Chapter 1

Section 1.2

The Real Number System

as a Number Field

Uses of Number

1. Naming 2. Ordering 3. Counting 4. Measuring Note: The counting numbers are not sufficient for measuring.

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.

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

Basic Operations

Addition +

Multiplication ∙

In elementary mathematics, particularly in arithmetic, we learned two basic operations on real numbers, namely

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.

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

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.

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.

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

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

Remark

Note that the set of counting (natural) numbers ℕ together with addition and multiplication satisfies all the axioms we have discussed so far.

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

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

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

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.

Remark

ℕ is not large enough to contain solutions even for such simple linear equation that we have seen.

Something must be done!

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

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

Solvable Equations

Is 3 + 𝑥 = 3 solvable in 𝕎?

Is 3 + 𝑥 = 2 solvable in ℕ? in 𝕎?

Additive Inverses

Existence of Additive Inverse

For every real number 𝑎, there is a number − 𝑎 such that

𝑎 + −𝑎 = 0

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.

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

Theorem

Theorem.

For 𝑎, 𝑏 ∈ ℝ, −𝑎 ∙ 𝑏 = − 𝑎 ∙ 𝑏

Corollary.

For 𝑎 ∈ ℝ, −1 ∙ 𝑎 = −𝑎

Theorem

Corollary.

−1 ∙ −1 = 1

Theorem.

For 𝑎, 𝑏 ∈ ℝ, −𝑎 ∙ −𝑏 = 𝑎 ∙ 𝑏

Theorem.

For 𝑎, 𝑏 ∈ ℝ, − 𝑎 + 𝑏 = −𝑎 + −𝑏

Definition

For any 𝑎, 𝑏 ∈ ℝ 𝑎 − 𝑏 = 𝑎 + −𝑏

Subtraction

For example: 9 − 7 = 9 + −7 = 2 2 − 2 = 2 + −2 = 0

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.

Subsets of ℤ

ℤ:: ℕ ℤ;: … ,−5,−4,−3,−2, −1

ℤ⊕: 𝕎

ℤ⊝: … ,−5,−4,−3,−2, −1,0

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?

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.

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

Theorem

Theorem

If 𝑎, 𝑏, are nonzero real numbers, 1

𝑎 ∙ 𝑏=

1

𝑎∙

1

𝑏

Definition

For any 𝑎, 𝑏 ∈ ℝ and 𝑎 ≠ 0 𝑏

𝑎= 𝑏 ∙

1

𝑎

Division

Remark: Division by 0 is undefined.

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

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

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

Theorem

Theorem.

If 𝑏

𝑎≠ 0, then

1𝑏

𝑎 =

𝑎

𝑏

Theorem.

If 𝑏

𝑎≠ 0, then

𝑑𝑐

𝑏𝑎 =

𝑑∙𝑎

𝑐∙𝑏

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

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

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

Solvable Equations

Is 𝑥2 = 9 solvable in ℚ?

Is 𝑥2 = 2 solvable in ℚ?

Why is there no rational number whose square is 2?

Definition

A real number which is not rational is called an irrational number.

Irrational Numbers

ℚ′ ≔ set of irrational numbers ℝ = ℚ ∪ ℚ′

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…

Decimals

• terminating or • nonterminating but repeating.

Rational Numbers

• nonterminating and non-repeating.

Irrational Numbers

Examples: 54 0. 33 = 1

3 −12

Examples: 𝑒 𝜋

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

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.

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.

Example

1. 16

2. 83

3. −273

4. 6254

5. −4

Determine the value of the following radicals

.= 4

.= 2

.= −3

.= 5

.undefined

Theorem

Theorem.

For a positive number 𝑝 and an even number 𝑛

𝑝𝑛𝑛 = 𝑝

Example.

164

= 244= 2

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.

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.

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.

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

Transitivity Axiom

Transitivity Axiom of Order

For any 𝑎, 𝑏, 𝑐 ∈ ℝ, and if 𝑎 < 𝑏 and 𝑏 <c, then 𝑎 < 𝑐.

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.

Multiplication Axiom

Multiplication Axiom of Order

If 𝑎, 𝑏, 𝑐 ∈ ℝ, and if 𝑎 < 𝑏 and 0 < 𝑐 then 𝑎 ∙ 𝑐 < 𝑏 ∙ 𝑐

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.

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

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

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

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.

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.

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

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.

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.

Example

Locate the following numbers on the real number line

1.1

2

2. −3

2

3. 6

4. 𝜋

0 11 22 3

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

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

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

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

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

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

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”

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

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.

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:

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

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

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

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.