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Chapter 2
Working with Real
Numbers
2-1
Basic Assumptions
CLOSURE PROPERTIES
a + b and ab are unique
7 + 5 = 12
7 x 5 = 35
COMMUTATIVE
PROPERTIES
a+b =b+a
ab = ba
2+6=6+2
2x6=6x2
ASSOCIATIVE
PROPERTIES
(a + b) + c = a + (b +c)
(ab)c = a(bc)
(5 + 15) + 20 = 5 + (15 +20)
(5·15)20 = 5(15·20)
Properties of
Equality
 Reflexive
Property - a = a
 Symmetric Property –
If a = b, then b = a
 Transitive Property –
If a = b, and b = c, then a = c
2-2
Addition on a Number Line
IDENTITY PROPERTIES
There is a unique real
number 0 such that:
a+0=0+a=a
-3 + 0 = 0 + -3 = -3
PROPERTY OF OPPOSITES
For each a, there is a unique
real number – a such that:
a + (-a) = 0 and (-a)+ a = 0
(-a) is called the opposite or
additive inverse of a

Property of the opposite of
a Sum
For all real numbers a and b:
-(a + b) = (-a) + (-b)
The opposite of a sum of real
numbers is equal to the sum of
the opposites of the numbers.
-(8 +2) = (-8) + (-2)
2-3
Rules for Addition
Addition Rules
1.
If a and b are both
positive, then
a + b = a + b
3 + 7 = 10
Addition Rules
2.
If a and b are both
negative, then
a + b = -(a + b)
(-6) + (-2) = -(6 +2) = -8
Addition Rules
3.
If a is positive and b is
negative and a has the
greater absolute value, then
a + b = a - b
6 + (-2) = (6 - 2) = 4
Addition Rules
4.
If a is positive and b is
negative and b has the
greater absolute value, then
a + b = -( b - a)
4 + (-9) = -(9 -4) = -5
Addition Rules
5.
If a and b are opposites, then
a+b=0
2 + (-2) = 0
2-4
Subtracting Real Numbers
DEFINITION of
SUBTRACTION
For all real numbers a and b,
a – b = a + (-b)
To subtract any real number,
add its opposite
Examples
1.
2.
3.
4.
5.
3 – (-4)
-y – (-y + 4)
-(f + 8)
-(-b + 6 – a)
m – (-n + 3)
2-5
The Distributive Property
DISTRIBUTIVE
PROPERTY
a(b + c) = ab + ac
(b +c)a = ba + ca
5(12 + 3) = 5•12 + 5 •3 = 75
(12 + 3)5 = 12• 5 + 3 • 5 = 75
Examples
1.
2.
3.
4.
5.
2(3x + 4)
5n + 7(n – 3)
2(x – 6) + 9
8 + 3(4 – y)
8(k + m) - 15(2k + 5m)
2-6
Rules for Multiplication
IDENTITY PROPERTY
of MULTIPLICATION
There is a unique real
number 1 such that for
every real number a,
a · 1 = a and 1 · a = a
MULTIPLICATIVE
PROPERTY OF 0
For every real number a,
a · 0 = 0 and 0 · a = 0
MULTIPLICATIVE
PROPERTY OF -1
For every real number a,
a(-1) = -a and (-1)a = -a
PROPERTY of OPPOSITES
in PRODUCTS
For all real number a and b,
-ab = (-a)(b)
and
-ab = a(-b)
Examples
1.
2.
3.
(-1)(3d – e + 8)
-6(7n – 6)
-[-4(x – y)]
2-7
Problem Solving:
Consecutive Integers
EVEN INTEGER
An integer that is the
product of 2 and any
integer.
…-6, -4, -2, 0, 2, 4, 6,…
ODD INTEGER
An integer that is not even.
…-5, -3, -1, 1, 3, 5,…
Consecutive Integers
Integers that are listed
in natural order, from
least to greatest
…,-2, -1, 0, 1, 2, …
Example
Three consecutive integers
have the sum of 24. Find all
three integers.
CONSECUTIVE EVEN
INTEGER
Integers obtained by
counting by twos beginning
with any even integer.
12, 14, 16
Example
Four consecutive even integers
have a sum of 36. Find all
four integers.
CONSECUTIVE ODD
INTEGER
Integers obtained by
counting by twos beginning
with any odd integer.
5,7,9
Example
There are three consecutive
odd integers. The largest
integer is 9 less than the
sum of the smaller two
integers. Find all three
integers.
2-8
The Reciprocal of a Real
Number
PROPERTY OF
RECIPROCALS
For each a except 0, there is a
unique real number 1/a such
that:
a · (1/a) = 1 and (1/a)· a = 1
1/a is called the reciprocal or
multiplicative inverse of a
PROPERTY of the
RECIPROCAL of the OPPOSITE
of a Number
For each a except 0,
1/-a = -1/a
The reciprocal of –a is -1/a
PROPERTY of the
RECIPROCAL of a PRODUCT
For all nonzero numbers a and
b,
1/ab = 1/a ·1/b
The reciprocal of the product of
two nonzero numbers is the
product of their reciprocals.
2-9
Dividing Real
Numbers
DEFINITION OF DIVISION
For every real number a and
every nonzero real number b,
the quotient is defined by:
a÷b = a·1/b
To divide by a nonzero number,
multiply by its reciprocal
1.
The quotient of two
positive numbers or two
negative numbers is a
positive number
-24/-3 = 8 and 24/3 = 8
2.
The quotient of two
numbers when one is
positive and the other
negative is a negative
number.
24/-3 = -8 and -24/3 = -8
PROPERTY OF DIVISION
For all real numbers a, b, and c
such that c 0,
a+b=a + b
and
c
c
c
a-b=a - b
c
c
c
Examples
4 ÷ 16
2. 8x
16
3. 5x + 25
5
1.
The End