1.2 the real number system
DESCRIPTION
MATH 17 - COLLEGE ALGEBRA AND TRIGONOMETRYTRANSCRIPT
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Chapter 1
Section 1.2
The Real Number System
as a Number Field
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Uses of Number
1. Naming 2. Ordering 3. Counting 4. Measuring Note: The counting numbers are not sufficient for measuring.
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Definition
Numbers that will be used for counting and measuring will be collectively called real numbers
The Set of Real Numbers
And the set of real numbers will be denoted by ℝ.
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Equality Axiom
Reflexive Property of Equality
For any 𝑎 ∈ ℝ, 𝑎 = 𝑎
Symmetric Property of Equality
For any 𝑎, 𝑏 ∈ ℝ, if 𝑎 = 𝑏 then 𝑏 = 𝑎
Transitive Property of Equality
For any 𝑎, 𝑏, 𝑐 ∈ ℝ, if 𝑎 = 𝑏 and 𝑏 = 𝑐 then 𝑎 = 𝑐
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Basic Operations
Addition +
Multiplication ∙
In elementary mathematics, particularly in arithmetic, we learned two basic operations on real numbers, namely
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Closure Axiom
Closure Property for Addition
For any 𝑎, 𝑏 ∈ ℝ, 𝑎 + 𝑏 ∈ ℝ
Closure Property for Multiplication
For any 𝑎, 𝑏 ∈ ℝ, 𝑎 ∙ 𝑏 ∈ ℝ
ℝ is closed under addition and multiplication.
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Example
Determine whether or not each of the following sets is closed under addition and multiplication.
1. ℕ 2. 𝐸 3. 𝑂 4. 𝑃 5. 𝐶
addition multiplication
YES YES
YES YES
NO YES
NO NO
NO YES
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Associativity Axioms
Associative Property for Addition
For any 𝑎, 𝑏, 𝑐 ∈ ℝ, 𝑎 + 𝑏 + 𝑐 = 𝑎 + 𝑏 + 𝑐
Associative Property for Multiplication
For any 𝑎, 𝑏, 𝑐 ∈ ℝ, 𝑎 ∙ 𝑏 ∙ 𝑐 = 𝑎 ∙ 𝑏 ∙ 𝑐
Addition and Multiplication are associative on any subset of ℝ.
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Commutativity Axioms
Commutative Property for Addition
For any 𝑎, 𝑏 ∈ ℝ, 𝑎 + 𝑏 = 𝑏 + 𝑎
Commutative Property for Multiplication
For any 𝑎, 𝑏 ∈ ℝ, 𝑎 ∙ 𝑏 = 𝑏 ∙ 𝑎
Addition and multiplication are commutative on any subset of ℝ.
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Distributivity Axiom
Distributive Property for Multiplication over Addition
For any 𝑎, 𝑏, 𝑐 ∈ ℝ,
𝑎 ∙ 𝑏 + 𝑐 = 𝑎 ∙ 𝑏 + 𝑎 ∙ 𝑐
These are satisfied by addition and multiplication on any subset of ℝ.
𝑎 + 𝑏 ∙ 𝑐 = 𝑎 ∙ 𝑐 + 𝑏 ∙ 𝑐
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Multiplicative Identity
Existence of Multiplicative Identity
There is a unique number 1 such that,
1 ∙ 𝑎 = 𝑎
𝑎 ∙ 1 = 𝑎
for any real number 𝑎
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Remark
Note that the set of counting (natural) numbers ℕ together with addition and multiplication satisfies all the axioms we have discussed so far.
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Equality Axiom
Addition Property of Equality (APE)
For any 𝑎, 𝑏, 𝑐 ∈ ℝ, if 𝑎 = 𝑏 then 𝑎 + 𝑐 = 𝑏 + 𝑐
Multiplication Property of Equality (MPE)
For any 𝑎, 𝑏, 𝑐 ∈ ℝ, if 𝑎 = 𝑏 then 𝑎 ∙ 𝑐 = 𝑏 ∙ 𝑐
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Substitution
If two real numbers are equal, then one may be substituted for the other in any algebraic expression
If 𝑥 = 5𝑦
then 2𝑥 + 3𝑦 = 2 5𝑦 + 3𝑦
Also, 𝑥3 = 5𝑦 3 = 125𝑦3
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Example
Tell which of the axioms of the real numbers justifies each of the following
statement.
1. 2 ∙ 3 + 2 ∙ 5 = 2 ∙ 3 + 5
2. 4 ∙ 10 ∙ 10 = 4 ∙ 10 ∙ 10
3. 10 ∙ 4 ∙ 10 = 4 ∙ 10 ∙ 10
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Solvable Equations
Consider 3 + 𝑥 = 5
𝑥 = 2 is a solution and 2 ∈ ℕ.
Thus, we say that the equation is solvable in ℕ
Now, consider 3 + 𝑥 = 0
This equation is not solvable in ℕ.
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Remark
ℕ is not large enough to contain solutions even for such simple linear equation that we have seen.
Something must be done!
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Additive Identity
Existence of Additive Identity
There is a unique number 0 such that,
0 + 𝑎 = 𝑎
𝑎 + 0 = 𝑎
for any real number 𝑎
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Identity Elements
Additive Identity 0
Multiplicative Identity 1
Remark: 0 ≠ 1 and 0 ∉ ℕ
Thus we form another set of numbers which contains the natural numbers and the additive identity. This set is called the set of whole numbers denoted by 𝕎
𝕎 = ℕ ∪ 0
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Solvable Equations
Is 3 + 𝑥 = 3 solvable in 𝕎?
Is 3 + 𝑥 = 2 solvable in ℕ? in 𝕎?
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Additive Inverses
Existence of Additive Inverse
For every real number 𝑎, there is a number − 𝑎 such that
𝑎 + −𝑎 = 0
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Example
What is the additive inverse of 0? By the existence of additive identity,
0 + 0 = 0 Therefore, −0 = 0.
What is the additive inverse of −𝑎? By the existence of additive identity,
𝑎 + −𝑎 = −𝑎 + 𝑎 = 0 Therefore, − −𝑎 = 𝑎.
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Theorem
Cancellation Law for Addition
For 𝑎, 𝑏, 𝑐 ∈ ℝ, if 𝑎 + 𝑐 = 𝑏 + 𝑐, then 𝑎 = 𝑏
Zero Property
For 𝑎 ∈ ℝ, 𝑎 ∙ 0 = 0
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Theorem
Theorem.
For 𝑎, 𝑏 ∈ ℝ, −𝑎 ∙ 𝑏 = − 𝑎 ∙ 𝑏
Corollary.
For 𝑎 ∈ ℝ, −1 ∙ 𝑎 = −𝑎
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Theorem
Corollary.
−1 ∙ −1 = 1
Theorem.
For 𝑎, 𝑏 ∈ ℝ, −𝑎 ∙ −𝑏 = 𝑎 ∙ 𝑏
Theorem.
For 𝑎, 𝑏 ∈ ℝ, − 𝑎 + 𝑏 = −𝑎 + −𝑏
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Definition
For any 𝑎, 𝑏 ∈ ℝ 𝑎 − 𝑏 = 𝑎 + −𝑏
Subtraction
For example: 9 − 7 = 9 + −7 = 2 2 − 2 = 2 + −2 = 0
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Set of Integers
ℤ = 𝕎 ∪ … ,−3,−2,−1 = … ,−3,−2,−1,0,1,2,3, …
Remark: ℤ is the smallest subset of ℝ that satisfies all the previous axioms.
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Subsets of ℤ
ℤ:: ℕ ℤ;: … ,−5,−4,−3,−2, −1
ℤ⊕: 𝕎
ℤ⊝: … ,−5,−4,−3,−2, −1,0
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Solvable Equations
Is 2 ∙ 𝑥 = −6 solvable in ℤ?
Is 2 ∙ 𝑥 = 1 solvable in ℕ? in 𝕎? In ℤ?
Is 2 ∙ 𝑥 = 3 solvable in ℕ? in 𝕎? In ℤ?
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Multiplicative Inverses
Existence of Multiplicative Inverses
For every real number 𝑎 ≠ 0, there is a
unique number 1
𝑎 such that
1
𝑎∙ 𝑎 = 1
The multiplicative inverse of 𝑎 is 1
𝑎.
Because of the commutative property for multiplication, the multiplicative inverse of 1
𝑎 is 𝑎.
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Theorem
Cancellation Law for Multiplication
For 𝑎, 𝑏 ∈ ℝ, and any nonzero number 𝑐 if 𝑎 ∙ 𝑐 = 𝑏 ∙ 𝑐, then 𝑎 = 𝑏
Theorem.
For 𝑎, 𝑏 ∈ ℝ, such that 𝑎 ∙ 𝑏 = 0, then either 𝑎 = 0 or 𝑏 = 0
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Theorem
Theorem
If 𝑎, 𝑏, are nonzero real numbers, 1
𝑎 ∙ 𝑏=
1
𝑎∙
1
𝑏
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Definition
For any 𝑎, 𝑏 ∈ ℝ and 𝑎 ≠ 0 𝑏
𝑎= 𝑏 ∙
1
𝑎
Division
Remark: Division by 0 is undefined.
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Definition
A real number 𝑥 which can be written as a quotient of an integer 𝑝 by a nonzero 𝑞 is called a rational number.
Rational Numbers
ℚ = 𝑥 𝑥 =𝑝
𝑞, 𝑝 ∈ ℤ, 𝑞 is a nonzero integer
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Theorem
Theorem.
For any number 𝑏, 𝑏
1= 𝑏.
Theorem. (Prove)
If 𝑐 is any nonzero real number then 𝑐
𝑐= 1
Theorem. (Prove)
𝑏
𝑎∙
𝑑
𝑐=
𝑏𝑑
𝑎𝑐, a ≠ 0, 𝑐 ≠ 0
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Theorem
Theorem.
If 𝑐 is a nonzero real number, 𝑎 ∙ 𝑐
𝑏 ∙ 𝑐=
𝑎
𝑏
Theorem. (Prove)
If 𝑏
𝑎=
𝑑
𝑎 then 𝑏 = 𝑑 and
If 𝑏
𝑎=
𝑑
𝑐 then 𝑏 ∙ 𝑐 = 𝑎 ∙ 𝑑, a ≠ 0, 𝑐 ≠ 0
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Theorem
Theorem.
If 𝑏
𝑎≠ 0, then
1𝑏
𝑎 =
𝑎
𝑏
Theorem.
If 𝑏
𝑎≠ 0, then
𝑑𝑐
𝑏𝑎 =
𝑑∙𝑎
𝑐∙𝑏
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Theorem
Theorem. ;𝑏
𝑎= −
𝑏
𝑎=
𝑏
;𝑎, and
;𝑏
;𝑎=
𝑏
𝑎, 𝑎 ≠ 0.
Theorem. 𝑏
𝑎+
𝑑
𝑐=
𝑏 ∙ 𝑐 + 𝑎 ∙ 𝑑
𝑎 ∙ 𝑐, a ≠ 0, 𝑐 ≠ 0
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Example
Perform the indicated operations
1. 2
3∙
5
12
2.4
5 20
25
3.2
5+
3
5
4.2
3+
4
5
.=2∙5
3∙2 .=
10
36 .=
5∙2
18∙2 .=
5
18
=4 ∙ 25
20 ∙ 5
=2 + 3
5
=2 ∙ 5 + 3 ∙ 4
3 ∙ 5
=100
100= 1
=5
5= 1
=22
15
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Theorem
Theorem.
For any number 𝑏, 𝑐 ∈ ℝ and any nonzero number 𝑎, there exists a unique solution to
the equation 𝑎 ∙ 𝑥 + 𝑏 = 𝑐
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Solvable Equations
Is 𝑥2 = 9 solvable in ℚ?
Is 𝑥2 = 2 solvable in ℚ?
Why is there no rational number whose square is 2?
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Definition
A real number which is not rational is called an irrational number.
Irrational Numbers
ℚ′ ≔ set of irrational numbers ℝ = ℚ ∪ ℚ′
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Decimals
A real number can have decimal representation that is:
terminating 2.36 or non-terminating 2.363636…
A non-terminating decimal representation can be
repeating 2.3636… or non-repeating π = 3.1415…
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Decimals
• terminating or • nonterminating but repeating.
Rational Numbers
• nonterminating and non-repeating.
Irrational Numbers
Examples: 54 0. 33 = 1
3 −12
Examples: 𝑒 𝜋
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Irrational Numbers
Some irrational numbers are of the form 𝑝𝑛 where 𝑝 is a real number
𝑝𝑛 : 𝑛𝑡ℎ root of 𝑝 (sometimes called radicals) 𝑝 is called the radicand 𝑛 is called the index
𝑝: square root of 𝑝
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nth Root
𝑝𝑛 is a solution to 𝑥𝑛 = 𝑝
−83
is a solution to 𝑥3 = −8. Since −2 is the only real number whose cube is − 8,
−83
= −2
4 is a solution to 𝑥2 = 4. Since −2 and 2 are real numbers whose
square is 4, we need to decide on which 4 is. Now we define the principal root.
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Definition
If 𝑛 is even and 𝑝 is a non-negative, we define 𝑝𝑛 as the positive 𝑛th root of 𝑝. If 𝑝 is negative, 𝑝𝑛 is undefined.
Principal Root
If 𝑛 is odd and 𝑝 is positive, −𝑝𝑛 = − 𝑝𝑛
Hence by definition, 4 = 2.
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Example
1. 16
2. 83
3. −273
4. 6254
5. −4
Determine the value of the following radicals
.= 4
.= 2
.= −3
.= 5
.undefined
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Theorem
Theorem.
For a positive number 𝑝 and an even number 𝑛
𝑝𝑛𝑛 = 𝑝
Example.
164
= 244= 2
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Order Axioms
Our objective now is to identify how the real numbers can be arranged on a line.
In order to do this, we refer to the order axioms that the set of real numbers satisfy.
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Less Than
First, we assume the existence of the relation “less than” between two numbers, denoted by the symbol <
𝑎 < 𝑏 is read as 𝑎 is less than 𝑏.
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Trichotomy Axiom
For any 𝑎, 𝑏 ∈ ℝ, one and only one of the following holds:
𝑎 < 𝑏 𝑎 = 𝑏 𝑏 < 𝑎
We will use this axiom to formally define positive and negative real numbers.
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Definition
A real number 𝑎 is said to be positive if 0 < 𝑎 and negative if 𝑎 < 0. The number 0 is neither positive nor negative.
The trichotomy axiom guarantees that every real number is either positive, negative or zero
Positive and Negative Numbers
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Transitivity Axiom
Transitivity Axiom of Order
For any 𝑎, 𝑏, 𝑐 ∈ ℝ, and if 𝑎 < 𝑏 and 𝑏 <c, then 𝑎 < 𝑐.
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Addition Axiom
Addition Axiom of Order
If 𝑎, 𝑏, 𝑐 ∈ ℝ, and 𝑎 < 𝑏 then 𝑎 + 𝑐 < 𝑏 + 𝑐
Theorem
For a positive real number 𝑎, c < 𝑎 + 𝑐.
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Multiplication Axiom
Multiplication Axiom of Order
If 𝑎, 𝑏, 𝑐 ∈ ℝ, and if 𝑎 < 𝑏 and 0 < 𝑐 then 𝑎 ∙ 𝑐 < 𝑏 ∙ 𝑐
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Definition
If 𝑎, 𝑏 ∈ ℝ, we say 𝑎 > 𝑏, read as “𝑎 is greater than b”, if and only if 𝑏 < 𝑎.
Greater Than
Thus, 𝑎 is positive if 𝑎 > 0 and negative if 0 > 𝑎.
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Theorem
Theorem.
The set of positive real numbers is closed under addition and multiplication
Theorem.
If 𝑎 > 0, then−𝑎 < 0 and if 𝑎 < 0, then −𝑎 > 0
Theorem.
If 𝑎 < 𝑏, then −𝑏 < −𝑎
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Theorem
Theorem.
If 𝑎 ∈ ℝ, either 𝑎2 = 0 or 𝑎2 > 0
Corollary.
1 > 0
Definition.
𝑎 < 𝑏 if and only if there exists a positive real number c such that a + c =b
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Theorem
Theorem.
If 𝑎, 𝑏, 𝑐 ∈ ℝ, and 𝑎 > 𝑏 and 𝑐 < 0, then ac < bc
Theorem.
If 𝑎 > 0 then 1
𝑎> 0
Theorem.
If 𝑎 > 𝑏, then −𝑏 > −𝑎
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Definition
An inequality of the form 𝑎 < 𝑥 < 𝑏 is called a continued inequality and is to be understood to be 𝑎 < 𝑥 and 𝑥 < 𝑏.
Continued Inequality
We shall also define 𝑎 ≤ 𝑏 to mean 𝑎 < 𝑏 or 𝑎 = 𝑏.
Similarly 𝑎 ≥ 𝑏 means 𝑎 > 𝑏 or 𝑎 = 𝑏.
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Real Number Line
The order properties of numbers allow us to arrange the real numbers on a line, called the real number line. We choose an arbitrary point to correspond to the number 0. To the right of 0, we place the positive real numbers while the negative real numbers are placed to the left of 0.
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Real Number Line
Moreover, a number 𝑥 is greater than another number 𝑦 if 𝑥 is found to the right of 𝑦. The real number line is shown below.
−5 −1 0 1 5
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Real Number Line
We can now say that every real number corresponds to a point on the line and every point corresponds to a real number. In other words, there is a one-to-one correspondence between the set of real numbers and the set of points on the line.
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Real Number Line
The one-to-one correspondence allows us to set-up a coordinate system on the line. Thus, to a given point on the line, there corresponds a real number which we shall call the coordinate of the point.
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Example
Locate the following numbers on the real number line
1.1
2
2. −3
2
3. 6
4. 𝜋
0 11 22 3
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Definition
If 𝑥1 and 𝑥2 are points on the real number line, the distance 𝑑(𝑥1, 𝑥2) between them
is given by 𝑑 𝑥1, 𝑥2 = 𝑥2 − 𝑥12
Distance
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Example
If the points 𝐴 and 𝐵 have the given coordinates respectively, find the distance between them.
1. 3, 7
2. −2,−5
𝑑 3,7 = 7 − 3 2 = 4
d −2,−5 = −2 − (−5) 2 = 3
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Definition
The absolute value of a number 𝑥 is defined as
𝑥 = 𝑥2 = 𝑥 if 𝑥 > 0−𝑥 if 𝑥 < 00 if 𝑥 = 0
With this definition, it follows that that the
distance 𝑑(𝑥1, 𝑥2) = |𝑥2 – 𝑥1| and it is always non-negative as expected.
Absolute Value
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Example
If the points 𝐴 and 𝐵 have the given coordinates respectively, find the distance between them.
1. 3, 7
2. −2,−5
𝑑 3,7 = 3 − 7 = 4
d −2,−5 = −2 − (−5) = 3
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Theorem
Theorem.
For any 𝑎 ∈ ℝ, − 𝑎 ≤ 𝑎 ≤ 𝑎
Theorem.
For any 𝑎, 𝑏 ∈ ℝ, 𝑎 ∙ 𝑏 = 𝑎 𝑏 and
𝑎𝑏 = 𝑎
𝑏
Triangle Inequality.
For any 𝑎, 𝑏 ∈ ℝ, 𝑎 + 𝑏 ≤ 𝑎 + 𝑏
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Example
Give an interpretation of the following statements: 1. 𝑥 = 5 2. 𝑥 − 2 = 3 3. 𝑥 < 3 4. 𝑥 > 3 5. 𝑥 − 1 < 2
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Interval Notations
For real numbers 𝑎 and 𝑏 we define the following sets:
𝑎, 𝑏 = 𝑥 ∈ ℝ 𝑎 < 𝑥 < 𝑏 𝑎, 𝑏 is read as ”open interval from 𝑎 to 𝑏”
𝑎, 𝑏 = 𝑥 ∈ ℝ 𝑎 ≤ 𝑥 ≤ 𝑏
𝑎, 𝑏 is read as ”closed interval from 𝑎 to 𝑏”
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Definition
The number 𝑢 is called an upper bound of a set 𝑆 if 𝑥 ≤ 𝑢 for all 𝑥 ∈ 𝑆. Similarly, the number 𝑣 is called a lower bound of a set 𝑆 if 𝑥 ≥ 𝑣, for all 𝑥 ∈ 𝑆
Upper and Lower Bounds
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Example
1. 7,0,8 − 2,3 Upper bounds: Lower bounds:
2. 1,1
2,1
3, …
Upper bounds: Lower bounds:
Give upper and lower bounds for the following sets.
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Example
3. 𝑥1
2< 𝑥 <
5
2
Upper bounds: Lower bounds: 4. 𝑥 𝑥 < −1 or 𝑥 > 1 Upper bounds: Lower bounds:
5. 𝑥 ∈ ℕ1
2≤ 𝑥 ≤
9
2
Upper bounds: Lower bounds:
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Definition
An upper bound 𝑢 is called the least upper bound (lub) if no upper bound is less than 𝑢. A lower bound 𝑣 is called the greatest lower bound (glb) if no lower bound is greater than 𝑣.
Least Upper Bound and Greatest Lower Bound
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Example
1. 7,0,8, −2,3 lub: glb:
2. 1,1
2,1
3, …
lub: glb:
Identify the lub and glb of the following
8
2
10
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Example
3. 𝑥1
2< 𝑥 <
5
2
lub: glb: 4. 𝑥 𝑥 < −1 or 𝑥 > 1 lub: glb:
5. 𝑥 ∈ ℕ1
2≤ 𝑥 ≤
9
2
lub: glb:
52
12
none
none
1 4
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Completeness Axiom
Completeness Axiom
Every subset 𝑆 of ℝ, that has an upper bound has a least upper bound in ℝ.
Similarly, every subset 𝑆 of ℝ, that has a lower
bound has a greatest lower bound in ℝ.