Properties of Real Numbers

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Properties of Real Numbers
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• 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.

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• 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.
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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
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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
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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
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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.
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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
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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

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• 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.
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EXAMPLE 1 Adding Two Real Numbers
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EXAMPLE 2 Adding Three Real Numbers
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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
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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
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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

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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
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PRACTICE
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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
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• 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
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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

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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
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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
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EXAMPLE 5 Subtracting Real Numbers
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Practice
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Warm-up Exercises
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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.

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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
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Evaluate the expression when x -7
Example 3
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USING MULTIPLICATION IN REAL LIFE
Displacement is the change in the position of an
object. Unlike distance, displacement can be
positive, negative or zero.
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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
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Test You Knowledge!!!!!
Practice
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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)
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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
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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.
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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.
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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
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(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
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WHAT you should learn.
Use division to simplify algebraic expressions.
8
1/8
-1/2
-2/1
-10
-1/10
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Practice a. b. c.
d.
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WORKING WITH ALGEBRAIC EXPRESSIONS
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Simplify the expression
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Evaluate the expression when a -2 and b -3
Evaluate the expression when x -5 and y -1
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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?
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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
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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
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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
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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