
multiplicative identity property
 1438
2016 + 0 = - 2016 is …
additive identity property
-
+ 50 = 50 + 25 is …
commutative property of addition
 25
- 5 ) 3 = 3 ( - 5 ) is …
commutative property of multiplication
(
 Decide
which property goes along with each
of the following examples




3 + ( 2 + 1 ) = ( 3 + 2 ) + 1 is …
associative property of addition
4 ( 3  2 ) = ( 4  3 )  2 is …
associative property of multiplication
- 9 + 9 = 0 is …
additive inverse property
1
- 15 ( - 15
) = 1 is …
multiplicative inverse property
 Decide
which property goes along with each
of the following examples
 5 ( 3 + 2 ) = 5  3 + 5  2 is …
distributive property
 If x,y  Z then x + y  Z is …
closure property of addition
 If 2  Z , 5  Z then ( 2 )( 5 )  Z is …
closure property of multiplication
 3 x + 2 x = ( 3 + 2 )  x is …
distributive property
 The
property that justify 5 ( 15 ) = 1 is called
A.
multiplicative identity property
B.
multiplicative inverse property
C.
commutative property of multiplication
D.
associative property of multiplication
 The
1
A. 11
B.
1
- 11
C.
11
D.
-11
additive inverse of the number 11 is
 The
A.
1
11
B.
1
- 11
C.
11
D.
-11
multiplicative inverse of the number 11 is
 The
A.
2
B.
- 12
C.
-2
D.
1
2
additive inverse of the number -
1
2
is
 The
A.
2
B.
- 12
C.
-2
D.
1
2
multiplicative inverse of the number - 12 is
 The
additive inverse of the number - 0.1 is
A.
- 0.1
B.
- 10
C.
10
D.
0.1
 What
A.
- 0.1
B.
- 10
C.
10
D.
0.1
is multiplicative inverse of - 0.1 ?
 Use
the distributive property to simplify :
5 ( 2x - 3 ) - 2 = …
A.
10 x + 17
B.
10 x - 6
C.
7 x - 17
D.
10 x - 17
 Use
properties of real numbers to simplify
algebraic expression :
3x+2y+4x-y
 Use
properties of real numbers to simplify
algebraic expression :
-3a-b-4a-5b
 Use
properties of real numbers to simplify
algebraic expression :
2(3x+y)+5(4x-y)
 Use
properties of real numbers to simplify
algebraic expression :
4 ( 3 x - y ) - 2 ( 5x + 4y )
 Use
properties of real numbers to simplify
algebraic expression :
1
1
-2(8x-2y)-3(6x-3y)
 Use
properties of real numbers to simplify
algebraic expression :
5 𝑦 − 2 − 5𝑦 + 3 − 3 3𝑦 − 2
 Use
properties of real numbers to simplify
algebraic expression :
𝑥− 𝑥− 𝑥− 𝑥−1
 What
is the identity element for addition of
real numbers?
A.
-1
B.
1
−1
C.
D.
0
 What
is the identity element for multiplication
of real numbers?
A.
-1
B.
1
−1
C.
D.
0
A
set S is said to be closed under addition if for
all elements a and b in S, a + b is also in S.
Which one of the following sets is closed under
addition?
A.
{0}
B.
{1}
C.
{0,1}

A set S is said to be closed under multiplication if for all
elements a and b in S, a × b is also in S.
Which of the following sets is not closed under multiplication?
A.
{0}
B.
{1}
C.
{ 1 , -1 }
D.
{1,2}

Closure: When you combine any two elements of the
set, the result is also included in the set.

If you add two even numbers (from the set of even
numbers), is the sum even?
A.
B.
Yes
No

Closure: When you combine any two elements of the
set, the result is also included in the set.

If you add two natural numbers (from the set of natural
numbers), is the sum natural number ?
A.
B.
Yes
No

A.
B.
If you divide two even numbers (from the set of even
numbers), is the quotient (the answer) even?
Yes
No