REAL NUMBERS
Dr. Gabriel Obed Fosu
Department of Mathematics
Kwame Nkrumah University of Science and Technology
https://scholar.google.com/citations?user=ZJfCMyQAAAAJ&hl=en&oi=ao
Google Scholar:
ResearchGate ID: https://www.researchgate.net/profile/Gabriel_Fosu2
Dr. Gabby (KNUST-Maths)
Real Numbers
1 / 26
Lecture Outline
1
Numbers
2
Standard subsets
Natural Numbers
Integers
Rational Numbers
Irrational and Real Numbers
3
Some Properties of Numbers
4
Relations on R
Inequalities
5
Absolute value
Dr. Gabby (KNUST-Maths)
Real Numbers
2 / 26
Numbers
Outline of Presentation
1
Numbers
2
Standard subsets
Natural Numbers
Integers
Rational Numbers
Irrational and Real Numbers
3
Some Properties of Numbers
4
Relations on R
Inequalities
5
Absolute value
Dr. Gabby (KNUST-Maths)
Real Numbers
3 / 26
Numbers
Introduction
Numbers
Numbers are vital mathematical objects used in various descriptive situations. For example
to describe weights, lengths, temperature, profits, loss, and many more.
Dr. Gabby (KNUST-Maths)
Real Numbers
4 / 26
Numbers
Introduction
Numbers
Numbers are vital mathematical objects used in various descriptive situations. For example
to describe weights, lengths, temperature, profits, loss, and many more.
Different quantities may require different kinds of numbers to describe them leading to
classification of numbers.
Dr. Gabby (KNUST-Maths)
Real Numbers
4 / 26
Standard subsets
Outline of Presentation
1
Numbers
2
Standard subsets
Natural Numbers
Integers
Rational Numbers
Irrational and Real Numbers
3
Some Properties of Numbers
4
Relations on R
Inequalities
5
Absolute value
Dr. Gabby (KNUST-Maths)
Real Numbers
5 / 26
Standard subsets
Natural Numbers
Natural Numbers
Definition
The set of natural (counting) numbers denoted by N is the set {1, 2, 3, 4, 5, 6, · · · }.
N admits some binary operations including addition and multiplication.
Dr. Gabby (KNUST-Maths)
Real Numbers
6 / 26
Standard subsets
Natural Numbers
Natural Numbers
Definition
The set of natural (counting) numbers denoted by N is the set {1, 2, 3, 4, 5, 6, · · · }.
N admits some binary operations including addition and multiplication.
Theorem
The set N is well-ordered, that is every subset of N has a least element.
Dr. Gabby (KNUST-Maths)
Real Numbers
6 / 26
Standard subsets
Natural Numbers
Natural Numbers
Properties of N under the operations
1
Commutativity: n + m = m + n and n × m = m × n.
Dr. Gabby (KNUST-Maths)
Real Numbers
7 / 26
Standard subsets
Natural Numbers
Natural Numbers
Properties of N under the operations
1
2
Commutativity: n + m = m + n and n × m = m × n.
Associativity: (n + m) + a = n + (m + a) and (n × m) × a = n × (m × a).
Dr. Gabby (KNUST-Maths)
Real Numbers
7 / 26
Standard subsets
Natural Numbers
Natural Numbers
Properties of N under the operations
1
2
3
Commutativity: n + m = m + n and n × m = m × n.
Associativity: (n + m) + a = n + (m + a) and (n × m) × a = n × (m × a).
Distribution: a × (n + m) = (a × n) + (a × m) and (n + m) × a = (n × a) + (m × a).
Dr. Gabby (KNUST-Maths)
Real Numbers
7 / 26
Standard subsets
Natural Numbers
Natural Numbers
Properties of N under the operations
1
2
3
4
Commutativity: n + m = m + n and n × m = m × n.
Associativity: (n + m) + a = n + (m + a) and (n × m) × a = n × (m × a).
Distribution: a × (n + m) = (a × n) + (a × m) and (n + m) × a = (n × a) + (m × a).
Multiplicative identity: There is some e ∈ N such that e × n = n = n × e for all
n ∈ N.
Dr. Gabby (KNUST-Maths)
Real Numbers
7 / 26
Standard subsets
Natural Numbers
Natural Numbers
Properties of N under the operations
1
2
3
4
5
Commutativity: n + m = m + n and n × m = m × n.
Associativity: (n + m) + a = n + (m + a) and (n × m) × a = n × (m × a).
Distribution: a × (n + m) = (a × n) + (a × m) and (n + m) × a = (n × a) + (m × a).
Multiplicative identity: There is some e ∈ N such that e × n = n = n × e for all
n ∈ N.
Cancellation: m = k if m + n = k + n or m × n = k × n.
Dr. Gabby (KNUST-Maths)
Real Numbers
7 / 26
Standard subsets
Integers
Integers
Definition
The set of integers denoted by Z is the set {· · · , −3, −2, −1, 0, 1, 2, 3, · · · }. Thus Z =
Z− ∪ {0} ∪ Z+
Z admits both addition and multiplication binary operations.
Dr. Gabby (KNUST-Maths)
Real Numbers
8 / 26
Standard subsets
Integers
Integers
Definition
The set of integers denoted by Z is the set {· · · , −3, −2, −1, 0, 1, 2, 3, · · · }. Thus Z =
Z− ∪ {0} ∪ Z+
Z admits both addition and multiplication binary operations.
Properties of Z under the operations
1
All the properties observed under N holds for Z.
Dr. Gabby (KNUST-Maths)
Real Numbers
8 / 26
Standard subsets
Integers
Integers
Definition
The set of integers denoted by Z is the set {· · · , −3, −2, −1, 0, 1, 2, 3, · · · }. Thus Z =
Z− ∪ {0} ∪ Z+
Z admits both addition and multiplication binary operations.
Properties of Z under the operations
1
2
All the properties observed under N holds for Z.
Additive and Multiplicative identity: ∃ e ∈ Z such that e + n = n = n + e and
e × n = n = n × e ∀ n ∈ Z respectively.
Dr. Gabby (KNUST-Maths)
Real Numbers
8 / 26
Standard subsets
Integers
Integers
Definition
The set of integers denoted by Z is the set {· · · , −3, −2, −1, 0, 1, 2, 3, · · · }. Thus Z =
Z− ∪ {0} ∪ Z+
Z admits both addition and multiplication binary operations.
Properties of Z under the operations
1
2
3
All the properties observed under N holds for Z.
Additive and Multiplicative identity: ∃ e ∈ Z such that e + n = n = n + e and
e × n = n = n × e ∀ n ∈ Z respectively.
There is an additive inverse n′ for all integers such that n′ + n = n + n′ = e.
Dr. Gabby (KNUST-Maths)
Real Numbers
8 / 26
Standard subsets
Integers
Integers
Definition
The set of integers denoted by Z is the set {· · · , −3, −2, −1, 0, 1, 2, 3, · · · }. Thus Z =
Z− ∪ {0} ∪ Z+
Z admits both addition and multiplication binary operations.
Properties of Z under the operations
1
2
3
All the properties observed under N holds for Z.
Additive and Multiplicative identity: ∃ e ∈ Z such that e + n = n = n + e and
e × n = n = n × e ∀ n ∈ Z respectively.
There is an additive inverse n′ for all integers such that n′ + n = n + n′ = e.
Dr. Gabby (KNUST-Maths)
Real Numbers
8 / 26
Standard subsets
Integers
Integers
Definition
The set of integers denoted by Z is the set {· · · , −3, −2, −1, 0, 1, 2, 3, · · · }. Thus Z =
Z− ∪ {0} ∪ Z+
Z admits both addition and multiplication binary operations.
Properties of Z under the operations
1
2
3
All the properties observed under N holds for Z.
Additive and Multiplicative identity: ∃ e ∈ Z such that e + n = n = n + e and
e × n = n = n × e ∀ n ∈ Z respectively.
There is an additive inverse n′ for all integers such that n′ + n = n + n′ = e.
Z admits subtraction operation by the definition a − b = a + (−b). It is evident that Z
is not commutative nor associative under this operation.
Dr. Gabby (KNUST-Maths)
Real Numbers
8 / 26
Standard subsets
Rational Numbers
Rational numbers
Definition
The set of rational numbers denoted by Q is the set of all numbers which can be written
in the form xy where x, y ∈ Z and y ̸= 0.
Alternatively a rational number can be written as a terminating or a repeating
decimal.
Dr. Gabby (KNUST-Maths)
Real Numbers
9 / 26
Standard subsets
Rational Numbers
Rational numbers
Definition
The set of rational numbers denoted by Q is the set of all numbers which can be written
in the form xy where x, y ∈ Z and y ̸= 0.
Alternatively a rational number can be written as a terminating or a repeating
decimal.
Addition and multiplication operations on Q is the usual addition and multiplication
operation on fractions.
Properties of Q under the operations
1
All the properties observed under Z holds here.
Dr. Gabby (KNUST-Maths)
Real Numbers
9 / 26
Standard subsets
Rational Numbers
Rational numbers
Definition
The set of rational numbers denoted by Q is the set of all numbers which can be written
in the form xy where x, y ∈ Z and y ̸= 0.
Alternatively a rational number can be written as a terminating or a repeating
decimal.
Addition and multiplication operations on Q is the usual addition and multiplication
operation on fractions.
Properties of Q under the operations
1
2
All the properties observed under Z holds here.
Every nonzero rational number has a multiplicative inverse.
Dr. Gabby (KNUST-Maths)
Real Numbers
9 / 26
Standard subsets
Irrational and Real Numbers
Irrational and Real Numbers
Irrational numbers
1
2
3
4
These are decimals which do not fit the description of rational numbers.
That is non-repeating nor terminating decimals.
Alternatively these are numbers which can not be written as ratio of integers.
The set of irrational numbers is denoted by Q.
Dr. Gabby (KNUST-Maths)
Real Numbers
10 / 26
Standard subsets
Irrational and Real Numbers
Irrational and Real Numbers
Irrational numbers
1
2
3
4
These are decimals which do not fit the description of rational numbers.
That is non-repeating nor terminating decimals.
Alternatively these are numbers which can not be written as ratio of integers.
The set of irrational numbers is denoted by Q.
Real Numbers
The set of real numbers denoted R is the union of Q and Q.
Dr. Gabby (KNUST-Maths)
Real Numbers
10 / 26
Standard subsets
Irrational and Real Numbers
Summary Page
Some standard subsets of R are:
1
N = {1, 2, 3, · · · } is the set of natural numbers.
2
W = {0, 1, 2, 3, · · · } is the set of whole numbers.
3
Z = {0, ±1, ±2, ±3, · · · } is the set of integers.
4
n
Q = {m
: n, m ∈ Z, m ̸= 0} is the set of rational numbers.
5
Q̄ is the set of irrational numbers or decimals which are not in Q.
6
R = Q ∪ Q̄ is the set of real numbers.
Dr. Gabby (KNUST-Maths)
Real Numbers
11 / 26
Standard subsets
Irrational and Real Numbers
Summary Page
Some standard subsets of R are:
1
N = {1, 2, 3, · · · } is the set of natural numbers.
2
W = {0, 1, 2, 3, · · · } is the set of whole numbers.
3
Z = {0, ±1, ±2, ±3, · · · } is the set of integers.
4
n
Q = {m
: n, m ∈ Z, m ̸= 0} is the set of rational numbers.
5
Q̄ is the set of irrational numbers or decimals which are not in Q.
6
R = Q ∪ Q̄ is the set of real numbers.
Figure 1: Numbers
Dr. Gabby (KNUST-Maths)
Real Numbers
11 / 26
Standard subsets
Dr. Gabby (KNUST-Maths)
Irrational and Real Numbers
Real Numbers
12 / 26
Some Properties of Numbers
Outline of Presentation
1
Numbers
2
Standard subsets
Natural Numbers
Integers
Rational Numbers
Irrational and Real Numbers
3
Some Properties of Numbers
4
Relations on R
Inequalities
5
Absolute value
Dr. Gabby (KNUST-Maths)
Real Numbers
13 / 26
Some Properties of Numbers
Closure
Definition
Let G be a set and let H be a subset of G. Then, for any two elements g, h ∈ H, then
their binary operation g ∗ h will be an element of G, which may or may not be in H. If
g ∗ h ∈ H for all g, h ∈ H , we say that H is closed under the binary operation of G.
Dr. Gabby (KNUST-Maths)
Real Numbers
14 / 26
Some Properties of Numbers
Closure
Definition
Let G be a set and let H be a subset of G. Then, for any two elements g, h ∈ H, then
their binary operation g ∗ h will be an element of G, which may or may not be in H. If
g ∗ h ∈ H for all g, h ∈ H , we say that H is closed under the binary operation of G.
1
The set of natural numbers is closed under + and ×, but not closed under − and
÷.
Dr. Gabby (KNUST-Maths)
Real Numbers
14 / 26
Some Properties of Numbers
Closure
Definition
Let G be a set and let H be a subset of G. Then, for any two elements g, h ∈ H, then
their binary operation g ∗ h will be an element of G, which may or may not be in H. If
g ∗ h ∈ H for all g, h ∈ H , we say that H is closed under the binary operation of G.
1
2
The set of natural numbers is closed under + and ×, but not closed under − and
÷.
The set of integers is closed under +, ×, and − but not closed under ÷.
Dr. Gabby (KNUST-Maths)
Real Numbers
14 / 26
Some Properties of Numbers
Closure
Definition
Let G be a set and let H be a subset of G. Then, for any two elements g, h ∈ H, then
their binary operation g ∗ h will be an element of G, which may or may not be in H. If
g ∗ h ∈ H for all g, h ∈ H , we say that H is closed under the binary operation of G.
1
2
3
The set of natural numbers is closed under + and ×, but not closed under − and
÷.
The set of integers is closed under +, ×, and − but not closed under ÷.
The set of rational numbers is closed under all the arithmetic operations +, −, ÷,
and ×.
In the case of closure under ÷, division by 0 must be avoided, as the result is either
indeterminate or infinity.
Dr. Gabby (KNUST-Maths)
Real Numbers
14 / 26
Some Properties of Numbers
Theorem (Q is dense in R)
For any two real numbers x and y, if x < y then there exists a rational number q such
that x < q < y.
Dr. Gabby (KNUST-Maths)
Real Numbers
15 / 26
Some Properties of Numbers
Theorem (Q is dense in R)
For any two real numbers x and y, if x < y then there exists a rational number q such
that x < q < y.
Archimedean Property
For any two positive real numbers x < y there exists n ∈ N such that y < nx.
Dr. Gabby (KNUST-Maths)
Real Numbers
15 / 26
Some Properties of Numbers
Theorem (Q is dense in R)
For any two real numbers x and y, if x < y then there exists a rational number q such
that x < q < y.
Archimedean Property
For any two positive real numbers x < y there exists n ∈ N such that y < nx.
Trichotomy
For any x and y, then exactly one of these applies
x < y,
Dr. Gabby (KNUST-Maths)
x = y,
Real Numbers
x>y
(1)
15 / 26
Relations on R
Outline of Presentation
1
Numbers
2
Standard subsets
Natural Numbers
Integers
Rational Numbers
Irrational and Real Numbers
3
Some Properties of Numbers
4
Relations on R
Inequalities
5
Absolute value
Dr. Gabby (KNUST-Maths)
Real Numbers
16 / 26
Relations on R
Relations on R
1
A binary relation on a set S describes how two elements s1 and s2 are related.
2
We use s1 Rs2 to mean ‘s1 is related to s2 ’.
3
For instance s1 Rs2 could mean ‘s1 is a friend of s2 ’ or ‘s1 is less than s2 ’.
Dr. Gabby (KNUST-Maths)
Real Numbers
17 / 26
Relations on R
Relations on R
1
A binary relation on a set S describes how two elements s1 and s2 are related.
2
We use s1 Rs2 to mean ‘s1 is related to s2 ’.
3
For instance s1 Rs2 could mean ‘s1 is a friend of s2 ’ or ‘s1 is less than s2 ’.
Let R be a binary relation on a set S.
1
R is reflexive if for any s ∈ S, sRs.
Dr. Gabby (KNUST-Maths)
Real Numbers
17 / 26
Relations on R
Relations on R
1
A binary relation on a set S describes how two elements s1 and s2 are related.
2
We use s1 Rs2 to mean ‘s1 is related to s2 ’.
3
For instance s1 Rs2 could mean ‘s1 is a friend of s2 ’ or ‘s1 is less than s2 ’.
Let R be a binary relation on a set S.
1
2
R is reflexive if for any s ∈ S, sRs.
R is symmetric if for any s1 , s2 ∈ S, s1 Rs2
asymmetric.
Dr. Gabby (KNUST-Maths)
Real Numbers
=⇒
s2 Rs1 . Otherwise, R is
17 / 26
Relations on R
Relations on R
1
A binary relation on a set S describes how two elements s1 and s2 are related.
2
We use s1 Rs2 to mean ‘s1 is related to s2 ’.
3
For instance s1 Rs2 could mean ‘s1 is a friend of s2 ’ or ‘s1 is less than s2 ’.
Let R be a binary relation on a set S.
1
2
3
R is reflexive if for any s ∈ S, sRs.
R is symmetric if for any s1 , s2 ∈ S, s1 Rs2 =⇒ s2 Rs1 . Otherwise, R is
asymmetric.
R is anti-symmetric if for any s1 , s2 ∈ S, s1 Rs2 and s2 Rs1 =⇒ s2 = s1 .
Dr. Gabby (KNUST-Maths)
Real Numbers
17 / 26
Relations on R
Relations on R
1
A binary relation on a set S describes how two elements s1 and s2 are related.
2
We use s1 Rs2 to mean ‘s1 is related to s2 ’.
3
For instance s1 Rs2 could mean ‘s1 is a friend of s2 ’ or ‘s1 is less than s2 ’.
Let R be a binary relation on a set S.
1
2
3
4
R is reflexive if for any s ∈ S, sRs.
R is symmetric if for any s1 , s2 ∈ S, s1 Rs2 =⇒ s2 Rs1 . Otherwise, R is
asymmetric.
R is anti-symmetric if for any s1 , s2 ∈ S, s1 Rs2 and s2 Rs1 =⇒ s2 = s1 .
R is transitive if for any s1 , s2 , s3 ∈ S, s1 Rs2 and s2 Rs3 =⇒ s1 Rs3 .
Dr. Gabby (KNUST-Maths)
Real Numbers
17 / 26
Relations on R
Example
Find the properties of the relations ‘is a f riend of ’, ‘≤’ and ‘=’.
Dr. Gabby (KNUST-Maths)
Real Numbers
18 / 26
Relations on R
Example
Find the properties of the relations ‘is a f riend of ’, ‘≤’ and ‘=’.
The relation ‘is a f riend of ’ is: reflexive, symmetric, transitive.
Dr. Gabby (KNUST-Maths)
Real Numbers
18 / 26
Relations on R
Example
Find the properties of the relations ‘is a f riend of ’, ‘≤’ and ‘=’.
The relation ‘is a f riend of ’ is: reflexive, symmetric, transitive.
The relation ‘≤’ is reflexive, anti-symmetric and transitive.
Dr. Gabby (KNUST-Maths)
Real Numbers
18 / 26
Relations on R
Example
Find the properties of the relations ‘is a f riend of ’, ‘≤’ and ‘=’.
The relation ‘is a f riend of ’ is: reflexive, symmetric, transitive.
The relation ‘≤’ is reflexive, anti-symmetric and transitive.
The relation ‘=’ is reflexive, anti-symmetric, symmetric and transitive.
Dr. Gabby (KNUST-Maths)
Real Numbers
18 / 26
Relations on R
Example
Find the properties of the relations ‘is a f riend of ’, ‘≤’ and ‘=’.
The relation ‘is a f riend of ’ is: reflexive, symmetric, transitive.
The relation ‘≤’ is reflexive, anti-symmetric and transitive.
The relation ‘=’ is reflexive, anti-symmetric, symmetric and transitive.
Definition (Order Relation)
A relation which is reflexive, anti-symmetric and transitive is an order relation.
For example (R, ≤) is ordered.
Dr. Gabby (KNUST-Maths)
Real Numbers
18 / 26
Relations on R
Example
Find the properties of the relations ‘is a f riend of ’, ‘≤’ and ‘=’.
The relation ‘is a f riend of ’ is: reflexive, symmetric, transitive.
The relation ‘≤’ is reflexive, anti-symmetric and transitive.
The relation ‘=’ is reflexive, anti-symmetric, symmetric and transitive.
Definition (Order Relation)
A relation which is reflexive, anti-symmetric and transitive is an order relation.
For example (R, ≤) is ordered.
Definition (Equivalence relation)
A relation which is reflexive, symmetric and transitive is an equivalence relation.
Dr. Gabby (KNUST-Maths)
Real Numbers
18 / 26
Relations on R
Order relation on N
We define the order relation < on N by
m < n, iff there is some k ∈ N such that m + k = n .
Dr. Gabby (KNUST-Maths)
Real Numbers
19 / 26
Relations on R
Order relation on N
We define the order relation < on N by
m < n, iff there is some k ∈ N such that m + k = n .
Order relation on Z
1
2
3
n + a < m + a iff n < m.
If a > 0, then na < ma iff n < m.
If a < 0, then na < ma iff n > m.
Dr. Gabby (KNUST-Maths)
Real Numbers
19 / 26
Relations on R
Order relation on N
We define the order relation < on N by
m < n, iff there is some k ∈ N such that m + k = n .
Order relation on Z
1
2
3
n + a < m + a iff n < m.
If a > 0, then na < ma iff n < m.
If a < 0, then na < ma iff n > m.
Order relation on Q
The order relation on Q is defined by a < b iff a − b < 0.
Dr. Gabby (KNUST-Maths)
Real Numbers
19 / 26
Relations on R
Inequalities
Inequalities
1
Let a, b ∈ R. If a − b is a positive number, we say that a is greater than b or b is
less than a. We write a > b or b < a.
Dr. Gabby (KNUST-Maths)
Real Numbers
20 / 26
Relations on R
Inequalities
Inequalities
1
2
Let a, b ∈ R. If a − b is a positive number, we say that a is greater than b or b is
less than a. We write a > b or b < a.
If the possibility exists that a equals b, then we write a ≥ b or b ≤ a.
Dr. Gabby (KNUST-Maths)
Real Numbers
20 / 26
Relations on R
Inequalities
Inequalities
1
2
3
Let a, b ∈ R. If a − b is a positive number, we say that a is greater than b or b is
less than a. We write a > b or b < a.
If the possibility exists that a equals b, then we write a ≥ b or b ≤ a.
If a, b ∈ R such that ab > 0, then either a > 0 and b > 0 (that is both are positive)
or a < 0 and b < 0 (that is both are negative).
Dr. Gabby (KNUST-Maths)
Real Numbers
20 / 26
Relations on R
Inequalities
Inequalities
1
2
3
4
Let a, b ∈ R. If a − b is a positive number, we say that a is greater than b or b is
less than a. We write a > b or b < a.
If the possibility exists that a equals b, then we write a ≥ b or b ≤ a.
If a, b ∈ R such that ab > 0, then either a > 0 and b > 0 (that is both are positive)
or a < 0 and b < 0 (that is both are negative).
If a < b then a + c < b + c
Dr. Gabby (KNUST-Maths)
Real Numbers
20 / 26
Relations on R
Inequalities
Inequalities
1
2
3
4
5
Let a, b ∈ R. If a − b is a positive number, we say that a is greater than b or b is
less than a. We write a > b or b < a.
If the possibility exists that a equals b, then we write a ≥ b or b ≤ a.
If a, b ∈ R such that ab > 0, then either a > 0 and b > 0 (that is both are positive)
or a < 0 and b < 0 (that is both are negative).
If a < b then a + c < b + c
If a < b and c < d then a + c < b + d.
Dr. Gabby (KNUST-Maths)
Real Numbers
20 / 26
Relations on R
Inequalities
Inequalities
1
2
3
4
5
6
Let a, b ∈ R. If a − b is a positive number, we say that a is greater than b or b is
less than a. We write a > b or b < a.
If the possibility exists that a equals b, then we write a ≥ b or b ≤ a.
If a, b ∈ R such that ab > 0, then either a > 0 and b > 0 (that is both are positive)
or a < 0 and b < 0 (that is both are negative).
If a < b then a + c < b + c
If a < b and c < d then a + c < b + d.
If a < b and c > 0 then ac < bc
Dr. Gabby (KNUST-Maths)
Real Numbers
20 / 26
Relations on R
Inequalities
Inequalities
1
2
3
4
5
6
7
Let a, b ∈ R. If a − b is a positive number, we say that a is greater than b or b is
less than a. We write a > b or b < a.
If the possibility exists that a equals b, then we write a ≥ b or b ≤ a.
If a, b ∈ R such that ab > 0, then either a > 0 and b > 0 (that is both are positive)
or a < 0 and b < 0 (that is both are negative).
If a < b then a + c < b + c
If a < b and c < d then a + c < b + d.
If a < b and c > 0 then ac < bc
If a < b and c < 0 then ac > bc
Dr. Gabby (KNUST-Maths)
Real Numbers
20 / 26
Relations on R
Inequalities
Inequalities
1
2
3
4
5
6
7
8
Let a, b ∈ R. If a − b is a positive number, we say that a is greater than b or b is
less than a. We write a > b or b < a.
If the possibility exists that a equals b, then we write a ≥ b or b ≤ a.
If a, b ∈ R such that ab > 0, then either a > 0 and b > 0 (that is both are positive)
or a < 0 and b < 0 (that is both are negative).
If a < b then a + c < b + c
If a < b and c < d then a + c < b + d.
If a < b and c > 0 then ac < bc
If a < b and c < 0 then ac > bc
1
1
If 0 < a < b then > .
a
b
Dr. Gabby (KNUST-Maths)
Real Numbers
20 / 26
Absolute value
Outline of Presentation
1
Numbers
2
Standard subsets
Natural Numbers
Integers
Rational Numbers
Irrational and Real Numbers
3
Some Properties of Numbers
4
Relations on R
Inequalities
5
Absolute value
Dr. Gabby (KNUST-Maths)
Real Numbers
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Absolute value
Absolute value
Definition
The absolute value of a real number x is the non-negative real number denoted by |x|
such that
x
if x ≥ 0
|x| =
(2)
−x if x < 0.
Dr. Gabby (KNUST-Maths)
Real Numbers
22 / 26
Absolute value
Absolute value
Definition
The absolute value of a real number x is the non-negative real number denoted by |x|
such that
x
if x ≥ 0
|x| =
(2)
−x if x < 0.
The following theorems hold for inequalities
For x, y ∈ R
1
|xy| = |x| |y|
Dr. Gabby (KNUST-Maths)
Real Numbers
22 / 26
Absolute value
Absolute value
Definition
The absolute value of a real number x is the non-negative real number denoted by |x|
such that
x
if x ≥ 0
|x| =
(2)
−x if x < 0.
The following theorems hold for inequalities
For x, y ∈ R
1
|xy| = |x| |y|
2
|x + y| ≤ |x| + |y| called the triangle inequality
Dr. Gabby (KNUST-Maths)
Real Numbers
22 / 26
Absolute value
Absolute value
Definition
The absolute value of a real number x is the non-negative real number denoted by |x|
such that
x
if x ≥ 0
|x| =
(2)
−x if x < 0.
The following theorems hold for inequalities
For x, y ∈ R
1
|xy| = |x| |y|
2
|x + y| ≤ |x| + |y| called the triangle inequality
3
|x − y| ≥ |x| − |y|
Dr. Gabby (KNUST-Maths)
Real Numbers
22 / 26
Absolute value
Absolute value
Definition
The absolute value of a real number x is the non-negative real number denoted by |x|
such that
x
if x ≥ 0
|x| =
(2)
−x if x < 0.
The following theorems hold for inequalities
For x, y ∈ R
1
|xy| = |x| |y|
2
|x + y| ≤ |x| + |y| called the triangle inequality
3
|x − y| ≥ |x| − |y|
4
|x| = 0 ⇔ x = 0
Dr. Gabby (KNUST-Maths)
Real Numbers
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Absolute value
Example
Let us rewrite the following without the symbol of absolute value.
1
|2| =
Dr. Gabby (KNUST-Maths)
Real Numbers
23 / 26
Absolute value
Example
Let us rewrite the following without the symbol of absolute value.
1
|2| =
Dr. Gabby (KNUST-Maths)
Real Numbers
23 / 26
Absolute value
Example
Let us rewrite the following without the symbol of absolute value.
1
|2| = 2 since 2 is positive.
2
|−4| =
Dr. Gabby (KNUST-Maths)
Real Numbers
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Absolute value
Example
Let us rewrite the following without the symbol of absolute value.
1
|2| = 2 since 2 is positive.
2
|−4| =
Dr. Gabby (KNUST-Maths)
Real Numbers
23 / 26
Absolute value
Example
Let us rewrite the following without the symbol of absolute value.
1
|2| = 2 since 2 is positive.
2
|−4| = −(−4) = 4 since −4 is negative.
3
|0| =
Dr. Gabby (KNUST-Maths)
Real Numbers
23 / 26
Absolute value
Example
Let us rewrite the following without the symbol of absolute value.
1
|2| = 2 since 2 is positive.
2
|−4| = −(−4) = 4 since −4 is negative.
3
|0| =
Dr. Gabby (KNUST-Maths)
Real Numbers
23 / 26
Absolute value
Example
Let us rewrite the following without the symbol of absolute value.
1
|2| = 2 since 2 is positive.
2
|−4| = −(−4) = 4 since −4 is negative.
3
|0| = 0
4
|x − 2| equals x − 2 if x − 2 ≥ 0. That is |x − 2| = x − 2 if x ≥ 2.
Dr. Gabby (KNUST-Maths)
Real Numbers
23 / 26
Absolute value
Example
Let us rewrite the following without the symbol of absolute value.
1
|2| = 2 since 2 is positive.
2
|−4| = −(−4) = 4 since −4 is negative.
3
|0| = 0
4
|x − 2| equals x − 2 if x − 2 ≥ 0. That is |x − 2| = x − 2 if x ≥ 2.
Dr. Gabby (KNUST-Maths)
Real Numbers
23 / 26
Absolute value
Example
Let us rewrite the following without the symbol of absolute value.
1
|2| = 2 since 2 is positive.
2
|−4| = −(−4) = 4 since −4 is negative.
3
|0| = 0
4
|x − 2| equals x − 2 if x − 2 ≥ 0. That is |x − 2| = x − 2 if x ≥ 2.
Otherwise, that is for x < 2, |x − 2| = −(x − 2) = −x + 2.
Dr. Gabby (KNUST-Maths)
Real Numbers
23 / 26
Absolute value
Example
Let us rewrite the following without the symbol of absolute value.
1
|2| = 2 since 2 is positive.
2
|−4| = −(−4) = 4 since −4 is negative.
3
|0| = 0
4
|x − 2| equals x − 2 if x − 2 ≥ 0. That is |x − 2| = x − 2 if x ≥ 2.
Otherwise, that is for x < 2, |x − 2| = −(x − 2) = −x + 2.
x−2
if x ≥ 2
Therefore |x − 2| =
−x + 2 if x < 2.
Dr. Gabby (KNUST-Maths)
Real Numbers
23 / 26
Absolute value
Example
Solve |−2x + 2| = 4.
1
Rewrite |−2x + 2| without the absolute value.
Dr. Gabby (KNUST-Maths)
Real Numbers
24 / 26
Absolute value
Example
Solve |−2x + 2| = 4.
1
2
Rewrite |−2x + 2| without the absolute value.
−2x + 2 ≥ 0 implies x ≤ 1 That is |−2x + 2| = −2x + 2 if x ≤ 1;
Dr. Gabby (KNUST-Maths)
Real Numbers
24 / 26
Absolute value
Example
Solve |−2x + 2| = 4.
1
2
3
Rewrite |−2x + 2| without the absolute value.
−2x + 2 ≥ 0 implies x ≤ 1 That is |−2x + 2| = −2x + 2 if x ≤ 1;
Otherwise |−2x + 2| = −(−2x + 2) = 2x − 2 for −2x + 2 < 0 =⇒ x > 0
Dr. Gabby (KNUST-Maths)
Real Numbers
24 / 26
Absolute value
Example
Solve |−2x + 2| = 4.
1
2
3
4
Rewrite |−2x + 2| without the absolute value.
−2x + 2 ≥ 0 implies x ≤ 1 That is |−2x + 2| = −2x + 2 if x ≤ 1;
Otherwise |−2x + 2| = −(−2x + 2) = 2x − 2 for −2x + 2 < 0 =⇒ x > 0
(
−2x + 2 = 4 if x ≤ 1
So we have
2x − 2 = 4
if x > 1
Dr. Gabby (KNUST-Maths)
Real Numbers
24 / 26
Absolute value
Example
Solve |−2x + 2| = 4.
1
2
3
4
5
Rewrite |−2x + 2| without the absolute value.
−2x + 2 ≥ 0 implies x ≤ 1 That is |−2x + 2| = −2x + 2 if x ≤ 1;
Otherwise |−2x + 2| = −(−2x + 2) = 2x − 2 for −2x + 2 < 0 =⇒ x > 0
(
−2x + 2 = 4 if x ≤ 1
So we have
2x − 2 = 4
if x > 1
For x ≤ 1, that is −2x + 2 = 4 =⇒ x = −1.
Dr. Gabby (KNUST-Maths)
Real Numbers
24 / 26
Absolute value
Example
Solve |−2x + 2| = 4.
1
2
3
4
5
Rewrite |−2x + 2| without the absolute value.
−2x + 2 ≥ 0 implies x ≤ 1 That is |−2x + 2| = −2x + 2 if x ≤ 1;
Otherwise |−2x + 2| = −(−2x + 2) = 2x − 2 for −2x + 2 < 0 =⇒ x > 0
(
−2x + 2 = 4 if x ≤ 1
So we have
2x − 2 = 4
if x > 1
For x ≤ 1, that is −2x + 2 = 4 =⇒ x = −1.
Dr. Gabby (KNUST-Maths)
Real Numbers
24 / 26
Absolute value
Example
Solve |−2x + 2| = 4.
1
2
3
4
5
6
Rewrite |−2x + 2| without the absolute value.
−2x + 2 ≥ 0 implies x ≤ 1 That is |−2x + 2| = −2x + 2 if x ≤ 1;
Otherwise |−2x + 2| = −(−2x + 2) = 2x − 2 for −2x + 2 < 0 =⇒ x > 0
(
−2x + 2 = 4 if x ≤ 1
So we have
2x − 2 = 4
if x > 1
For x ≤ 1, that is −2x + 2 = 4 =⇒ x = −1.
Since −1 ≤ 1 then it is a valid solution.
For x > 1, that is 2x − 2 = 4 =⇒ x = 3.
Dr. Gabby (KNUST-Maths)
Real Numbers
24 / 26
Absolute value
Example
Solve |−2x + 2| = 4.
1
2
3
4
5
6
Rewrite |−2x + 2| without the absolute value.
−2x + 2 ≥ 0 implies x ≤ 1 That is |−2x + 2| = −2x + 2 if x ≤ 1;
Otherwise |−2x + 2| = −(−2x + 2) = 2x − 2 for −2x + 2 < 0 =⇒ x > 0
(
−2x + 2 = 4 if x ≤ 1
So we have
2x − 2 = 4
if x > 1
For x ≤ 1, that is −2x + 2 = 4 =⇒ x = −1.
Since −1 ≤ 1 then it is a valid solution.
For x > 1, that is 2x − 2 = 4 =⇒ x = 3.
Dr. Gabby (KNUST-Maths)
Real Numbers
24 / 26
Absolute value
Example
Solve |−2x + 2| = 4.
1
2
3
4
5
6
7
Rewrite |−2x + 2| without the absolute value.
−2x + 2 ≥ 0 implies x ≤ 1 That is |−2x + 2| = −2x + 2 if x ≤ 1;
Otherwise |−2x + 2| = −(−2x + 2) = 2x − 2 for −2x + 2 < 0 =⇒ x > 0
(
−2x + 2 = 4 if x ≤ 1
So we have
2x − 2 = 4
if x > 1
For x ≤ 1, that is −2x + 2 = 4 =⇒ x = −1.
Since −1 ≤ 1 then it is a valid solution.
For x > 1, that is 2x − 2 = 4 =⇒ x = 3.
Since 3 > 1 then it is a valid solution as well.
In summary, the solutions are {−1, 3}.
Dr. Gabby (KNUST-Maths)
Real Numbers
24 / 26
Absolute value
Exercise
1
We denote by Z− the set of non-positive integers. Fill the table below with True or False
(N, +)
(Z− , −)
(Z, ×)
(Q \ {0}, ×)
Closed
Commutativity
Associativity
Identity exists
Inverse exists
2
Let ∝ be a binary relation on R defined by x ∝ y if and only if there exists q ∈ Q \ {0}
such that x = qy. Find the properties of ∝ . Is this an equivalence or an order relation?
3
Rewrite the following without the absolute value:
1
2
|x + 5| = 10
|−x + a| = 3a − 2,
−3 |1 − 2x| = −9
3
Dr. Gabby (KNUST-Maths)
Real Numbers
25 / 26
END OF LECTURE
THANK YOU
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Real Numbers
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