a2 - real number system

The Real Number System and Integer Exponents

Mathematics 17

Institute of Mathematics, University of the Philippines-Diliman

Lecture 2

Math 17 (UP-IMath) R and Integer Exponents Lec 2 1 / 33

Outline

1 The Real Number SystemAxioms on ROrder in RThe Real Number LineInterval NotationThe Absolute Value

2 Integer ExponentsLaws of Exponents

The Real Number System

Recall: R, the set of real numbers

The real number system consists of R and two operations on its elements:

addition and multiplication

Operation Symbol ResultAddition + Sum a + b

Multiplication · Product a · b

The Real Number System

Recall: R, the set of real numbers

The real number system consists of R and two operations on its elements:

addition and multiplication

Operation Symbol ResultAddition + Sum a + b

Multiplication · Product a · b

Axioms on Equality

Axioms: logical statements that are assumed to be true

For any real numbers a, b, c,

Reflexive : a = a

Symmetric : If a = b, then b = a.

Transitive : If a = b and b = c, then a = c.

Additive : If a = b, then a + c = b + c.

Multiplicative : If a = b, then ac = bc.

Axioms on Equality

Axioms: logical statements that are assumed to be true

For any real numbers a, b, c,

Reflexive : a = a

Symmetric : If a = b, then b = a.

Transitive : If a = b and b = c, then a = c.

Additive : If a = b, then a + c = b + c.

Multiplicative : If a = b, then ac = bc.

Axioms for Addition and Multiplication

For any real numbers a, b and c

Axiom Addition Multiplication

Closure a + b ∈ R a · b ∈ RAssociativity (a + b) + c = a + (b + c) (a · b) · c = a · (b · c)

Commutativity a + b = b + a a · b = b · a

Distributivity: c · (a + b) = c · a + c · b

Let a ∈ R.

Identity Axiom for Addition :There exists a real number, zero (0), such thata + 0 = 0 + a = a

0 is the identity element for addition.

Identity Axiom for Multiplication :There exists a real number, one (1), such thata · 1 = 1 · a = a

1 is the identity element for multiplication.

Let a ∈ R.

Inverse Axiom for Addition :There exists a real number, −a, such thata + (−a) = (−a) + a = 0−a is the additive inverse of a.

Inverse Axiom for Multiplication :

If a 6= 0, there exists a real number,1a

, such that

a · 1a

=1a· a = 1

is the multiplicative inverse of a.

Cancellation Laws

For any a, b, c ∈ R,

Addition :If a + c = b + c, then a = b.If c + a = c + b, then a = b.

Multiplication :If ac = bc, c 6= 0, then a = b.If ca = cb, c 6= 0, then a = b.

For any real numbers a and b,

a · 0 = 0

If ab = 0, then a = 0 or b = 0

−(−a) = a

(−a)b = −ab

(−a)(−b) = ab

−(a + b) = (−a) + (−b)

(−1)a = −a

Subtraction

Definition

If a, b ∈ R, then subtraction is the operation that assigns to a and b a realnumber, a− b, the difference of a and b, where

a− b = a + (−b)

a− a = 0a− (−b) = a + b

a(b− c) = ab− ac

Subtraction

Definition

If a, b ∈ R, then subtraction is the operation that assigns to a and b a realnumber, a− b, the difference of a and b, where

a− b = a + (−b)

a− a = 0a− (−b) = a + b

a(b− c) = ab− ac

Division

Definition

If a, b ∈ R and b 6= 0, then division is the operation that assigns to a and b

a real number, a÷ b =a

b, the quotient of a and b, where

b= a · 1

For any a, b, c, d ∈ R, with b, d 6= 0

a= 1 if a 6= 0

) = a for a 6= 0

afor a 6= 0

d⇔ ad = bc

bcif c 6= 0

−b= −a

Division

Definition

b= a · 1

For any a, b, c, d ∈ R, with b, d 6= 0

a= 1 if a 6= 0

) = a for a 6= 0

afor a 6= 0

d⇔ ad = bc

bcif c 6= 0

−b= −a

Division

Definition

b= a · 1

For any a, b, c, d ∈ R, with b, d 6= 0a

a= 1 if a 6= 0

) = a for a 6= 0

afor a 6= 0

d⇔ ad = bc

bcif c 6= 0

−b= −a

Division

Definition

b= a · 1

a= 1 if a 6= 0

) = a for a 6= 0

afor a 6= 0

d⇔ ad = bc

bcif c 6= 0

−b= −a

Division

Definition

b= a · 1

a= 1 if a 6= 0

) = a for a 6= 0

afor a 6= 0

d⇔ ad = bc

bcif c 6= 0

−b= −a

Division

Definition

b= a · 1

a= 1 if a 6= 0

) = a for a 6= 0

afor a 6= 0

d⇔ ad = bc

bcif c 6= 0

−b= −a

Division

Definition

b= a · 1

a= 1 if a 6= 0

) = a for a 6= 0

afor a 6= 0

d⇔ ad = bc

bcif c 6= 0

−b= −a

Division

Definition

b= a · 1

a= 1 if a 6= 0

) = a for a 6= 0

afor a 6= 0

d⇔ ad = bc

bcif c 6= 0

−b= −a

Division

Definition

b= a · 1

a= 1 if a 6= 0

) = a for a 6= 0

afor a 6= 0

d⇔ ad = bc

bcif c 6= 0

−b= −a

Division

Definition

b= a · 1

a= 1 if a 6= 0

) = a for a 6= 0

afor a 6= 0

d⇔ ad = bc

bcif c 6= 0

−b= −a

Other properties of division: For a, b, c, d ∈ R, b, d 6= 0

Sum :a

ad + bc

Difference :a

b− c

ad− bc

Product :a

b· cd

Quotient :a

b· dc

bcprovided c 6= 0

Order Axioms of R

For any real number a,

a is positive if and only if a > 0.

a is negative if and only if a < 0.

For any real numbers a, b, c

Trichotomy One and only one of the following relations holds:

a = b, a > b or a < b

Transitive If a > b and b > c then a > c.

Addition If a > b, then a + c > b + c.

Multiplication If a > b and c > 0, then ac > bc.

Order Axioms of R

For any real number a,

a is positive if and only if a > 0.

a is negative if and only if a < 0.

For any real numbers a, b, c

Trichotomy One and only one of the following relations holds:

a = b, a > b or a < b

Transitive If a > b and b > c then a > c.

Addition If a > b, then a + c > b + c.

Multiplication If a > b and c > 0, then ac > bc.

For any a, b, c ∈ RThe set of positive real numbers is closed under addition andmultiplication.

If a > b, then −a < −b.

a2 ≥ 0If a > b and c < 0, then ac < bc.

If a > 0, then1a

If a > b > 0, then1a

If a > 0, then1a

a2 ≥ 0

If a > b and c < 0, then ac < bc.

If a > 0, then1a

The Real Number Line

There is a one-to-one correspondence between

the points on the line l and R.

All real numbers can be put in sequence

on a line.

Interval Notation

Let a, b ∈ R such that a < b.

The open interval (a, b) is the set {x ∈ R | a < x < b}.

The closed interval [a, b], is the open interval (a, b) together with itstwo endpoints.

a is called the left endpoint

b is called the right endpoint

Interval Notation

Interval Notation Set Notation Number Line Graph

[a, b] {x | a ≤ x ≤ b}

(a, b) {x | a < x < b}

[a, b) {x | a ≤ x < b}

(a, b] {x | a < x ≤ b}

[a, +∞) {x | x ≥ a}

(a, +∞) {x | x > a}

(−∞, b] {x | x ≤ b}

(−∞, b) {x | x < b}

Example.

1 [−4, 1] ∪ (−2, 3]

= [−4, 3]2 [−4, 1] ∩ (−2, 3]

= (−2, 1]

Example.

1 [−4, 1] ∪ (−2, 3]

= [−4, 3]2 [−4, 1] ∩ (−2, 3]

= (−2, 1]

Example.

1 [−4, 1] ∪ (−2, 3]

= [−4, 3]2 [−4, 1] ∩ (−2, 3]

= (−2, 1]

Example.

1 [−4, 1] ∪ (−2, 3]

= [−4, 3]2 [−4, 1] ∩ (−2, 3]

= (−2, 1]

Example.

1 [−4, 1] ∪ (−2, 3]

= [−4, 3]2 [−4, 1] ∩ (−2, 3]

= (−2, 1]

Example.

1 [−4, 1] ∪ (−2, 3] = [−4, 3]

2 [−4, 1] ∩ (−2, 3]

= (−2, 1]

Example.

1 [−4, 1] ∪ (−2, 3] = [−4, 3]2 [−4, 1] ∩ (−2, 3]

= (−2, 1]

Example.

1 [−4, 1] ∪ (−2, 3] = [−4, 3]2 [−4, 1] ∩ (−2, 3]

= (−2, 1]

Example.

1 [−4, 1] ∪ (−2, 3] = [−4, 3]2 [−4, 1] ∩ (−2, 3] = (−2, 1]

The Absolute Value

If x is any real number, the absolute value of x, written as |x|, is definedas:

−x if x < 00 if x = 0x if x > 0

Example. | − 7| = |7| = 7

The Absolute Value

If x is any real number, the absolute value of x, written as |x|, is definedas:

−x if x < 00 if x = 0x if x > 0

Example. | − 7| = |7| = 7

Let x, y ∈ R. Then

|x| ≥ 0| − x| = |x||xy| = |x| · |y|∣∣∣∣xy∣∣∣∣ = |x||y| , y 6= 0

Example:

∣∣∣∣3− 95

∣∣∣∣ = ∣∣∣∣−65

∣∣∣∣ = | − 6||5|

Let x, y ∈ R. Then

|x| ≥ 0| − x| = |x||xy| = |x| · |y|∣∣∣∣xy∣∣∣∣ = |x||y| , y 6= 0

Example:

∣∣∣∣3− 95

∣∣∣∣ = ∣∣∣∣−65

∣∣∣∣ = | − 6||5|

Geometric Interpretation of Absolute Value

|x| represents the distance of the point corresponding to x from the pointcorresponding to 0.

Let a > 0|x| = a if and only if x = ±a

|x| < a if and only if − a < x < a

|x| > a if and only if x < −a or x > a

Distance between Real Numbers

The distance between two real numbers x and y is

d = |x− y| = |y − x|

Example: The distance between −11 and 5 is

| − 11− 5| = | − 16| = 16

d = |x− y| = |y − x|

| − 11− 5| = | − 16| = 16

d = |x− y| = |y − x|

| − 11− 5| = | − 16| = 16

For x, y ∈ R,

|x− y| ≥ 0|x− y| = 0 if and only if x = y

Triangle Inequality: |x + y| ≤ |x|+ |y|

Integer Exponents

Let a ∈ R, n ∈ Z. The nth power of a is denoted by an.

If n > 0, then an = a · a · a · · · · · a︸︷︷︸n times

If a 6= 0, then a0 = 1 and a−1 =1a

If n > 0 and a 6= 0, then a−n =1an

Integer Exponents

If a 6= 0, then a0 = 1 and a−1 =1a

Integer Exponents

If a 6= 0, then a0 = 1

and a−1 =1a

Integer Exponents

If a 6= 0, then a0 = 1 and a−1 =1a

Integer Exponents

If a 6= 0, then a0 = 1 and a−1 =1a

Laws of Exponents: Product Law

Let n, m ∈ Z, a ∈ R, then an · am = an+m.

Examples:

42 · 43 = 42+3

= 1024

x2 · x−5 = x−3

Examples:

42 · 43 = 42+3

= 1024

x2 · x−5 = x−3

Examples:

42 · 43 =

= 1024

x2 · x−5 = x−3

Examples:

42 · 43 = 42+3

= 1024

x2 · x−5 = x−3

Examples:

42 · 43 = 42+3

= 1024

x2 · x−5 = x−3

Examples:

42 · 43 = 42+3

= 1024

x2 · x−5 = x−3

Examples:

42 · 43 = 42+3

= 1024

x2 · x−5 =

Examples:

42 · 43 = 42+3

= 1024

x2 · x−5 = x−3

Laws of Exponents: Power raised to a Power Law

Let n, m ∈ Z, a ∈ R, then (an)m = anm.

Examples:

(23)4 = 23·4

= 4096

(x−1)3 = x−3

(y2)−3 = y−6

Examples:

(23)4 = 23·4

= 4096

(x−1)3 = x−3

(y2)−3 = y−6

Examples:

(23)4 =

= 4096

(x−1)3 = x−3

(y2)−3 = y−6

Examples:

(23)4 = 23·4

= 4096

(x−1)3 = x−3

(y2)−3 = y−6

Examples:

(23)4 = 23·4

= 4096

(x−1)3 = x−3

(y2)−3 = y−6

Examples:

(23)4 = 23·4

= 4096

(x−1)3 = x−3

(y2)−3 = y−6

Examples:

(23)4 = 23·4

= 4096

(x−1)3 =

(y2)−3 = y−6

Examples:

(23)4 = 23·4

= 4096

(x−1)3 = x−3

(y2)−3 = y−6

Examples:

(23)4 = 23·4

= 4096

(x−1)3 = x−3

(y2)−3 =

Examples:

(23)4 = 23·4

= 4096

(x−1)3 = x−3

(y2)−3 = y−6

Laws of Exponents: Power of a Product Law

Let n ∈ Z, a, b ∈ R. Then (ab)n = anbn.

Examples:

(−2 · 3)2 = (−2)2 · (3)2

= 4 · 9= 36

25 · 55 = (2 · 5)5

= 100000

Examples:

(−2 · 3)2 = (−2)2 · (3)2

= 4 · 9= 36

25 · 55 = (2 · 5)5

= 100000

Examples:

(−2 · 3)2 =

(−2)2 · (3)2

= 4 · 9= 36

25 · 55 = (2 · 5)5

= 100000

Examples:

(−2 · 3)2 = (−2)2 · (3)2

= 4 · 9= 36

25 · 55 = (2 · 5)5

= 100000

Examples:

(−2 · 3)2 = (−2)2 · (3)2

= 4 · 9

25 · 55 = (2 · 5)5

= 100000

Examples:

(−2 · 3)2 = (−2)2 · (3)2

= 4 · 9= 36

25 · 55 = (2 · 5)5

= 100000

Examples:

(−2 · 3)2 = (−2)2 · (3)2

= 4 · 9= 36

25 · 55 =

(2 · 5)5

= 100000

Examples:

(−2 · 3)2 = (−2)2 · (3)2

= 4 · 9= 36

25 · 55 = (2 · 5)5

= 100000

Examples:

(−2 · 3)2 = (−2)2 · (3)2

= 4 · 9= 36

25 · 55 = (2 · 5)5

= 100000

Examples:

(−2 · 3)2 = (−2)2 · (3)2

= 4 · 9= 36

25 · 55 = (2 · 5)5

= 100000

Laws of Exponents: Quotient Law

Let n, m ∈ Z, a ∈ R. If a 6= 0, thenan

am= an−m.

Examples:

x6= x10−6 = x4

y−7= y2−(−7) = y9

1110= 110 = 1

am= an−m.

Examples:

x6= x10−6 = x4

y−7= y2−(−7) = y9

1110= 110 = 1

am= an−m.

Examples:

x6= x10−6 = x4

y−7= y2−(−7) = y9

1110= 110 = 1

am= an−m.

Examples:

x6= x10−6 = x4

y−7= y2−(−7) = y9

1110= 110 = 1

am= an−m.

Examples:

x6= x10−6 = x4

y−7= y2−(−7) = y9

1110= 110 = 1

Laws of Exponents: Power of a Quotient Law

Let n ∈ Z, a, b ∈ R. If b 6= 0, then(a

Examples.

Laws of Exponents

Let n, m ∈ Z, a, b ∈ R, then

1 an · am = an+m

2 (an)m = anm

3 (ab)n = anbn

4 If a 6= 0, thenan

am= an−m

5 If b 6= 0, then(a

Example.

20x8y2z5

−5x2y7z5

=20−5· x

x2· y

y7· z

= −4 · x8−2 · y2−7 · z5−5

= −4 · x6 · y−5 · 1

= −4x6

Example.20x8y2z5

−5x2y7z5

=20−5· x

x2· y

y7· z

= −4 · x8−2 · y2−7 · z5−5

= −4 · x6 · y−5 · 1

= −4x6

Example.20x8y2z5

−5x2y7z5=

20−5· x

x2· y

y7· z

= −4 · x8−2 · y2−7 · z5−5

= −4 · x6 · y−5 · 1

= −4x6

Example.20x8y2z5

−5x2y7z5=

20−5· x

x2· y

y7· z

= −4 · x8−2 · y2−7 · z5−5

= −4 · x6 · y−5 · 1

= −4x6

Example.20x8y2z5

−5x2y7z5=

20−5· x

x2· y

y7· z

= −4 · x8−2 · y2−7 · z5−5

= −4 · x6 · y−5 · z0

= −4x6

Example.20x8y2z5

−5x2y7z5=

20−5· x

x2· y

y7· z

= −4 · x8−2 · y2−7 · z5−5

= −4 · x6 · y−5 · 1

= −4x6

Example.20x8y2z5

−5x2y7z5=

20−5· x

x2· y

y7· z

= −4 · x8−2 · y2−7 · z5−5

= −4 · x6 · y−5 · 1

= −4x6

Exercises:

1 Let the universal set be R, X be the half-open interval[−3

2 , 5),

Y = {x | 4 ≤ |x|}, and Z be the interval (2, +∞). Find the followingsets and express them in interval notation:

1 Y ∩ Zc

2 (X ∪ Y )c

3 Xc ∩ (Z ∪X)4 X\(Y ∩ Z)

2 Simplify the following expressions:

1a−1 + b−1

(a + b)−1 2

(185 · 203x−1y2z4

159x−3y−2z−1

a2 - real number system

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