Working With Real Numbers
Algebra I
Name ______________
Date ______________
Basic Assumptions, Addition and Subtraction: To use number properties to simplify expressions,
add real numbers using a number line or properties about opposites, add real numbers using rules for addition,
subtract real numbers and to simplify expressions involving differences.
Vocabulary
Terms
Factors
Opposite Signs
Properties of Real Numbers
Closure Properties
The sum and product of any two real
numbers are also real numbers and
they are unique
Commutative Properties
The order in which you add or multiply
any two real numbers does not affect
the result
Associative Properties
When you add or multiply any three
real numbers, the grouping (or
association) of the numbers does not
affect the answer
Addition
Multiplication
Example 1 Simplify each of the following.
a. 75 + 13 + 25 + 47
f.
b. 4  7  25  3
c. 1 13  16 54  2 23  3 15
d. 0.8 + 3. 7 + 0.2 + 5.3
e. 6 + 8n + 4 + 7n
(3w)(2x)(4y)(5z)
Properties
Identity Property of Addition
The sum of a real number and 0 is identical the number itself
a + 0 = a and 0 + a = a
Properties of Opposites
Every real number has an opposite. The sum of a real
number and its opposite is 0.
a + (-a) = 0 and (-a) + a = 0
Property of the Opposite of a Sum
For all real numbers a and b:
-(a + b) = (-a) + (-b)
Rules for Addition
Examples
If two numbers have the same sign, add their absolute
values and put their common sign before the result.
If two numbers have opposite signs, subtract the lesser
absolute value from the greater and put the sign of the
number having the greater absolute value before the
result.
If two numbers are opposites, then their sum is zero.
Example 2 Simplify
a. 6 + 2
Examples
d. (-8 + 5) + 2
b. -4 + -7
e. – 4 + (-14) + 4
c. -3 + 3
f. -3 + (-9) + 7 + (-5)
Example 3 Simplify
a. -2 + x + (-6) + 3
b. -5 + 2a + 3 + (-3)
c. 17 + 8b + (-15) + (-10)
d. –(-7) + 3y + (-6) + 4
Example 4 Evaluate each expression if x = -2, y = 5 and z = -3
a. y + z + (-2)
c. 1 + (-y) + x
b. -11 + (-x) + (-y)
d. –x + (-y) + (-15)
Definition of Subtraction
To subtract a real bumber b, add the opposite of b.
a – b = a + (-b)
Example 5 Simplify
a. -8 – (-3)
b. 56 – (45 – 32)
f. –(b – 6)
c. (32- 24) – (-6 – 9)
g. 6 – (y + 4)
d. 3 – 4 + 7 – 15 + 21
e. –(x + 2)
h. x – (x – 2)
Distributive Property: To use the distributive property
Distributive Property
Distributive Property (with respect to addition)
a(b + c) = ab + ac
OR
(b + c)a = ba + ca
Distributive Property (with respect to subtraction)
a(b - c) = ab - ac
OR
(b - c)a = ba - ca
Example
Example 1 Distribute the following.
a. 3(6n + 2)
c. (3x + 4)5
b. 8(5n – 3)
d. (3x – 4y)8
Example 2 Simplify the following
a. 6a + 4a
e. 7n + 1 + 3n
b. 15y - 6y
f. 3x + 8 – 2x
c. -4n + 9n
g. 10n – 7 + 6n
d. 2a + 9a – 5a
Multiplication: To multiply real numbers and to write equations to represent relationships
among integers
Properties
Identity Property of Multiplication
The product of a number and 1 is identical to the
number itself.
a1=a
and
1a=a
Multiplication Property of Zero
When one of the factors of a product is zero, the
product itself is zero.
a0=0
and
0a=0
Multiplication Property of -1
For every real number a:
a(-1) = -a
and
(-1)a = -a
Property of Opposites in Products
For all real numbers a and b:
(-a)(b) = -ab
a(-b) = -ab
(-a)(-b) = ab
Examples
Rules for Multiplication
1. If two numbers have the same sign, their product is ______________.
If two numbers have opposite signs, their product is ______________.
2. The product of an even number of negative numbers is _____________.
The product of an odd number of negative numbers is _____________.
Example 1 Simplify
a. (-12)(-3)
f. (-4e)(7f)
b. (4)(-7)(10)
g. -7a +(-8a)
c. 5(-2)(-8)(-5)
h. -6(x – 2y)
d. (-3a)(-4b)
i.
6x – 2(x + 3)
e. 2p(-5q)
j.
(-1)(2x – y – 3)