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The Game of Algebra or The Other Side of Arithmetic

Lesson 2

× ÷

Order of

+ -

Operations

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.

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

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

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 F

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… next © 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).

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 =

next

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.

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)

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)”. © 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…

Start with Multiply by 2 Add 3 Multiply by 4 Subtract 6 Divide by 2 7 14 17 68 62

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 ( ) ( ) or 23 14

12 3

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..

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.

Stated a bit differently, the output of each step is the input of the next step. That is…

Start with Multiply by 2 Add 3 Multiply by 4 Subtract 6 Divide by 2 7 7 × 2 (7 × 2) + 3 ((7 × 2) + 3)) × 4 (((7 × 2) + 3)) × 4) - 6 ((((7 × 2) + 3)) × 4) – 6) ÷ 2

Trying to match the parentheses correctly is confusing. Therefore, we often agree to use grouping symbols other than parentheses.

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.

If you are intimidated by an expression such as…

Key Point

{ [( x × 2) + 3] × 4} – 6 ÷ 2 but not by the “recipe”

Start with Multiply by 2 Add 3 Multiply by 4 Subtract 6 Divide by 2

x y you have a language problem; not a math problem!!

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.

In summary…

Start with Multiply by 2 Add 3 Multiply by 4 Subtract 6 Divide by 2

x 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

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”.

From the previous problem, we can rewrite the question in the form… For what value of x is it true that…?

Start with 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…

× 2 + 3 × 4 – 6 ÷ 2

…and the answer would be 51 .

To find the value of

, w

e could start with the answer ( 51 ) and successively undo each step of the recipe to obtain… The Indicated Process

Start with Multiply by 2 Add 3 Multiply by 4 Subtract 6 Divide by 2

x 12

12 24 27 108 102 The answer is Unmultiply ( ÷) by 2 “Unadd” ( ) 3 “Unmultiply” ( ÷ ) by 4 “Unsubtract” ( + ) 6 “Undivide” ( × ) by 2 The answer is

51 51

Start with

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

becomes…

-6 ÷2

51

÷ 2

-3

÷4

108

+6

102

×2

51

As a check we see that…

Start with Multiply by 2 Add 3 Multiply by 4 Subtract 6 Divide by 2 The answer

24 27 108 102 51

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 next © 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 next © 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 next © 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 © 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 © 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 © 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:

arentheses,

xponents,

ultiplication,

ivision,

ddition and

ubtraction. 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.

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 H uron, O ntario, M ichigan, E rie, and S uperior.

A Brief Review of Exponents The notation 2 3 is an abbreviation for the product of three 2’s. That is, 2 3 = 2 × 2 × 2. This should not be confused with 2 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 3 2 = 3 × 3 or 9.

3 (or In the expression 2 3 , 2 is called the base and 3 is called the exponent . We read 2 3 as “2 to the third” or “2 to the third power”; and we refer to 2 1 , 2 2 , 2 3 , etc as the powers of 2.

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: 10 1 10 2 = 1 0 ×1 0 = 1 00 10 3 = 1 0 × 1 0 × 1 0 =1, 000 and observe that for any positive number, n , 10 n whole in place value notation is a 1 followed by n zeroes.

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.

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) © 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.

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.

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)

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 © 2007 Herbert I. Gross 3 + (4 × 5), with 23 as the answer.

next

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.

Solving a Problem

As practice in using our PEMDAS agreement, what number is named by… 3 × 5 + 4 + 7 × 2 + 3 ?

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 next © 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.

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.

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 .

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.

• 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.

• 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”.

Slide 1

Transcript Slide 1

The Game of Algebra or The Other Side of Arithmetic

Lesson 2

Order of

Operations

Review

6 × 5 + 4 =

Key Point

(6 × 5) + 4

6 × (5 + 4)

Note

7 × 2 + 3 × 4 – 6 ÷ 2

( 7 ( ) ( ) or 23 14

12 3

Key Point

× 2 + 3 × 4 – 6 ÷ 2

× 2 + 3 × 4 – 6 ÷ 2

To find the value of

, w

× 2 +3 ×4

-6 ÷2

51

÷ 2

-3

÷4

+6

×2

51

Key Point

Note

Key Point

Rule

Caution

Note

Solving a Problem

Note

Summary

Directory