Transcript Slide 1

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 2
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross
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Order
× ÷ of + Operations
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© 2007 Herbert I. Gross
Review
In our first presentation we looked at a
simple one operation “recipe” such as…
The “Feet to Inches” Recipe
Input the number of feet
Multiply by 12 (inches per foot)
The output is the number of inches.
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© 2007 Herbert I. Gross
The instructions are unambiguous. In
terms of a formula we let I denote the
number of inches and F denote the number
of feet; and the formula becomes I = 12 × F.
If we are given the number of feet, say 5,
and want to find the number of inches we
obtain the direct computation I = 12 × 5.
And if we are given the number of inches,
say 72, and want to find the corresponding
number of feet, the formula becomes the
indirect computation 72 = 12 × F, which we
paraphrase as the direct computation
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F = 72 ÷ 12
© 2007 Herbert I. Gross
However, as soon as more than one
operation is contained in the “recipe”, it
becomes a bit more “tricky” to express
the recipe in terms of a formula.
For example, in the previous lesson we
wrote the recipe for finding the cost of
6 pounds of candy from a catalog in which
the price was $5 per pound plus a
one-time $4 charge for shipping and
handling. Written in recipe format we
computed the total cost as follows…
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© 2007 Herbert I. Gross
Start with 6 as the input (the number of
pounds).
Multiply by 5 (the cost per pound).
Add 4 (the shipping and handling).
The output is the cost in dollars
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© 2007 Herbert I. Gross
So far there is no ambiguity because the
recipe gives us the exact sequence of
steps to be followed. The same thing is
true if we use a calculator and are given
the exact sequence of key strokes.
6
×
5
+
4
=
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© 2007 Herbert I. Gross
The problem occurs if all we see is a
formula such as C = 6 × 5 + 4 where
no explanation is given with respect to the
order of operations. Namely, as written we
have a choice between first multiplying 6
by 5 and then adding 4 or first adding
5 and 4 and then multiplying by 6.
Key Point
In mathematics, we use the agreement that
any expression that is enclosed in
parentheses is to be treated as one
number.
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© 2007 Herbert I. Gross
Thus if our intent was first to multiply 6 by
5 and then add 4, we would write…
(6 × 5) + 4
…but if we first wanted to add 5 and 4,
we would have written…
6 × (5 + 4)
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© 2007 Herbert I. Gross
Note
In mathematics we use parentheses the way we
use hyphens in English. For example, consider
the ambiguous phrase “the high school building”.
Is this a one story building that houses grades 9
through 12 or is it a multi-storied college building?
If we mean the former we write “the high-school
building” but if we mean the latter we write “the
high school-building”. If we were to use
parentheses instead of hyphens we would rewrite
“the high-school building” as “the (high school)
building” and we would rewrite “the high schoolbuilding” as “the high (school building)”. next
© 2007 Herbert I. Gross
However, when more and more operations
are involved expressions become very
cumbersome even with the grouping
symbols. For example, consider the
following recipe…
© 2007 Herbert I. Gross
Start with
7
Multiply by 2
14
Add 3
17
Multiply by 4
68
Subtract 6
62
Divide by 2
31
answer
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If we now write the steps in the order in
which they appear, we get the expression…
7×2+3×4–6÷2
The question now arises: how can we be
sure that there will be no ambiguity?
For example, if a person looks at the
expression and decides to replace
7 × 2 by 14, 3 × 4 by 12, and 6 ÷ 2 by 3,
the expression becomes…
( 7 × 2 ) + (3 × 4 ) – (6 ÷ 2 ) or 23
14
© 2007 Herbert I. Gross
12
3
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And if the person had instead elected to
replace 2 + 3 by 5 the expression would
become…
7×5×4–6÷2
…and if we now performed the operations
in the given order we would obtain..
7 × 5 × 4 = 140
140 – 6 = 134
134 ÷ 2 = 67
answer
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© 2007 Herbert I. Gross
To make sure that everyone reads the
expression in the way we meant it, we
would have to write the cumbersome
expression…
((((7 × 2)+ 3)) × 4) - 6) ÷ 2
In this format we start with the innermost
set of parentheses (7 × 2) and work our
way outward. In terms of how this
compares with our written recipe, notice
that the expression in each step names
one number.
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© 2007 Herbert I. Gross
Stated a bit differently, the output of
each step is the input of the next step.
That is…
Start with
7
Multiply by 2
7×2
Add 3
(7 × 2) + 3
Multiply by 4
((7 × 2) + 3)) × 4
Subtract 6
(((7 × 2) + 3)) × 4) - 6
Divide by 2
((((7 × 2) + 3)) × 4) – 6) ÷ 2
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© 2007 Herbert I. Gross
Trying to match the parentheses correctly
is confusing. Therefore, we often agree to
use grouping symbols other than
parentheses.
For Example
Instead of writing…
((((7 × 2) + 3)) × 4) – 6) ÷ 2
…we might write…
{ [(7 × 2) + 3] × 4} – 6 ÷ 2
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© 2007 Herbert I. Gross
In this way, we look to match the symbols…
For Example
When we see “(” we match it with “)”.
When we see “[” we match it with “]”.
When we see “{” we match it with “}”.
etc.
© 2007 Herbert I. Gross
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If you are intimidated by Key Point
an expression such as…
{ [(x × 2) + 3] × 4} – 6 ÷ 2
but not by the “recipe”
Start with
x
Multiply by 2
you have a language
problem; not a math
problem!!
Add 3
Multiply by 4
Subtract 6
Divide by 2
y
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© 2007 Herbert I. Gross
More specifically, to convert
{ [(x × 2) + 3] × 4} – 6 ÷ 2
into the recipe format, we start with x, and
we see that since x is inside the
parentheses, we first multiply by 2. The
result is inside the brackets, so we next
add 3. We are still within the braces, so
we next multiply by 4. We are still within
the angle brackets, so we next subtract 6.
And finally we divide by 2.
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© 2007 Herbert I. Gross
In summary…
Start with
x
Multiply by 2
Add 3
Multiply by 4
Subtract 6
Divide by 2
y
Starting after x, in terms of entering a
sequence of key strokes on a calculator, the
recipe would look like…
×
© 2007 Herbert I. Gross
2
+
3
×
4
–
6
÷
…and y represents the answer.
2
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If we are comfortable using the language
of algebra, that's good. However, if
that's not the case, it's okay to translate
from algebra into “Plain English”.
For Example
Suppose we want to determine the value
of x for which…
{ [(x × 2) + 3] × 4} – 6 ÷ 2 = 51
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© 2007 Herbert I. Gross
From the previous
problem, we can
rewrite the question
in the form…
For what value of x
is it true that…?
Start with
x
Multiply by 2
Add 3
Multiply by 4
Subtract 6
Divide by 2
The answer
51
Or in the language of calculators, starting
with x, the problem could be written as…
×
© 2007 Herbert I. Gross
2
+
3
×
4
–
6
÷
…and the answer would be 51.
2
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To find the value of x, we could start
with the answer (51) and successively
undo each step of the recipe to obtain…
The Indicated Process
12
The answer is
Multiply by 2
12
Unmultiply (÷) by 2
Add 3
24
“Unadd” (-) 3
Multiply by 4
27
“Unmultiply” (÷) by 4
Subtract 6
108
“Unsubtract” (+) 6
Divide by 2
102
“Undivide” (×) by 2
51
Start with
Start with
The answer is
x
51
The “Unprocess”
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© 2007 Herbert I. Gross
And in terms of using a calculator, we
could undo the sequence by starting with
the 51 as the input, and then undoing each
step in succession. That is…
x
×2
+3
×4
÷2
51
÷4 +6 ×2
27 108 102
51
-6
becomes…
÷2
12
x
12
-3
24
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© 2007 Herbert I. Gross
As a check we see that…
Start with
12
Multiply by 2
24
Add 3
27
Multiply by 4
108
Subtract 6
102
Divide by 2
51
The answer
51
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© 2007 Herbert I. Gross
Once you are able to read the algebraic
equation directly, there is no need to
take the time to write the equation in terms
of a verbal recipe. The basic idea is that to
convert from an indirect to a direct
computation, the last operation we did is
the first operation we undo. With this in
mind let's revisit the equation…
{ [(x × 2) + 3] × 4} – 6 ÷ 2 = 51
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© 2007 Herbert I. Gross
{ [(x × 2) + 3] × 4} – 6 ÷ 2 = 51
The idea is that we want to “isolate” x.
Since the last step in arriving at 51 was to
divide by 2, we begin by multiplying by 2;
and to keep the equation balanced if we
multiply one side by 2 we have to multiply
the other side by 2. Hence we may
multiply both sides of the above equation
by 2 to obtain the equivalent equation…
{ [(x × 2) + 3] × 4} – 6 = 102
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© 2007 Herbert I. Gross
{ [(x × 2) + 3] × 4} – 6 = 102
Remember that we use grouping symbols
only when an ambiguity arises if we omit
them; and since nothing is outside the
angle brackets, we can remove them to
rewrite the above equation as…
{ [(x × 2) + 3] × 4} – 6 = 102
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© 2007 Herbert I. Gross
{ [(x × 2) + 3] × 4} – 6 = 102
From this equation, we see that the last
thing we did was to subtract 6; so to undo
this we add 6 to both sides to obtain…
{ [(x × 2) + 3] × 4} = 108
and since the braces can be omitted…
[(x × 2) + 3] × 4 = 108
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© 2007 Herbert I. Gross
[(x × 2) + 3] × 4 = 108
To “unblock” the brackets, we first have to
“get rid of” the 4; and since the last step
we did was to multiply by 4, we now divide
both sides of the above equation by 4 to
obtain…
(x × 2) + 3 = 27
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© 2007 Herbert I. Gross
(x × 2) + 3 = 27
The last thing we did in the above equation
was to add 3, so we now subtract 3 from
both sides to obtain…
x × 2 = 24
And finally, we divide both sides by 2 to
obtain…
x = 12
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© 2007 Herbert I. Gross
Key Point
To reduce the need for grouping symbols,
we use an agreement determining the order
of operations, summarized in the acronym
PEMDAS. The letters in PEMDAS stand
for: parentheses, exponents, multiplication,
division, addition and subtraction. This
acronym is simply a mnemonic device for
remembering the order in which we
perform arithmetic operations. This avoids
any ambiguity in our calculating an
expression.
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© 2007 Herbert I. Gross
Note
Such mnemonic devices are used
elsewhere as well. For example, the
word “HOMES” is often used to help us
remember that the names of the five
Great Lakes are Huron, Ontario,
Michigan, Erie, and Superior.
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© 2007 Herbert I. Gross
A Brief Review of Exponents
The notation 23 is an abbreviation for the
product of three 2’s. That is, 23 = 2 × 2 × 2.
This should not be confused with 2 × 3 (or
3 × 2). That is: there is a big difference
between multiplying three 2’s and adding
three 2’s. For example, 3 × 2 =6, but
32 = 3 × 3 or 9.
In the expression 23, 2 is called the base and
3 is called the exponent. We read 23 as “2 to
the third” or “2 to the third power”; and we
refer to 21, 22, 23, etc as the powers of 2.
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© 2007 Herbert I. Gross
We will discuss exponents in greater detail
later in the course, but for now we want to
focus on the powers of 10. That is:
101 = 10
102 == 10
10 ×10
×10 == 100
100
10 3 = 10 × 10 × 10 =1,000
and observe that for any positive whole
number, n, 10n in place value notation is a 1
followed by n zeroes.
Thus, for example, 107 is an abbreviation
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for 10000000
or 10 million.
, ,
© 2007 Herbert I. Gross
We could make up several plausible
explanations as to why the agreement
PEMDAS was chosen; but from our
point of view, the most natural way is to
revisit how we write a place value
numeral in expanded notation.
For Example
2,345
becomes…
2 × 1,000 + 3 × 100 + 4 × 10 + 5 × 1
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© 2007 Herbert I. Gross
2 × 1,000 + 3 × 100 + 4 × 10 + 5
In reading the above equation, it is
understood that the multiplication is taking
place before the addition. In other words, if
we were to use grouping symbols, the
equation would become…
(2 × 1,000) + (3 × 100) + (4 × 10) + (5 × 1)
…which in terms of our exponent notation
may be rewritten as…
(2 × 10³) + (3 × 10²) + (4 × 10¹) + (5)
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© 2007 Herbert I. Gross
Key Point
This helps to motivate why we do what's
inside the grouping symbols first; and why
when multiplication and addition appear in
the same expression, we perform all of the
multiplications before we perform any of
the additions.
Because addition and subtraction are so
closely related, we treat them as being
“equal” and we do the same with
multiplication and division. Thus our rule
becomes…
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© 2007 Herbert I. Gross
Rule
When the four basic operations occur in
the same expression, we perform the
multiplications and divisions first;
and then we do the additions and
subtractions.
In any “string” of terms that involves
only addition and subtraction (or only
multiplication and division), we proceed
through the string from left to right.
© 2007 Herbert I. Gross
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Caution
An exponent refers only to the number
immediately to its left. For example…
2 ×10³ means 2 × (10)³.
If we wanted first to multiply 2 by 10
and then raise that product to the 3rd
power, we would have to write…
(2 × 10)³
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© 2007 Herbert I. Gross
Note
A problem would occur if we
wanted to follow a different agreement,
such as starting at the left and proceeding
left to right. For example, consider the
expression 3 + 4 × 5. The left-to-right
agreement would yield 35 (that is 3 + 4 = 7
and 7 × 5 = 35) as the correct answer.
On the other hand, the PEMDAS agreement
tells us that we must multiply before we
add, and this leads to the grouping
3 + (4 × 5), with 23 as the answer. next
© 2007 Herbert I. Gross
Thus it's not that one convention is more
logical than the other. Rather, it's that we
can't have two conventions that contradict
one another. It would be like having
two rules in baseball; one of which said
3 strikes is an out, and the other saying
that 4 strikes is an out.
Neither rule is more logical than the other;
but if we tried to play a game according to
both rules at the same time, there
would be an unsolvable impasse when
a batter got to 3 strikes.
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© 2007 Herbert I. Gross
Solving a Problem
As practice in using our PEMDAS
agreement, what number is named by…
3×5+4+7×2+3?
Answer: 36
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© 2007 Herbert I. Gross
Solution for the Problem
3×5+4+7×2+3
The agreement is that we do all the
multiplications first. In other words, the
agreement takes the place of our having to
write the expression as…
(3 × 5) + 4 + (7 × 2) + 3
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© 2007 Herbert I. Gross
Solution for the problem
(3 × 5) + 4 + (7 × 2) + 3
We then do the arithmetic within the
parentheses to obtain…
15 + 4 + 14 + 3
We then add the numbers in the expression
to obtain 36 as our final answer to the
problem.
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© 2007 Herbert I. Gross
Note
In adding the numbers in 15 + 4 + 14 + 3
we knew from our study of arithmetic that
addition was both commutative and
associative (that is, we could add
in any order that we wished).
However, subtraction doesn’t have these
“convenient” properties.
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© 2007 Herbert I. Gross
For Example
Without a specific agreement, there would be
two different answers for the value of an
expression such as 9 – 6 – 2. That is, we
might read it from left to right as if it were
written as (9 – 6) – 2, or we could read it
from right to left as 9 – (6 – 2). In the first
case the answer is 1 and in the second case
the answer is 5.
In terms of PEMDAS, since there are no
grouping symbols in 9 – 6 – 2, we read it
from left to right; that is, as (9 – 6) – 2.
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© 2007 Herbert I. Gross
Summary
Using the PEMDAS convention, we evaluate
expressions as follows…
• Start by doing what's inside each set of
parentheses.
• Then proceed to working with
exponents, remembering that the exponent
refers only to the number immediately to
next
its left.
© 2007 Herbert I. Gross
• Then perform all the multiplications
and divisions, proceeding from left to
right if there are ambiguities.
• Finally, do all the additions and
subtractions, again proceeding from left
to right if there are ambiguities.
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© 2007 Herbert I. Gross
• In evaluating expressions, we start with
the innermost grouping symbols
and work our way out.
• However, in solving equations, we start
with the outermost grouping
symbols and work our way inward to
isolate the “unknown”.
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© 2007 Herbert I. Gross