3.
Multiplicative Counting Skills and Practices
How many elements are there in a finite set of equi-sized groups?
Product Rule - Counting or Denoting How Many in Equal-Sized Groups
The number of elements in M groups, each of size N, gives a count denoted
by M × N and called M times N or the product of M and N.
For example 1 times N is N and 2 × N is N +N, 3 × N is N+N+N. We will
see later how to denote counts by decimals and how to mutliply
decimals.
Commutative Law for Multiplication
Geometric Reason for 5 × 3 = 3 × 5
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5 columns (groups) of three pluses
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can also be viewed of 3 rows (groups) of 5 pluses without losing
or gaining pluses - the tally marks. Therefore 5 × 3 = 3 × 5 =
two ways of counting them. Here we use the assumption that
any two ways of counting the same set of elements must
yield the same number.
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Geometric Reason for Product Commutative Law M × N = N × M
Now if M groups, each of size of N, are stacked in N equi-height
columns, where each group element occupies a single square in that
column, then all squares being the same size, we obtain a
rectangle. Now we divide the total number present into N rows
(groups) of M elements each. So that implies the total count or tally
initially given by M × N is also given by N × M.
The foregoing as is or recast implies the commutative law of
multiplication namely
Commutative Law: N × M =M × N whenever M and N are whole
numbers.
Students of pure mathematics can recast the foregoing
into set theory form where the number of elements in the Cartesian
product A × B of two sets is M × N if A and B have M and N elements,
respectively. The number of elements in the Cartesian product defines
M × N.
Associative Law for Multiplication
In general, for any three whole numbers M, N and P,
N(MP) = (NM)P)
Proof: Suppose we divide a set of cubes T = N(MP) cubes into
N one-cube high layers of M by P cubes, and stack the layers. The result
is a rectangular box of cubes N cubes high, M cubes wide and P
cubes long. Now we may recount the by viewing the box
as NM columns, each P cubes long.. That yield T = (NM)P
as well.
Corollary: N(MP) = (NM)P = (MN)P = M(NP) =
P(MN) = (PM)N
We can take the value of NMP to be the common value of the six ways to
compute the number of small boxes (cubes with sides of unit length) in a
larger box with dimensions N units, M units and P units.
Remark. In more advanced mathematics, the foregoing may be
rewritten algebraically to give a formal proof. By mathematically
induction, we may argue that a product of r whole number factors yields
a single result independent of the order and grouping. Thus
grouping and regrouping view as different ways to count a r-dimensional
hyper-box implies order of multiplication does not affect
results.
Consequences of Associative Law:
In our discussion of decimal methods below, we will need the property
that
(a 10m) × (b 10n) = (ab)
10m+n
That follows as (a 10m) × (b
10n)
= (a × 10m) × (b 10n) due to a change in
notation
= a × (10m × (b 10n)) due to a 1st
use of associative law
= a × (b × (10m x 10n)) due
to a 2nd use of associative law or six ways to compute the product
10m × (b 10n)
= a × (b × (10m+n)) as 10m+n
=10m x 10n
= (a b) x
10m+n due to a 3rd use of associative law
= (a b) × 10m+n due to a change in notation
Note: Some authors might mention the commutative law in going
from one way to compute the product 10m × (b
10n)to another.
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