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Real Numbers and Algebraic Expressions

HamestGevorgyanDepartmen of Mathematics and CS

hgevorgyan@nccu.edu

• Recognize subsets of the real number.•Use inequality symbols.•Evaluate absolute value to express distance.•Algebraic expressions and their evaluation•Properties of algebraic operations•Understand and use of integer exponents•Properties of exponents•Simplify exponential expressions•Use of scientific notation.

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Algebraic expression

A combination of variables and numbers using

the operations of addition, subtraction, division,

or multiplication, as well as powers or roots, is

called an algebraic expression.

Example: 7x, x+11, x-9, x4 -10,

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Definition of a Natural Number Exponent

If b is a real number and n is a natural number,

bn b b b ... b

bn is read “the nth power of b” or “ b to the nth power.” Thus, the nth power of b is defined as the product of n factors of b. Furthermore, b1 = b

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1. Perform operations within the innermost parentheses and work outward. If the algebraic expression involves division, treat the numerator and the denominator as if they were each enclosed in parentheses.

2. Evaluate all exponential expressions.

3. Perform multiplication or division as they occur, working from left to right.

4. Perform addition or subtraction as they occur, working from left to right.

The Order of Operations Agreement

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A set is a collection of objects whose contents can be clearly determined.

The set {1, 3, 5, 7, 9} has five elements.

• The objects in a set are called the elements of the set.

• We use braces to indicate a set and commas to separate the elements of that set.

The set of counting numbers can be represented by {1, 2, 3, … }.

The set of even counting numbers are {2, 4, 6, …}.

The set of even counting numbers is a subset of the set of counting numbers, since each element of the subset is also contained in the set.

For example,

The Basics About Sets

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Set operations

The union of two sets is the set of all elements formed by combining all the elements of set A and all the elements of set B into one set.

The symbolism used is The Venn Diagram representing the union of A and B is the entire region shaded yellow.

{ }A B x x A or x B AA

BB

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Union of Sets

Combination of everything in both sets A = {all tall children} B = {all girls} A union B = {all girls OR tall children} = {all

girls and all tall boys}

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Intersection of sets A and B

The intersection of sets A and B is the set of elements that is common to both sets A and B. It is symbolized as

{ x l x ∈A and x ∈B }

Represented by Venn Diagrams:

A B

AA BB

Intersection Intersection

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Intersection of Sets

What they have in common A = {all tall children} B = {all girls} A intersect B = {all tall girls} All children that are girls AND are tall

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A group of biology majors are taking Biology I & Chem. I. A group of chemistry majors are taking Calculus, Chem. I and Physics I. The Physics majors enrolled in Calculus, Physics I, and Chem I. What is the intersection of the 3 groups?

1. Students in biology, chemistry, & physics.2. Students in chemistry.3. Students in calculus.4. Students in physics.

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-15, -7, -4, 0, 4, 7{…, -2, -1, 0, 1, 2, 3, …}

Add the negative natural numbers to the whole numbers

Integers

Z

0, 4, 7, 15{0, 1, 2, 3, … }

Add 0 to the natural numbers

Whole Numbers

W

4, 7, 15{1, 2, 3, …}

These are the counting numbers

Natural Numbers

N

ExamplesDescriptionName

Important Subsets of the Real Numbers

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This is the set of numbers whose decimal representations are neither terminating nor repeating. Irrational numbers cannot be expressed as a quotient of integers.

Irrational NumbersI

These numbers can be expressed as an integer divided by a nonzero integer:Rational numbers can be expressed as terminating or repeating decimals.

Rational NumbersQ

ExamplesDescriptionName

Important Subsets of the Real Numbers

2 1.414214

3 1.73205

3.142

2

1.571

17 17

1, 5 5

1, 3, 2

0,2,3,5,17

2

50.4,

2

3 0.666666... 0.6

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Rational numbers Irrational numbers

Integers

Whole numbers

Natural numbers

The set of real numbers is formed by combining the rational numbers and the irrational numbers.

The Real Numbers

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Negative numbers

Units to the left of the origin are negative.

Positive numbers

Units to the right of the origin are

positive.

The real number line is a graph used to represent the set of real numbers. An arbitrary point, called the origin, is labeled 0;

-4 -3 -2 -1 0 1 2 3 4

the Origin

The Real Number Line

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Real numbers are graphed on the number line by placing a dot at the location for each number. –3, 0, and 4 are graphed below.

-4 -3 -2 -1 0 1 2 3 4

Graphing on the Number Line

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On the real number line, the real numbers increase from left to right. The lesser of two real numbers is the one farther to the left on a number line. The greater of two real numbers is the one farther to the right on a number line.

Since 2 is to the left of 5 on the number line, 2 is less than 5. 2 < 5

Since 5 is to the right of 2 on the number line, 5 is greater than 2. 5 > 2

-2 -1 0 1 2 3 4 5 6

Ordering the Real Numbers

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Because -5 = -5-5 > -5

Because 7 > 37 > 3b is greater than or equal to a.b > a

Because 7 =77 < 7

Because 3 < 73 < 7a is less than or equal to b.a < b

ExplanationExampleMeaningSymbols

Inequality Symbols

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Absolute value describes the distance from 0 on a real number line. If a represents a real number, the symbol |a| represents its absolute value, read “the absolute value of a.”

-3 -2 -1 0 1 2 3 4 5

The absolute value of –3 is 3 because –3 is 3 units from 0 on the number line.

|–3| = 3

The absolute value of 5 is 5 because 5 is 5 units from 0

on the number line.

|5| = 5

Absolute Value

For example, the real number line below shows that

|-3| = 3 and |5| = 5.

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The absolute value of x is given as follows:

|x| =x if x > 0-x if x < 0{

Definition of Absolute Value

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For all real number a and b,1. |a| > 0

5. = , b not equal to 0

2. |-a| = |a| 3. a < |a|

4. |ab| = |a||b|

6. |a + b| < |a| + |b| (the triangle inequality)

|a||b|

ab

Properties of Absolute Value

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Example

Find the following: |-3| and |3|.

Solution:

| -3 | = 3 and | 3 | = 3Distance Between Two Points on the Real Number Line

If a and b are any two points on a real number line, then the distance between a and b is given by

|a – b| or |b – a|

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Find the distance between –5 and 3 on the real number line.

Solution Because the distance between a and b is given by |a – b|, the distance between –5 and 3 is |-5 – 3| = |-8| = 8.

-5 -4 -3 -2 -1 0 1 2 3

We obtain the same distance if we reverse the order of subtraction:|3 – (-5)| = |8| = 8.

8

Text Example

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A combination of variables and numbers using the operations of addition, subtraction, multiplication, or division, as well as powers or roots, is called an algebraic expression.

Here are some examples of algebraic expressions:

x + 6, x – 6, 6x, x/6, 3x + 5.

Algebraic Expressions

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1. Perform operations within the innermost parentheses and work outward. If the algebraic expression involves division, treat the numerator and the denominator as if they were each enclosed in parentheses.

2. Evaluate all exponential expressions.

3. Perform multiplication or division as they occur, working from left to right.

4. Perform addition or subtraction as they occur, working from left to right.

The Order of Operations Agreement

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The algebraic expression 2.35x + 179.5 describes the population of the United States, in millions, x years after 1980. Evaluate the expression when x = 20. Describe what the answer means in practical terms.

Solution We begin by substituting 20 for x. Because x = 20, we will be finding the U.S. population 20 years after 1980, in the year 2000.

2.35x + 179.5 Replace x with 20.

= 2.35(20) + 179.5

= 47 + 179.5 Perform the multiplication.

= 226.5 Perform the addition.

Thus, in 2000 the population of the United States was 226.5 million.

Text Example

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• 3 + ( 8 + x)

= (3 + 8) + x

= 11 + x

If 3 real numbers are added, it makes no difference which 2 are added first.(a + b) + c = a + (b + c)

Associative Property of Addition

• x · 6 = 6x Two real numbers can be multiplied in any order.ab = ba

Commutative Property of Multiplication

• 13 + 7 = 7 + 13

• 13x + 7 = 7 + 13x

Two real numbers can be added in any order.a + b = b + a

Commutative Property of Addition

ExamplesMeaningName

Properties of the Real Numbers

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• 0 + 6x = 6x Zero can be deleted from a sum.a + 0 = a0 + a = a

Identity Property of Addition

• 5 · (3x + 7)

= 5 · 3x + 5 · 7

= 15x + 35

Multiplication distributes over addition.a · (b + c) = a · b + a · c

Distributive Property of Multiplication over Addition

• -2(3x) = (-2·3)x = -6xIf 3 real numbers are multiplied, it makes no difference which 2 are multiplied first.(a · b) · c = a · (b · c)

Associative Property of Multiplication

ExamplesMeaningName

Properties of the Real Numbers

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• 2 · 1/2 = 1The product of a nonzero real number and its multiplicative inverse gives 1, the multiplicative identity.a · 1/a = 1 and 1/a · a = 1

Inverse Property of Multiplication

• (-6x) + 6x = 0The sum of a real number and its additive inverse gives 0, the additive identity.a + (-a) = 0 and (-a) + a = 0

Inverse Property of Addition

• 1 · 2x = 2x One can be deleted from a product.a · 1 = a and 1 · a = a

Identity Property of Multiplication

ExamplesMeaningName

Properties of the Real Numbers

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Let a and b represent real numbers.

Subtraction: a – b = a + (-b)

We call –b the additive inverse or opposite of b.

Division: a ÷ b = a · 1/b, where b = 0

We call 1/b the multiplicative inverse or reciprocal of b. The quotient of a and b, a ÷ b, can be written in the form a/b, where a is the numerator and b the denominator of the fraction.

Definitions of Subtraction and Division

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Simplify: 6(2x – 4y) + 10(4x + 3y).

Solution

6(2x – 4y) + 10(4x + 3y)

= 6 · 2x – 6 · 4y + 10 · 4x + 10 · 3y Use the distributive property.

= 12x – 24y + 40x + 30y Multiply.

= (12x + 40x) + (30y – 24y) Group like terms.

= 52x + 6y Combine like terms.

Text Example

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Properties of Negatives

Let a and b represent real numbers, variables, or

algebraic expressions.1. (-1)a = -a2. -(-a) = a3. (-a)(b) = -ab4. a(-b) = -ab5. -(a + b) = -a - b6. -(a - b) = -a + b = b - a

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Definition of a Natural Number Exponent

If b is a real number and n is a natural number,

bn b b b ... b

bn is read “the nth power of b” or “ b to the nth power.” Thus, the nth power of b is defined as the product of n factors of b. Furthermore, b1 = b

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The Negative Exponent Rule

If b is any real number other than 0 and n is a natural number, then

b n 1

bn

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The Zero Exponent Rule

If b is any real number other than 0,b0 = 1.

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The Product Rule

b m · b n = b m+n

When multiplying exponential expressions with the same base, add the exponents. Use this sum as the exponent of the common base.

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The Power Rule (Powers to Powers)

(bm)n = bm•n

When an exponential expression is raised to a power, multiply the exponents. Place the product of the exponents on the base and remove the parentheses.

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The Quotient Rule

When dividing exponential expressions with the same nonzero base, subtract the exponent in the denominator from the exponent in the numerator. Use this difference as the exponent of the common base.

bm

bn bm n

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Example

Find the quotient of 43/42

4444

4 1232

3

Solution:

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Products to Powers

(ab)n = anbn

When a product is raised to a power, raise each factor to the power.

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Simplify: (-2y)4.

(-2y)4 = (-2)4y4 = 16y4

Text Example

Solution

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Quotients to Powers

When a quotient is raised to a power, raise the numerator to that power and divide by the denominator to that power.

n

nn

b

a

b

a

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Example

Simplify by raising the quotient (2/3)4 to the given power.

81

16

3

2

3

24

44

Solution:

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Properties of Exponents

1. b n 1

bn 2. b0 1 3. bm bn bmn 4. (bm)n bmn

5.bm

bn bm n 6. (ab)n anbn 7.a

b

n

an

bn

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QUIZ#1

If the sum of q-6, q-3, and q is 0, what is the

value of q.

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The number 5.5 x 1012 is written in a form called scientific notation. A number in scientific notation is expressed as a number greater than or equal to 1 and less than 10 multiplied by some power of 10. It is customary to use the multiplication symbol, x, rather than a dot in scientific notation.

Scientific Notation

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Scientific Notation

A number is written in scientific notation when

it is expressed in the form

a x 10n

where the absolute value of a is greater than or

equal to 1 and less than 10, and n is an integer.

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Text Example

Write each number in decimal notation: a. 2.6 X 107

b. b. 1.016 X 10-8

a. 2.6 x 107 can be expressed in decimal notation by moving the decimal point in 2.6 seven places to the right. We need to add six zeros.

2.6 x 107 = 26,000,000.

Solution:

b. 1.016 x 10-8 can be expressed in decimal notation by moving the decimal point in 1.016 eight places to the left. We need to add seven zeros to the right of the decimal point.

1.016 x 10-8 = 0.00000001016.

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To convert from decimal notation to scientific notation, we reverse the procedure.

• Move the decimal point in the given number to obtain a number greater than or equal to 1 and less than 10.

• The number of places the decimal point moves gives the exponent on 10; the exponent is positive if the given number is greater than 10 and negative if the given number is between 0 and 1.

Scientific Notation

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Write each number in scientific notation. a. 4,600,000 b. 0.00023

Solution

a. 4,600,000 = 4.6 x 10? 4.6 x 106Decimal point moves 6 places

b. 0.00023 = 2.3 x 10? 2.3 x 10-4Decimal point moves 4 places

Text Example