The Game of Algebra or The Other Side of Arithmetic Lesson 2 by Herbert I. Gross & Richard A. Medeiros © 2007 Herbert I. Gross next

Order of Operations - × ÷ + next © 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. next next © 2007 Herbert I. Gross

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 next © 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… 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). The output is the cost in dollars next next next next © 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 = next © 2007 Herbert I. Gross

5 and 4 and then multiplying by 6. 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. next next © 2007 Herbert I. Gross

…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) next next © 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 school-building” 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… Start with 7 Multiply by 2 14 Add 3 17 Multiply by 4 68 Subtract 6 62 Divide by 2 31 31 answer next next next next next next next next © 2007 Herbert I. Gross

the expression becomes… 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 12 3 next next next next next next © 2007 Herbert I. Gross

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 next next next next next © 2007 Herbert I. Gross

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. next next next © 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 next © 2007 Herbert I. Gross

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 next next next © 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. next next © 2007 Herbert I. Gross 17

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 next next next © 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. next © 2007 Herbert I. Gross

…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. next next © 2007 Herbert I. Gross 20

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 next next © 2007 Herbert I. Gross

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. next next next © 2007 Herbert I. Gross 22

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” next next next © 2007 Herbert I. Gross 23

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 next next next © 2007 Herbert I. Gross 24

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 next © 2007 Herbert I. Gross 25

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 26

{ [(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 27

{ [(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 28

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 next next © 2007 Herbert I. Gross 29

[(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 next next © 2007 Herbert I. Gross 30

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 next next next © 2007 Herbert I. Gross 31

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. next © 2007 Herbert I. Gross 32

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. next © 2007 Herbert I. Gross 33

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. next next © 2007 Herbert I. Gross 34

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 0000000 , , or 10 million. next next next next © 2007 Herbert I. Gross 35

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 × 1,000 + 3 × 100 + 4 × 10 + 5 × 1 next next next © 2007 Herbert I. Gross 36

…which in terms of our exponent notation may be rewritten as… 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) next next © 2007 Herbert I. Gross 37

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… next next © 2007 Herbert I. Gross 38

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. next © 2007 Herbert I. Gross 39

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 next © 2007 Herbert I. Gross 40

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 41

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. next © 2007 Herbert I. Gross 42

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 × 5 + 4 + 7 × 2 + 3 ? Answer: 36 next next © 2007 Herbert I. Gross 43

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 44

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. next © 2007 Herbert I. Gross 45

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. next © 2007 Herbert I. Gross 46

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. next © 2007 Herbert I. Gross 47

• 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. next next next © 2007 Herbert I. Gross 48

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. next next © 2007 Herbert I. Gross 49

• 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”. next next © 2007 Herbert I. Gross 50