MAE3811Sp. 2010 Mahoney1
Im worried Im not going to sleep for the next three
days
So last time we learned about
The set of _ numbers or _,
Where
Special notes about positive and negatives:
MAE3811Sp. 2010 Mahoney2
How did we add and multiply:
To Add:
To W
MAE3811Sp. 2010 Mahoney1
From Last time. Division In Depth
Recall, it was recent, that for any integers a and b, with b not equal to 0, a b = c if
and only if c is the unique whole number such that bc = a.
We say b divides a, written b | a, if, and only i
MAE3811Sp. 2010 Mahoney1
Division for the Integers
It is defined exactly the same as it was for whole numbers
WHOLE #: Formally, for any whole numbers a and b, with b not equal to 0, a b = c
if and only if c is the unique whole number such that bc = a.
IN
MAE3811Sp. 2010 Mahoney1
Last Time Multiplication (Number Line Model part III)
Example 1:
Move East at 2 mph
4 hours from now
Example 2:
Move East at 3 mph
5 hours ago
Example 3:
Move West at 2 mph
5 hour from now
Example 4:
Moving West at 3 mph
4 hours a
MAE3811Sp. 2010 Mahoney1
So far.
Weve seen how multiplication can be thought of as repeated additions
or as the area of a rectangular array
How the area model of multiplication implies the distribution property
How the whole numbers really arent that good
MAE3811Sp. 2010 Mahoney1
So far.
Weve seen how multiplication can be thought of as repeated additions
or as the area of a rectangular array
Today well discuss distribution and division
MAE3811Sp. 2010 Mahoney2
Distribution
The distribution property over
MAE3811Sp. 2010 Mahoney1
So far.
We looked at pencil and paper algorithms for addition, they rely on our
numerical systems place holder properties
Weve discussed the group properties and weve looked at two operations on
the whole numbers.
Addition has
MAE3811–Sp. 2010 Mahoney–1
Last Time….
• We defined Subtraction in terms of addition [and order]
• We looked at ways to interpret subtraction as “Take Away” and “What to add”
• We looked at pencil and paper algorithms for addition
Today we’ll look at penc
MAE3811Sp. 2010 Mahoney1
Computation
We will now spend a lot of time looking out how computation is performed. Time
willing I will also describe the abstract side of computation and algebra. So now is
a good time to review the Group Properties.
A Binary O
MAE3811Sp. 2010 Mahoney2
Lets practice that some more
Lets start simpler
To convert any number from base to another we need to know the place value in
both bases:
Table of common base values:
System
Base
b6
b5
b4
b3
b2
b1
b0
Binary
2
64
32
16
8
4
2
1
Quin
MAE3811Sp. 2010 Mahoney1
Recap.
So last time we finished on this slide
All modern number systems are modeled and generalized off the Hindu-Arabic,
and the Mayan and Babylonian systems would technically fit in this category.
All require.
A Base
A set of s
MAE3811Sp. 2010 Mahoney1
Recap.
So last time we introduced number systems and looked at some examples.
Number systems are extensions of [everyday] language that allow us to
count things. They consists of symbols and rules (properties).
They initially on
MAE3811Sp. 2010 Mahoney1
First, Some Leftovers.
MAE3811Sp. 2010 Mahoney2
Example: A pollster interviewed 900 university seniors who owned credit
cards. She reported that 550 owned a gold card, 570 owned a platinum card,
and 560 owned a standard card. Of t
MAE3811Sp. 2010 Mahoney1
Last time we learned about set operations and the master set diagrams.
Recall the two set Master Diagram:
We can write each region as the intersection of A, B, and their compliments.
We can also write A and B as the union of certa
MAE3811Sp. 2010 Mahoney1
How many subsets does a set have?
Remember, the empty set is always a subset, in fact its usually a _
subset.
Is a set a subset of itself?
Consider the set
= cfw_1, 2, 3. Let S be a subset of this set. Lets examine
the possibiliti
MAE3811Sp. 2010 Mahoney1
Last time we talked about Arguments and Quantifiers
All Chihuahuas are dogs.
Pepper is a Chihuahua.
Therefore, Pepper is a dog.
In the broad sense, the Formal logic weve just learned is considered to be a
Foundation of Mathematics
MAE 3811 Sp. 2010 Mahoney
So far we have avoided and important but tricky logical idea. To
illustrate these idea better, each of these connectives will have
pictures to accompany it.
Quantifies include the words/phrases
All
Every
MAE 3811 Sp. 2010 Mahon
MAE 3811 Sp. 2010 Mahoney
Last time we saw that the following pairs of compound statements were
equivalent. The last town are known as De Morgans Laws
We can use these laws like formulas to change one compound statement in
one form to another
MAE 3811 Sp.
MAE3811Sp. 2010 Mahoney1
Divisibility Test for 2 (Even Numbers)
Theorem: An integer is divisible by 2 if, and only if, its units digit is divisible by 2
that is
Proof:
MAE3811Sp. 2010 Mahoney2
Divisibility Test for 2, Concrete Example
Lets examine the pro
MAE3811Sp. 2010 Mahoney1
Divisibility Test for 3 and 9 , part 1
Is 333 divisible by 3? Lets check the old fashioned way.
Is 333 divisible by 9? Again, Lets check the old fashioned way.
MAE3811Sp. 2010 Mahoney2
Divisibility Test for 3 and 9 , part 2
Observ
MAE3811Sp. 2010 Mahoney1
Prime Numbers and Composite Numbers
Theorem: There are infinitely many primes.
I think well skip the proof youve all been such good sports!
Fundamental Theorem of Arithmetic: Each _ number can be written
as a product of _ in one,
MAE3811Sp. 2010 Mahoney1
Wrapping the Integers around the Clock
Last time we saw that the clock, with its special clock addition, behaved like the
integers. Both are _ because you can do simple algebra on them
both. We can wrap the integers around the clo
MAE3811Sp. 2010 Mahoney1
A Clock
Lets say there is only one hand on the clock,
the hour hand. Ill draw it when its needed.
If its 6 Oclock Now.
5 hours from now it will be.?
How could we write that as an arithmetic expression?
5 hours ago it was?
How coul
MAE3811Sp. 2010 Mahoney5
The GCD Euclidean Algorithm Practical
The last slide was the proof and origin of the original long division algorithm. The ancient
mathematician Euclid invented it to determine the GCD of two numbers with out having to
figure out
MAE3811Sp. 2010 Mahoney1
Determining if a Number is Prime [More]
The Sieve of Eratosthenes
This is Tedious. Thats why it was last!
This is an ancient algorithm for determining if a number is prime. You need a big
table first.
Locate the first Prime
number
MAE3811Sp. 2010 Mahoney1
Prime Numbers and Composite Numbers
Theorem: There are infinitely many primes.
I think well skip the proof youve all been such good sports!
Fundamental Theorem of Arithmetic: Each _ number can be written
as a product of _ in one,
MAE3811Sp. 2010 Mahoney1
Divisibility Test for 3 and 9 , part 1
Is 333 divisible by 3? Lets check the old fashioned way.
Is 333 divisible by 9? Again, Lets check the old fashioned way.
MAE3811Sp. 2010 Mahoney2
Divisibility Test for 3 and 9 , part 2
Observ
MAE3811Sp. 2010 Mahoney1
Divisibility Test for 2 (Even Numbers)
Theorem: An integer is divisible by 2 if, and only if, its units digit is divisible by 2
that is
Proof:
MAE3811Sp. 2010 Mahoney2
Divisibility Test for 2, Concrete Example
Lets examine the pro
MAE3811Sp. 2010 Mahoney1
From Last time. Division In Depth
Recall, it was recent, that for any integers a and b, with b not equal to 0, a b = c if
and only if c is the unique whole number such that bc = a.
We say b divides a, written b | a, if, and only i
MAE3811Sp. 2010 Mahoney1
Division for the Integers
It is defined exactly the same as it was for whole numbers
WHOLE #: Formally, for any whole numbers a and b, with b not equal to 0, a b = c
if and only if c is the unique whole number such that bc = a.
IN