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Visual Day Models of
Number Systems
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Todays Travel Plans
What did we talk about yesterday?
A Visual Model for the Whole Numbers and
Operations
A Visual Model for the Integers and
Operations
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What did we talk about yesterday?
We saw a really abstract way of defining the whole
numbers in terms of sets.
How did we do that exactly?
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What did we talk about yesterday?
How did we define the operation of
+
-
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A Visual Model of the Whole Numbers
So from our definition yesterday we know that 0 = { }
Visually, Let 0 be represented by empty space.
From yesterday, 1 = {0}. What is the bottom line though?
Is this set the only representation of 1?
What does a set need to have in order to be a representation
of the number 1?
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A Visual Model of the Whole Numbers
We can let a single block represent the value 1.
Therefore, two blocks represents 2
and three blocks represent 3
and so on and so forth
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A VM for W#s: Operations - Addition
How do we model addition in this context?
a + b is modeled as the merging of two sets of blocks, one
representing a and the other representing b, into a single set
of blocks.
The number of blocks in the merged sets is the sum a + b
Examples
What are we really doing when grouping the two sets of
blocks into one set, were taking a Union.
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A VM for W#s: Operations Addition Prop.
What properties does Addition have?
If you have two sets of blocks does it matter what order they are
in when you merge them?
If you have three sets of blocks that you intend to merge, does it
matter which two sets you choose to merge first?
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A VM for W#s: Operations Addition Prop.
Can you merge a set a blocks with the set of empty space?
If so, what would the result be?
The last three properties are known in order as
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A VM for W#s: Operations - Multiplication
How do we model multiplication in this context?
Recall the definition of multiplication from yesterday
What does this statement mean in terms of our diagrams?
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A VM for W#s: Operations - Multiplication
Example:
Observe the blocks form a rectangle that is ___ blocks high and ___
blocks across.
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A VM for W#s: Operations - Multiplication
An alternative definition of multiplication now follows
Why would we want to define multiplication again?
If a and b are whole numbers then
a b = Number of blocks, or ______, in a rectangle a blocks
high and b blocks across.
The main point here is that for the whole numbers,
multiplication is synonymous with finding ____________.
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A VM for W#s: Operations Mult. Prop.
What properties does multiplication have?
Is multiplication of whole numbers commutative?
Is it associative?
Is there an identity property?
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A VM for W#s: Operations Mult. Prop.
Do Multiplication and Addition get along?
This is known as the ________________ property of
multiplication over addition [for Whole numbers].
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A VM for W#s: Operations Subtraction
Using our visual mode we can define subtraction with
out resorting directly to the definition of addition (i.e.
and inverse process)
This is known as the ______________ or ______________ Model of
subtraction.
Example:
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A Visual Model of the Integers
The integers are the set
Integers extend the whole numbers
can Integers be formally defined by using special sets
of whole numbers [I may show you it later]
Well consider a visual construction
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A VM of the Integers Via M&Ms
Suppose you have a container of X-mas M&Ms. What
colors are the colorful candies?
Suppose the bag of M&Ms was jumbo sized and you
wanted to know which color M&M occurred more in
that bag with out counting them. How could you do it.
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A VM of the Integers Via M&Ms
Lets say you have a scale and youre going to weigh green
M&Ms on the right against red M&Ms on the left.
Observe the scale doesnt have units. Thats not a problem
because the only thing we are weighing is M&Ms.
Note: On this scale the weight of just one color of M&M in this
case is the exact number of M&Ms on the scale.
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A VM of the Integers Via M&Ms
How does the weight of one red M&M compare with one green
M&M? [Use common sense here.]
What will tip the scale to the right?
What will tip the scale to the left?
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A VM of the Integers Via M&Ms
The following all have net weight 0.
The following all have net weight 1 Green M&M
The following all have net weight 1 Red M&M
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A VM of the Integers M&M Equivalence
We can say two containers of M&Ms are equivalent if and only if
they have the _________ net weight of Red or Green M&Ms. Thus
As a consequence, every container of M&Ms is equivalent to a
container that has only ___________ color of M&M inside it. Thus
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A VM of the Integers M&M Paranoia
Suppose we believed that
Green M&Ms were good (because it counts as eating your
Greens?)
Red M&Ms were bad (because red dyes often turn out to be
carcinogenic?)
If a container of M&Ms had more Red then Green then emotionally
we might feel ______________ so the net weight in this case
would be a _______________ value.
On the other hand, if a container of M&Ms had more Green then
Red then emotionally we might feel ________________ so the net
weight in this case would be a __________________ value.
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A VM of the Integers M&M Paranoia
Suppose we believed that
Green M&Ms were good (because it counts as eating your
Greens?)
Red M&Ms were bad (because red dyes often turn out to be
carcinogenic?)
If a container of M&Ms had more Red then Green then emotionally
we might feel ______________ so the net weight in this case
would be a _______________ value.
On the other hand, if a container of M&Ms had more Green then
Red then emotionally we might feel ________________ so the net
weight in this case would be a __________________ value.
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A VM of the Integers M&M Integers
Let
Observe that there are two copies of the whole numbers in
the integers.
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A VM of the Integers Addition
When Adding the same type
When Adding different types
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A VM of the Integers Subtraction
Define Subtraction as Taking Away (as in the Whole numbers)
Example 1:
Example 2:
Example 3:
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A VM of the Integers Upgrade!
The Integers inherit all the properties of the whole numbers and
gain two new ones
Additive Inverses: For every a in
that
there is an element ____ such
Example:
Subtraction is Closed and Identical to Addition of Opposites
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