Download Multiplication with Integers

Math 952
2.4 "Multiplication, Division, and Order of Operations with Integers"
Objectives:
*
Know how to multiply integers.
*
Know how to divide integers.
*
Understand that division by 0 is not de…ned.
*
Be able to apply the rules for order of operations to expressions with integers.
Preliminaries:
Recall that multiplication with whole numbers is a shorthand form of repeated addition. We can use this same idea to
multiply integers.
For example:
Repeated addition with positive integers,
Repeated addition with negative integers,
6 + 6 + 6 + 6 + 6 = 5 6 = 30
( 3) + ( 3) + ( 3) + ( 3) = 4 ( 3)
Note: The example above shows that the product of a positive integer and a negative integer is
To multiply two negative integers, we use the fact that
For example:
a=
:
1 (a) :
( 2) ( 5) =
Note: The example above shows that the product of two negative integers is
:
Multiplication with Integers
Rules for Multiplication with Integers:
If a and b are two positive integers, then
1. The product of two positive integers is positive:
:
2. The product of two negative integers is positive:
:
3. The product of a positive integer and a negative integer is negative:
and
4. The product of 0 and any integer is 0:
NOTE:
and
:
:
When multiplying integers we can use the commutative and associative laws. Thus, to …nd the product of more
than two integers, we can multiply any two and continue to multiply until all the integers have been multiplied.
Page: 1
Notes by Bibiana Lopez
Prealgebra by Franklin Wright
2.4
Example 1: (Multiplying two integers)
Find each product:
a) 5 ( 3)
b) 8 (4)
c)
19 (0)
d)
10 ( 2)
Example 2: (Multiplying more than two integers using the commutative and associative laws).
Find each product:
a) 3 ( 20) ( 1) ( 1)
b) 9 (7) ( 2)
3
c) ( 2) ( 2) ( 2) (3)
0
3
2
d) ( 5) ( 1) ( 1) ( 1)
Division with Integers
In this section we’ll indicate division such as a
b in the fraction form,
a
:
b
a
=x:
b
For integers a, b, and x (where b 6= 0) we have
Another meaning of
Example 3: (Dividing integers by expressing division as multiplication)
Find each quotient.
30
+ 40
+32
b)
c)
a)
+ 10
5
+8
Page: 2
d)
35
5
Notes by Bibiana Lopez
Prealgebra by Franklin Wright
2.4
Rules for Division with Integers:
If a and b are positive integers, then
1. The quotient of two positive integers is positive:
:
2. The quotient of two negative integers is positive:
:
3. The quotient of a positive integer and a negative integer is negative:
and
In summary:
When the signs are the same, the quotient is
:
:
When the signs are not the same, the quotient is
:
Division by 0 is Not De…ned:
The following mnemonics can help us remember the rules for division by 0.
0
is "0K". 0 can be in the numerator.
K
k
is "K0" (knockout). 0 cannot be in the denominator.
0
Example 4: (Dividing integers)
Find the following quotients (if possible).
39 + 3
a)
b)
3
Example 5:
If the quotient of
28
4
c)
4
56 and 8 is subtracted from the product of
12 and
Page: 3
0
7
d)
35
0
3, what is the di¤erence?
Notes by Bibiana Lopez
Prealgebra by Franklin Wright
2.4
Rules of Order of Operation
Rules for Order of Operation
1: Do all calculations within grouping symbols before operations outside (P )
2: Evaluate all exponential expressions (E)
3: Do all multiplications and divisions in order from left to right (M D)
4: Do all additions and subtractions in order from left to right (AS)
"P EM DAS"
Example 6: (Using the rules of order of operations)
Find the value of each expression by following the rules for order of operation.
a) 25 ( 8) 2 7 + 4 ( 3)
b) 9 12 5 32
c) 2
2 + 32
6
5 23
d) (7
Page: 4
10) [49
2+5
( 7) + 20 3
4 15
( 10)]
Notes by Bibiana Lopez