P.1 Real Numbers and Algebraic Expressions

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

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

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

Be especially careful when the numbers are negatively signednegatively signed. The relationship will be the oppositeopposite of what it would be if positive.Ex: 5<7, yet –5>-7. Always “grab” the larger (further to the right) number.Ex: - ______ -3.14 (ans: < since 3.14 ends & neg.)

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.”

For example, the real number line below shows that |-3| = 3 and |5| = 5.

-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

The absolute value of x is given as follows:

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

Definition of Absolute Value

Note: the absolute value of any real number will always be positive or zero. If it is originally negatively signed (less than zero), we must take the opposite sign. (|-3| = - (-3)) If it is originally positively signed (greater than zero) we want to keep it positively signed.

Ex: Find | \/7 - | (ans: pi-root7 since neg)

Ex: Evaluate the given expression for x = -5 and y = 7|x-y|/(x-y) (ans: -1)

Q: If |x| = 5, what possible values could x equal? (ans5 or –5)

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|

Show how we could write the above using integers and absolute value symbols. (ans: |80-20| and if WEST: |80-(-20)|)

Distance Between Two Points on the Real Number Line

You live 80 miles east off I-4, your partner lives 20 miles east off I-4.

What is the distance between you and your partner?

If your partner lived 20 miles WEST off I-4 what would the distance be?

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.

?

Ex:

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. Note: there is NO EQUAL SIGN, and we can only SIMPLIFY, not SOLVE!

Here are some examples of algebraic expressions:

x + 6, x – 6, 6x, x/6, 3x + 5.

Algebraic Expressions

Note: In none of these cases can you SOLVE for the variable, the best you can do is SIMPLIFY. In what case could you “solve”? What would that be called (instead of an expression?)

1. Perform operations within the innermost parentheses (or any other grouping symbol such as a radical sign, absolute value, or division bar) 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.

• That is: remove parenthesis and use order of operations to combine “like” terms.

Ex: Simplify 123•2 (ans: 8, NOT 2!)

The Order of Operations Agreement

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 in 1980 + 20 or 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 the year 2000 the population of the United States was 226.5 million.

Note the FORM of the simplification. Your test papers should look like this.

Text Example

Simplify: 6(2x – 4y) + 10(4x + 3y).

Solution 6(2x – 4y) + 10(4x + 3y)

= Use the distributive property.

= Multiply.

= Combine like terms.

Example

Note: when we add “like” terms, we add the coefficient parts, but leave the variable part (the like part) the same.(ans: 53x+6y)

Think: 2(ab) vs 2(a+b): In which case do we use the distributive law and why? (2nd only, 1st is all mult.)

• 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,

From this point I have provided these sheets as review only. Please review and answer the

questions. Answers will be discussed next class.

The Basics About Sets

-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

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

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

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

Q: How could we graph –13/5 on the real number line?

(ans: -13/5 = -2and 3/5, so it would be a little over half way from –2 to –3.)

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

Example• Find the following:

|-3| and |3|.

|-5•7| and |-5+7|

(Ans: 3 and 3, 35 and 2)

• 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

Think: How is 3(8+x) different and why can’t we use the associative property? (Ans: two operations: mult & addtn.)

• 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

• 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

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

Rewrite using multiplication:

10 2 10 (1/2)

(ans: 10*(1/2) and 10 * (2/1) or 10*2)

Properties of Negatives

• Let a and b represent real numbers, variables, or algebraic expressions.

1. (-1)a = -a

2. -(-a) = a

3. (-a)(b) = -ab

4. a(-b) = -ab

5. -(a + b) = -a - b

6. -(a - b) = -a + b = b - a

Algebra Translation DictionaryYou should add to this list as you discover new translations.

“Of” means use parenthesis (see p15 #111 ~118 to practice.)

Translation: Example:

+ add, increased by, sum, more than

Five times the sum of 4 and n: 5(4+n)

- difference, decreased by, less, minus, less than

n decreased by 5: n-5

p less than 7: 7-p

x product, times, double (etc.), half (etc.)

twice a number, decreased by 5: 2n-5

quotient, divided by, divided into

n divided by 5: n/5

p divided into 7: 7/p