Mathematics 96 (3581)
CA (Class Addendum) 1: Commutativity
Mt. San Jacinto College
Menifee Valley Campus
Spring 2013
__________________
Name
This class handout is worth a maximum of five (5) points. It is due no later than the end
of class on Monday, 4 February.
Important: You should study this entire handout carefully several times before you
begin the exercises it contains. You’ll probably need to study example exercises
carefully several times before you attempt the exercise sets that follow them. Also, the
order in which the exercises occur may not necessarily be the order in which you
complete them. If the solutions to a particular exercise set elude you, skip to another one.
You are being given (almost) two weeks to complete this handout because you’ll
probably need to study it, attempt some of the exercises and then take a break, continuing
with it a day or two later.
The operations of addition and multiplication are commutative. That is, the order in
which constants or variables are combined utilizing either operation does not change the
result. For example, the constants 3 and 5 add to 8 regardless of whether the 3 is written
first (3 + 5) or the 5 is written first (5 + 3). The equality of these two sums expresses the
commutativity of addition:
3 + 5 = 5 + 3.
Consider the term x and the term 3y. Because addition is commutative, the sum of these
two unlike terms is the same regardless of which of the terms appears first:
x + 3y = 3y + x.
Since multiplication is commutative, utilizing x and 3y as factors, we have:
x(3y) = (3y)x.
The following three equations express the commutativity of addition:
-2 + 5 = 5 + (-2)
(a + b) + c = c + (a + b)
□+◊ = ◊+□
The following three equations express the commutativity of multiplication:
-2 (5) = 5 (-2)
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(a + b) c = c (a + b)
□◊ = ◊□
Utilizing variables, we can express the commutativity of addition and multiplication as
follows:
The Commutative Property of Addition
If x and y are real numbers, then
x+y = y+x
(1)
The Commutative Property of Multiplication
If x and y are real numbers, then
xy = yx
(2)
Example 1. Express the commutativity of addition utilizing the terms 4a and 7b.
Solution: We are asked to display that no matter which term is written first, the resulting
sum is the same. We can complete the exercise by following the pattern displayed in
formula (1). That is, starting with formula (1), we’ll remove the placeholders x and y, and
then insert parentheses in their locations. Next, we’ll insert 4a into the parentheses in the
x location and 7b into the parentheses in the y location. Then formula (1)
x+y = y+x
becomes
(4a) + (7b) = (7b) + (4a).
Note that the parentheses are redundant. First, removing them does not change the value
of the expressions 4a + 7b or 7b + 4a. Therefore, the equation is true with or without the
parentheses, so we’ll choose the simpler form (i.e. without parentheses) for each
expression. Second, removing the parentheses still leaves a clear expression of
commutativity, as required by the exercise instructions. That is, one can (still) clearly see
that the equation 4a + 7b = 7b + 4a is an expression of the commutativity of addition
without the parentheses.
One answer is: 4a + 7b = 7b + 4a. The other answer is: 7b + 4a = 4a + 7b.
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Example 2. Express the commutativity of multiplication utilizing the factors 3u and 5v.
Solution: We are asked to display that no matter which factor is written first, the product
is the same. Since the operation is multiplication rather than addition, we will utilize
formula (2), first removing the variables x and y and inserting parentheses in their place.
Next, we’ll insert 3u into the parentheses in the x location and 5v into the parentheses in
the y location. Then formula (2)
xy = yx
becomes
(3u)(5v) = (5v)(3u).
Here, some of the parentheses are not redundant. While removing all parentheses would
still yield a true equation (the expression on each side of the equal sign would still have
the same value), removing all of them would mask the commutativity required by the
exercise instructions. Since multiplication should still be “performed” left to right (as is
the case in arithmetic), we can drop the first set of parentheses in each expression and
still retain an expression of commutativity.
One answer is: 3u(5v) = 5v(3u). The other answer is: 5v(3u) = 3u(5v).
(NOTE: As the discussion in the solution to the example exercise above indicates, you
must use grouping symbols (e.g. parentheses) whenever “following” the order of
operations would combine numbers in an order different from that intended by a
commutative law. For instance, in the expression 3u(5v), the v is to be multiplied first
while in the expression 3u5v, the v would be multiplied last. Therefore, to simply write
3u5v = 5v3u would not be a correct answer to Example 2. While true, it would not
express the commutativity of 3u multiplied to 5v, as requested.)
Exercise 1. (To receive full credit (one point), you must complete at least three of the
following four parts correctly).
a. Express the commutativity of addition utilizing the constants -9 and 8.
b. Express the commutativity of multiplication utilizing the constants -5 and 3.
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c. Express the commutativity of addition utilizing the terms 2d and 7t.
d. Express the commutativity of multiplication utilizing the factors 5 y and 6m 2 .
(Hint: See “NOTE” above!)
There is nothing special about the letters x and y utilized in formulas (1) and (2) above.
In Exercise 2 you will be asked to express the two commutative laws utilizing a variety of
symbols. Here are two examples.
Example 3. Express the commutativity of addition using the variables t and u.
Solution: One way to interpret this request is that we are being asked to express the
Commutative Property of Addition using t and u rather than x and y used in formula (1).
Therefore, removing x and y, inserting parentheses in their locations, and inserting t into
the parentheses in the x location and u into the parentheses in the y location,
x + y = y + x
becomes
(t) + (u) = (u) + (t)
Since all the parentheses are redundant, they may be removed.
One solution is: t + u = u + t. The other solution is: u + t = t + u.
Example 4. Express the commutativity of multiplication utilizing the symbols □ and ◊.
Solution: As in Example 3, we can interpret this request as a restatement of formula (2)
using □ and ◊ rather than x and y. That is, with □ substituted for x and ◊ replacing y,
xy = yx
becomes
(□) (◊) = (◊) (□)
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Again, all parentheses are redundant and may be dropped.
One solution is: □ ◊ = ◊ □. The other solution is: ◊ □ = □ ◊.
Exercise 2. (To receive full credit (one point), you must complete at least three of the
following four parts correctly).
a. Express the commutativity of addition utilizing the variables u and z.
b. Express the commutativity of multiplication utilizing the variable x and the variable
expression b - w. (Hint: Not all parentheses are redundant!)
c. Express the commutativity of addition utilizing the symbol £ and the expression β - 6.
(Hint: Not all parentheses are redundant!)
d. Express the commutativity of multiplication utilizing the expression c + 8 and the
symbol α.
(Hint: Not all parentheses are redundant!)
Notice that Exercise 2 provides four additional ways to express commutativity! That is,
(a) and (c) are simply restatements of formula (1) and (b) and (d) are just restatements of
formula (2).
In Exercise 3, you will be asked to “finish” applications of one of the two commutative
laws. In other words, as written, each equation will be missing a symbol (or two) that
you must insert to create an expression of commutativity. Here are two examples.
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Example 5. Insert the missing symbol(s) (e.g. parentheses, a constant or a variable
expression) to create an expression of commutativity.
3+4 =
+3
Solution: Since we see addition and the equal sign, it appears we are to complete an
application of the Commutative Property of Addition. The left side of the equation, 3 +
4, looks complete, in that it could already be one side of an equation that expresses
commutativity (of addition). The right side, + 3, appears to be missing something!!
That is, if we inserted the constant 4 between the equal sign the plus sign, we’d have the
expression 4 + 3. The entire equation would then become:
3 + 4 = 4 + 3.
This equation takes the form
a+b = b+a
where a corresponds to the constant 3 and b corresponds to the constant 4. But the
equation a + b = b + a states that addition is commutative (i.e. it’s really The
Commutative Law of Addition, substituting the variables a and b in formula (1) for x and
y). Therefore, inserting a 4 between the equal sign and the plus sign completes an
application of the Commutative Property of Addition and thus completes the exercise.
Example 6. Insert the missing symbol(s) (e.g. parentheses, a constant or a variable
expression) to create an expression of commutativity.
34z = (4z)3
Solution: Since the lack of any other operation symbol (e.g. a plus, minus or division
sign) implies the operation of multiplication, it appears we are to complete an application
of the Commutative Property of Multiplication. Moreover, on the right hand side of the
equation the parentheses indicate a product of the factor 4z with the factor 3. That is, it
looks like the right hand side is complete, in that it could already be one side of an
equation that expresses commutativity (of multiplication). As written, the left hand side
doesn’t appear equivalent to the right hand side. That is, 34z and (4z)3 = 12z aren’t
equivalent values (unless z = 0). Therefore, we must insert a symbol, or symbols, to
make the left hand and right hand sides not only equal but to make the entire equation an
expression of commutativity.
Placing parentheses around the 4z does the trick. By placing parentheses around the 4z
on the left hand side of the original equation, we now have
3(4z) = (4z)3
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Notice that this equation now has the form
a b =
b a
where a corresponds to the constant 3 and b corresponds to the variable expression 4z.
But the equation ab = ba states that multiplication is commutative (i.e. it’s really
formula (2) in The Commutative Law of Multiplication utilizing the variables a and b
instead of x and y). Therefore, inserting parentheses as we’ve done completes an
application of the Commutative Property of Multiplication and thus completes the
exercise.
Exercise 3. Insert the missing symbol(s) (e.g. parentheses, a constant or a variable
expression) to create an expression of commutativity. (To receive full credit (one point),
you must complete at least three of the following four parts correctly). PLEASE USE A
PENCIL OR INK OTHER THAN BLACK!
a.
q + j =
b.
(5u)(-14)
c.
(
d.
5 2
+ q
= -14 5
= 17 + (g – 9)
- 9 ) + 17
=
(2a) 5
In order to recognize an application of a real number property, it is necessary to
determine how a mathematical expression, say in an exercise, corresponds to a variable in
the formula for the property. The following examples utilize the commutative properties
to illustrate this correspondence.
Example 7. The following equation is an expression of the commutativity of addition.
4 + (5d – g) = (5d – g) + 4
Comparing it to formula (1), which expression in the equation corresponds to the variable
x in formula (1)?
Solution: The equation above and formula (1), x + y = y + x, are both expressions of the
commutativity of addition. One way to determine the correspondence requested in
Example 7 is write down both equations in a vertical format as follows:
4 + (5d – g) = (5d – g) + 4
x+
y
=
y +x
Notice that as we read both equations simultaneously as we would read a book (from left
to right), we see that the 4 in the upper equation always corresponds to the variable x in
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the lower equation. That is, both are always aligned vertically. This suggests that the
equation
4 + (5d – g) = (5d – g) + 4
takes the form of formula (1)
x+
y
=
y
+x
where the 4 corresponds to the x (and the expression 5d – g corresponds to the y).
Therefore, the answer is: 4 corresponds to x (or x corresponds to 4).
Example 8. The following equation is an expression of the commutativity of
multiplication.
9(3 – x) = (3 – x)9
Comparing it to formula (2), which expression in the equation corresponds to the variable
y in formula (2)?
Solution: The equation above and formula (2), xy = yx, are both expressions of the
commutativity of multiplication. One way to determine the correspondence requested in
Example 8 is write down both equations in a vertical format as follows:
9(3 – x) = (3 – x)9
x y = y x
Notice that as we read both equations simultaneously as we would read a book (from left
to right), we see that the expression 3 – x in the upper equation always corresponds to the
variable y in the lower equation. That is, both are always aligned vertically. This
suggests that the equation
9(3 – x) = (3 – x)9
takes the form of formula (2)
xy = yx
where the expression 3 - x corresponds to the y (and 9 corresponds to the x). Therefore,
the answer is: 3 - x corresponds to y (or y corresponds to 3 - x).
Exercise 4. To receive full credit (two points), you must complete at least three of the
following four parts correctly. You’ll receive one point if you complete two parts
correctly. Completing less than two parts correctly will earn zero points.
a. The following equation is an expression of the commutativity of addition.
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k + 5z = 5z + k
Comparing it to formula (1), which expression in the equation corresponds to the variable
x in formula (1)?
b. The following equation is an expression of the commutativity of multiplication.
4(v - a) = (v – a)4
Comparing it to formula (2), which expression in the equation corresponds to the variable
y in formula (2)?
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c. The following equation is an expression of the commutativity of addition.
(g + 6) + (3 – n) = (3 – n) + (g + 6)
Comparing it to formula (1), which expression in the equation corresponds to the variable
y in formula (1)?
d. The following equation is an expression of the commutativity of multiplication.
(11e)(p + r) = (p + r)(11e)
Comparing it to formula (2), which expression in the equation corresponds to the variable
x in formula (2)?
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