1. The Game of Algebra or The Other Side of Arithmetic
Lesson 2 by
Herbert I. Gross & Richard A. Medeiros
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2. Order of
Operations - × ÷ +
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3. 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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4. The instructions are unambiguous
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
F = 72 ÷ 12
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5. 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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6. 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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7. 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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8. 5 and 4 and then multiplying by 6.
The problem occurs if all we see is a formula such as C = 6 × 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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9. …but if we first wanted to add 5 and 4, we would have written…
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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10. 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 school-building” as “the high (school building)”.
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11. However, when more and more operations are involved expressions become very cumbersome even with the grouping symbols. For example, consider the following recipe…
Start with 7
Multiply by 2 14
Add 3 17
Multiply by 4 68
Subtract 6 62
Divide by 2 31 31
answer
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12. the expression becomes…
If we now write the steps in the order in which they appear, we get the expression…
7 × × 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 × × 4 – 6 ÷ 2 ) ( ) ( )
or 23 14 12 3
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13. 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 67
answer
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14. would have to write the cumbersome expression…
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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15. 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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16. Trying to match the parentheses correctly is confusing
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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17. 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.
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18. Key Point If you are intimidated by an expression such as…
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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19. 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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20. …and y represents the answer.
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… × 2 + 3 × 4 – 6 ÷ 2
…and y represents the answer.
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21. Suppose we want to determine the value of x for which…
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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22. From the previous problem, we can rewrite the question in the form…
Start with x
Multiply by 2
Add 3
Multiply by 4
For what value of x is it true that…?
Subtract 6
Divide by 2
The answer 51
Or in the language of calculators, starting with x, the problem could be written as… × 2 + 3 × 4 – 6 ÷ 2
…and the answer would be 51.
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23. 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
Start with x 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
The answer is 51 51
Start with
The “Unprocess”
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24. 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 -6 ÷2 51
becomes… x 12
÷ 2 -3 ÷4 +6 ×2 51 12 24 27
108
102
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25. As a check we see that… Start with 12 Multiply by 2 24 Add 3 27
108
Subtract 6
102
Divide by 2 51
The answer 51
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26. 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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27. { [(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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28. { [(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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29. and since the braces can be omitted…
{ [(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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30. [(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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31. And finally, we divide both sides by 2 to obtain…
(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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32. 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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33. Great Lakes are Huron, Ontario, Michigan, Erie, and Superior.
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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34. 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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35. Thus, for example, 107 is an abbreviation for 1 0000000 , ,
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 = 100
103 = 10 × 10 × 10 =1,000
101 = 10
102 = 10 ×10 = 100
103 = 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 for 1 , ,
or 10 million.
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36. revisit how we write a place value numeral in expanded notation.
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, × × × 1
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37. …which in terms of our exponent notation may be rewritten as…
2 × 1, × ×
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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38. 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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39. and then we do the additions and subtractions.
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.
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40. Caution 2 ×10³ means 2 × (10)³. If we wanted first to multiply 2 by 10
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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41. 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 × 5. The left-to-right agreement would yield 35 (that is = 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.
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42. two rules in baseball; one of which said
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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43. As practice in using our PEMDAS agreement, what number is named by…
Solving a Problem
As practice in using our PEMDAS agreement, what number is named by…
3 × × ?
Answer: 36
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44. Solution for the Problem
3 × × 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) (7 × 2) + 3
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45. Solution for the problem (3 × 5) + 4 + (7 × 2) + 3
We then do the arithmetic within the parentheses to obtain…
We then add the numbers in the expression to obtain 36 as our final answer to the problem.
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46. Note
In adding the numbers in 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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47. 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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48. • Start by doing what's inside each set of parentheses.
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 its left.
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49. to right if there are ambiguities.
• 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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50. • However, in solving equations, we start with the outermost grouping
• 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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