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Properties of Real Numbers

2

- Chapter 2 is about Real Numbers
- How to add, subtract, multiply, and divide real

numbers. - How to determine the likelihood of an event using

probability and odds.

- The Real Number Line

2.1

Real numbers can be pictures as points on a

horizontal line called a real number line. The

point labeled 0 is the origin. Points to the

left are negative numbers and points to the right

of zero represent positive numbers. Zero is

neither negative nor positive.

Negative

Positive

The point that corresponds to a number is the

graph of the number, and drawing the point is

called graphing the number or plotting the point.

Graphing Real Numbers

Graph the numbers ½ and -2.3 on a number line.

The point that corresponds to ½ is one half unit

to the right of zero. The point that corresponds

to -2.3 is 2.3 units to the left of zero.

-2.3

1/2

Comparing Real Numbers

Graph the numbers -2.3 and ½ and then write and

inequality to compare the numbers.

-2.3

1/2

SOLUTION

One the graph -2.3 is to the left of ½, so -2.3

is LESS THAN ½.

-2.3 lt 1/2

Ordering Real Numbers

Write the following numbers in increasing order

-2, 4, 0, 1.5, ½, -3/2

SOLUTION

Graph the numbers on a number line.

-2 -3/2 0 ½ 1.5 4

Finding Opposites and Absolute Values

Two points that are the same distance from the

origin but on opposite sides of the origin are

opposites. The absolute value of a real number is

the distance between the origin and the point

representing the real number. The symbol a

represent the absolute value of a number a.

Absolute value is never negative.

Velocity indicates both speed and direction (up

is positive and down is negative). The speed of

an object is the absolute value of its

velocity. If the velocity is -10 feet per

second, what is the direction?? What is the

speed??

Direction is down (-) Speed is positive (10 feet

per second)

In mathematics, to prove that a statement is

true, you need to show that is true for all

examples. To prove that a statement is false,

you need to show that it is not true for only one

example, called a counterexample. Counterexample

See example 9

What you Should Learn

- Adding real numbers using a number line or

addition rules.

Using Addition in Real Life

- Use addition of real numbers to solve real-life

problems such as finding the profit of a business

in Example 5

- Addition of Real Numbers

2.2

- Addition can be modeled with movements on a

number line. - You add a positive number by moving to the right.
- You add a negative number by moving to the left.

Negative

Positive

The point that corresponds to a number is the

graph of the number, and drawing the point is

called graphing the number or plotting the point.

EXAMPLE 1 Adding Two Real Numbers

EXAMPLE 2 Adding Three Real Numbers

The rules of addition show how to add two real

numbers without a number line

RULES OF ADDITION

TO ADD TWO NUMBERS WITH THE SAME SIGN STEP 1 Add

their absolute values STEP 2 Attach the common

sign. Example -4 (-5) Step 1 -4 -5

9 Step 2 -9 TO ADD TWO NUMBERS WITH OPPOSITE

SIGNS STEP 1 Subtract the smaller absolute value

from the larger absolute value STEP 2 Attach the

sign of the number with the larger absolute

value Example 3 (-9) Step 1 -9 - 3 6

Step 2 -6

PROPERTIES OF ADDITION

COMMUTATIVE PROPERTY The order in which two

numbers are added does not change the sum. a b

b a example 3 (-2)

-2 3 ASSOCIATIVE PROPERTY The way you group

three numbers when adding does not change the

sum. (a b) c a (b c) example (-5

6) 2 -5 (6 2) IDENTITY PROPERTY The sum

of a number and 0 is the number. a 0 a

example -4 0

-4 PROPERTY OF ZERO (INVERSE PROPERTY) The sum

of a number and its opposite is 0 a -a a

example 5 (-5) 0

(No Transcript)

EXAMPLE 4 Adding Rule Numbers

On a standardized test, students receive a score

of 1 for each correct answer and -1 for each

incorrect answer. Those questions that are left

blank are given a zero. In order to get a grade

above average on the test, a students total

score must be above 35. Would the following

students score above average on the test?

- Sylvia 7 incorrect, 12 blank, 41 correct.
- Nathan 8 incorrect, 5 blank, 47 correct.
- a. NO
- b. YES

EXAMPLE 5 Finding the total profit

A child care center had the following monthly

results after comparing income and expenses. Add

the monthly profits and losses to find the

overall profit or loss during the four-month

period.

Answer The center had a profit of 7,168.33

PRACTICE

What you Should Learn

- Subtract real numbers using the subtraction rule.

Using Subtraction in Real Life

- Use subtraction of real numbers to solve

real-life problems such as the difference in

stock market prices.

WARMUP

- Subtraction of Real Numbers

2.3

Some addition expressions can be modeled with

subtraction. Example 5 (-3) 2

5 3 2 9

(-6) 3 9 6

3 Adding the opposite of a number is the same

as subtraction.

SUBTRACTION RULE

To subtract b from a, add the opposite of b to

a a b a (-b) example 3

5 3 (-5) The result is the difference of

a and b

EXAMPLE 1 Using the Subtraction Rule

- Find the difference.
- -4 3 b. 10 11 c. 11-10 d. -3/2 (-1/2)

- -4 -3 -4 (-3)
- -7
- 10-11 10 (-11)
- -1
- 11-10 11 (-10)
- 1
- -3/2 (-1/2) -3/2 ½
- -1

EXAMPLE 2 Evaluating Expressions with More than

One Subtraction

Evaluate the expression 3 (-4) 2

8 Solution 3 (-4) 2 8 3 4 2 8

7 (-2) 8

5 8 13

EXAMPLE 3 Finding the Terms of an Expression

Find the terms of -9 2x Solution Use the

subtraction Rule -9 (-2x) In this form you can

see the two terms are -9 and -2x

EXAMPLE 4 Evaluating a Function

Evaluate the function y-5 x for these values

of x -2, -1, 0, and 1 Organize your results in a

table and describe the pattern.

Solution

EXAMPLE 5 Subtracting Real Numbers

Practice

Warm-up Exercises

Multiplication of Real Numbers

Remember that multiplication can be modeled as

repeated addition. For example 3 (-2) (-2)

(-2) (-2) 6

- TIP for Students
- A product is negative if it has an odd number of

negative factors - A product is positive if it has an even number of

negative factors.

Example 1

Multiplying Real Numbers

- (-3) (4) (-2) (-12) (-2) 24 Two

negative factors positive product - (-1/2) (-2) (-3) (1) (-3) -3 Three

negative factors negative product -

Four negative factors positive product

Example 2

Two negative signs

Positive Product

Three negative signs

Negative Product

Three negative signs

Negative Product

One negative sign

Negative Product

Evaluate the expression when x -7

Example 3

USING MULTIPLICATION IN REAL LIFE

Displacement is the change in the position of an

object. Unlike distance, displacement can be

positive, negative or zero.

A leaf floats down from a tree at a velocity of

-12 cm/sec. Find the vertical displacement in

4.2 seconds.

Example 4

A store runs a special on spinach, normally they

sell a bag for 1.69, today they are selling two

for the price of one. How much do they lose if

they sell 798 free bags?

Example 5

(No Transcript)

Test You Knowledge!!!!!

Practice

MODELING THE DISTRIBUTIVE PROPERTY

You can use algebra tiles to model algebraic

expressions.

1-tile

x-tile

1

1

1

x

This 1-by-1 square tile has an area of 1 square

unit.

This 1-by-x square tile has an area of x

square units.

Model the Distributive Property using Algebra

Tiles

Area 3(x 2)

Area 3(x ) 3(2)

The product of a and (b c)

a(b c) ab ac

2(x 5)

2(x) 2(5)

2x 10

(b c)a ba ca

(x 5)2

(x)2 (5)2

2x 10

y(1 y)

y(1) y(y)

y y 2

(1 5x)2

(1)2 (5x)2

2 10x

Remember that a factor must multiply each term of

an expression.

Distribute the 3.

(3)(1 x)

(3)(1) (3)(x)

Simplify.

3 3x

Distribute the 2.

(y 5)(2)

(y)(2) (5)(2)

Simplify.

2y 10

a 1 a

(7 3x)

(1)(7) (1)(3x)

Simplify.

7 3x

Forgetting to distribute the negative sign when

multiplying by a negative factor is a common

error.

SOLUTION

You are shopping for CDs. You want to buy six CDs

for 11.95 each.

The mental math is easier if you think of 11.95

as 12.00 .05.

Write 11.95 as a difference.

6(11.95) 6(12 0.05)

6(12) 6(0.05)

Use the distributive property.

Use the distributive property to calculate the

total cost mentally.

72 0.30

Find the products mentally.

71.70

Find the difference mentally.

SIMPLIFYING BY COMBINING LIKE TERMS

Each of these terms is the product of a number

and a variable.

Like terms have the same variable raised to the

same power.

y2 x2 3y3 5 3 3x2 4y3 y

(8 3)x

8x 3x

Use the distributive property.

11x

Add coefficients.

4x2 2 x2

4x2 x2 2

Group like terms.

3x2 2

Combine like terms.

3 2(4 x)

3 (2)(4 x)

Rewrite as addition expression.

3 (2)(4) (2)(x)

Distribute the 2.

3 (8) (2x)

Multiply.

5 (2x)

Combine like terms and simplify.

5 2x

WHAT you should learn.

Use division to simplify algebraic expressions.

8

1/8

-1/2

-2/1

-10

-1/10

Practice a. b. c.

d.

WORKING WITH ALGEBRAIC EXPRESSIONS

Simplify the expression

Evaluate the expression when a -2 and b -3

Evaluate the expression when x -5 and y -1

You are descending in a hot-air balloon. You

descend 500 feet in 40 seconds. What is your

velocity?

DOMAIN

INPUTS

What number can be input to get a solution and

what numbers cant be used?

PROBABILITY AND ODDS

The probability of an event is a measure of the

likelihood that the event will occur. It is a

number between 0 and 1, inclusive. When you do a

probability experiment, the different possible

results are called outcomes. When an experiment

has N equally likely outcomes, each of them

occurs with the probability 1/N, For example, in

the roll of a six sided dice, the possible

outcomes are 1,2,3,4,5,6, the probability

associated with each outcome is 1/6. An event

consists of a collection of outcomes. In the

roll of a six-sided number cube, an even roll

consists of the outcomes 2,4,6. The theoretical

probability of an even roll is 3/6. The outcomes

for an event you wish to happen is called the

favorable outcome. Theoretical probability P

Number of favorable outcomes/total number of

outcomes

You have 2 red and 2 black socks in a drawer.

You reach in and pick two without looking. What

is the probability P that they do not match? 2

chances favorable 4 total chances

P 1/2

PROBABILITY AND ODDS

The ODDS of an event happening can be found by

divided the number of favorable outcomes by the

number of unfavorable.

You randomly choose a letter from the word

SUMMER. What are the odds that the letter is a

vowel. S M M R Not vowels U E vowels

favorable /

unfavorable 2 / 4

The probability that a randomly chosen household

has a cat is .27. What are the odds that a

household has a cat? Favorable .27 Not

Favorable 1 - .27 .73 Answer .27

/.73

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