Download IMA101-Lecture2

IMA101 Basic Math
LECTURE 2
OCTOBER 28, 2010
Lecture Outline
2
 Whole numbers (continued)
 Multiplication
 Division
 Order of operations
 Integers
 Negative numbers
 Absolute value
 Ordered values
IMA101 1/2010- Lecture 2
October 28, 2010
Multiplication (continued)
3
IMA101 1/2010- Lecture 2
October 28, 2010
Multiplication Examples
4
Times 10
Times a number with 2
digits
 13 * 10 =
 435 *23 =
 13 * 100 =
 253 * 406
 13 * 1,000 =
IMA101 1/2010- Lecture 1
October 26, 2010
Division
5
IMA101 1/2010- Lecture 2
October 28, 2010
Division: the opposite of multiplication
6
 Take a number of items (24) and DIVIDE them into
a number of groups of a certain size. How many
groups are there?
 24 divided by 4 = 6

4 groups of 6
IMA101 1/2010- Lecture 1
October 26, 2010
Division: Symbols
7
Example
Symbol
÷
/
division sign
12 ÷ 4
long division
4 12
fraction bar
12
4
1242
23
12/4
1242/23
back-slash(typed)
IMA101 1/2010- Lecture 1
1242 ÷ 23
23 1242
October 26, 2010
Division: properties
8
Zero
One
 0 divided by any
 Any number divided by
number is 0

0/4=0
 Any number divided by
zero is undefined


52 / 0 is undefined
0/0 is undefined
IMA101 1/2010- Lecture 1
1 is that number

14 /1 = 14
 Any nonzero number
divided by itself is
equal to 1

14/14 = 1
October 26, 2010
Division: Examples
9
Long division
Numbers ending in zero
 246 / 6 =
 80 / 10 =
 With a remainder
 169/7 =
 168/5 =
 40,700 / 100 =
IMA101 1/2010- Lecture 1
October 26, 2010
Tests for Divisibility: A number is divisible by
10
 2 if

It’s last digit is divisible by 2

Example: 45,692
 3 if

the sum of its digits is divisible by 3

Example: 29,874
 4 if

it’s last 2 digits form a number divisible by 4

Example: 8,316
 5 if

It’s last digit is a 0 or 5
 10 if

It’s last digit is 0
IMA101 1/2010- Lecture 1
October 26, 2010
Division and Multiplication Table
11
11
110
2
4
6
18
3
10
24
40
5
25
8
7
60
56
96
63
4
9
12
IMA101 1/2010- Lecture 1
54
108
132
October 26, 2010
Division and Multiplication Table: Answers
12
4
9
7
3
5
11
8
6
10
2
12
11
44
99
77
33
55
121
88
66
110
22
132
2
8
18
14
6
10
22
16
12
20
4
24
6
24
54
42
18
30
66
48
36
60
12
72
3
12
27
21
9
15
33
24
18
30
6
36
10
40
90
70
30
50
110
80
60
100
20
120
5
20
45
35
15
25
55
40
30
50
10
60
8
32
72
56
24
40
88
64
48
80
16
96
7
28
63
49
21
35
77
56
42
70
14
84
4
16
36
28
12
20
44
32
24
40
8
48
9
36
81
63
27
45
99
72
54
90
18
108
12
48
108
84
36
60
132
96
72
120
24
144
IMA101 1/2010- Lecture 1
October 26, 2010
Prime Factors and Exponents
13
 Factors are:
 Numbers multiplied together

2 * 6 = 12, so 2 and 6 are factors of 12
 Prime number:
 A whole number that is greater than 1 and has only 1 and itself
as factors
 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, …
The only even prime number is 2
 There is no pattern to prime numbers

 Composite number:
 non-prime whole numbers
IMA101 1/2010- Lecture 1
October 26, 2010
Prime factorizations
14
 Write 90 as a product of only prime numbers
 Different paths can lead to the same result
 What are the prime factors of 17640?
 IMPORTANT:

Any composite number has exactly one set of prime factors
IMA101 1/2010- Lecture 1
October 26, 2010
Exponents
15
 Used to represent repeated factors
 The exponent tells us how many times a base is used
as a factor
 Base = 2, exponent = 3

23 = 2 * 2 * 2
IMA101 1/2010- Lecture 1
October 26, 2010
Order of Operations
16
First do all calculations within parentheses:

1.
2.
3.
When all grouping symbols are removed:



Evaluate all powers
Do multiplications and divisions as they occur from left to
right
Do all additions and subtractions as they occur from left to
right
Repeat steps 1 -3 as before
If a fraction bar is present, do the operations above
and below separately, then divide the number
IMA101 1/2010- Lecture 1
October 26, 2010
Order of Operations: Examples
17
2(13) - 2
3
3(2 )
IMA101 1/2010- Lecture 1
October 26, 2010
Order of Operations: Examples
18
(4  2)  7
5(2  4)  7
3
IMA101 1/2010- Lecture 1
October 26, 2010
Quiz!
19
 Please take out a sheet of paper and solve the
following:
 What are the prime factors of 32760?

write in exponent form
 (13877  2503) 
 Solve 100  

(
5642

5462
)


3
IMA101 1/2010- Lecture 1
October 26, 2010
Integers
20
LET’S NOT BE TOO NEGATIVE
IMA101 1/2010- Lecture 2
October 28, 2010
Extending the number line
21
 Positive and negative numbers
 Zero is neither
 As value of number gets higher, negative numbers
get smaller
 Examples of negative numbers:
 Temperature, debt, elevation
IMA101 1/2010- Lecture 2
October 28, 2010
Ordering of numbers and inequality
22
Meaning
Symbol
IMA101 1/2010- Lecture 2
<
Less than
>
Greater than
≤
Less than or equal
≥
Greater than or equal
≠
Does not equal
October 28, 2010
Absolute Value
23
 The distance from a number to zero
 |-4| = |4| = 4
 On the number line:
IMA101 1/2010- Lecture 2
October 28, 2010
Opposite
24
 The negative of a number is the point on the number
line the same distance from zero but on the other
side of zero
 On the number line:
 Negative of 4 is -4, and the negative of -4 is 4
IMA101 1/2010- Lecture 2
October 28, 2010
The “-” symbol
25
 Can be used to show a
–1
 Can be used to show the
–(-1) = 1
number less than zero
negative of a number
 Can be used to subtract
numbers
 equivalent to
multiplying by negative
one
IMA101 1/2010- Lecture 2
or –(1) = –1
5 – 4 = 1 or 5 – (–4) = 9
5–4
= 5 + (–4)
=5 + (–1*4)
October 28, 2010
Integer Addition
26
 If they are both positive: same as before
 If they are both negative: same as before and make
negative
 Examples can be seen on the number line

4 + 6 = 10

-4 + (-6) = -10
IMA101 1/2010- Lecture 2
October 28, 2010
Integer Addition
27
 If one is positive and the other is negative:
 Which has a smaller absolute value?
 Examples: 7 + (-5)


If the number with the larger absolute value is positive, then the
answer will be positive. On number line)
Example: 5 + (-7)

If the number with the larger absolute value is negative, then the
answer will be negative. On number line)
IMA101 1/2010- Lecture 2
October 28, 2010
Integer Subtraction
28
 Analogous to adding the negative of a number
 Examples:




6 – 7 = 6 + (-7) =
6 – (-7) = 6 + 7 =
-6 – 7 = -6 + (-7) =
-6 –(-7) = -6 + 7 =
IMA101 1/2010- Lecture 2
October 28, 2010