Download Working With Real Numbers

Working With Real Numbers
Algebra I
Name ______________
Date ______________
2.1-2.4: 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
Unique
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
d. 0.8 + 3. 7 + 0.2 + 5.3
b. 4  7  25  3
e. 6 + 8n + 4 + 7n
c. 1 13  16 54  2 23  3 15
f.
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)
(3w)(2x)(4y)(5z)
Examples
Rules for Addition
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.
Examples
Example 2 Simplify
a. 6 + 2
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)
e. –(x + 2)
b. 56 – (45 – 32)
f. –(b – 6)
c. (32- 24) – (-6 – 9)
g. 6 – (y + 4)
d. 3 – 4 + 7 – 15 + 21
h. x – (x – 2)
2. 5 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
Matrices Operations
A matrix is identified by the number of rows and the number of columns. For example the matrix
 3 7 1
 2 0 8  is represented as 2 x 3, as there are 2 rows and 3 columns.


Each number in the matrix is called the element.
Adding and Subtracting Matrices
To add or subtract a matrix, simply add or subtract each element in the first matrix by its corresponding
element in the other matrix. Matrices must be the same size (rows x columns) in order to add or
subtract them. The answer matrix will be the same size as the matrices in the original problem.
 21 12   37 11
Example 1 Find the sum of the matrices: 


15 25  22 18 
 21 12  37 11
Example 2 Find the difference of the matrices: 


15 25  22 18 
Multiplying Matrices by a Scalar
To multiply a matrix by a scalar, simply multiply each element by the scalar. The answer
matrix will be the same size as the matrix in the problem.
 6 2 
Example 3 If A = 
 , find 2A.
 3 1
Example 4 The following matrix represents the sales total, in dollars, of two pet store owners
at their respective shops. If there is a 5% sales tax, how much tax does will there be for each
store for each pet?
Ralph Doug
Iguanas
 25 14 
17 13 
Rats


Parakeets  33 11


2. 6, 2.7 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)
2. 8 Division: To simplify expressions involving reciprocals, divide real numbers and to
simplify expressions involving quotients
Reciprocals: Two numbers whose product is 1 are called reciprocals.
Properties
Property of Reciprocals
Every nonzero real number a has a reciprocal
1
1
1
a , such that a  a  1 and a  a  1
Examples
Property of the Reciprocal of the Opposite
of a Number
For every nonzero number a.
1
a
  1a
Property of the Reciprocal of a Product
For all nonzero numbers a and b.
1
ab
 1a  b1
Example 1 Simplify
a. (-42)(  17 )
d. 72ab19 
(24)( 14 )
e.
 42ac  17 
c.  60 12 13 
f.
1
2
b.
1
2
(8a  10)
g.  14 (24 g  32h)
h.
 26e  52 f  131 
i.
 40 x  56 y  18 
j.
 5a  30b 15 
Definition of Division
To divide by a nonzero real number b, multiply by the reciprocal of b.
a  b  ba  a  b1
Rules for Division
If two numbers have the same sign, their quotient is positive.
If two numbers have opposite signs, their quotient is negative
CAUTION!!
You can’t divide by zero since zero has no reciprocal
Division is not commutative
Divisions is not associative
Example 2 Simplify.
a. 42 ÷ 14
g.
8
 18
h.
36 x
6
d. 12  ( 14 )
i.
 10 x2 
e. 0   72 
j.
144b
12
 20
k.
w
8
b. -32 ÷ (-8)
c.
f.
100
5
1
5
Example 3 Find the average of the given numbers.
a. -12, 5, -10, -7
b. 15, -21, -8, 6
8