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

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

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Review In our first presentation we looked at a simple one operation “recipe” such as… © 2007 Herbert I. Gross next 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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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 © 2007 Herbert I. Gross next

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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… © 2007 Herbert I. Gross next

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© 2007 Herbert I. Gross next Multiply by 5 (the cost per pound). Add 4 (the shipping and handling). The output is the cost in dollars Start with 6 as the input (the number of pounds). next

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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. © 2007 Herbert I. Gross next 6×5+4=

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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. © 2007 Herbert I. Gross next In mathematics, we use the agreement that any expression that is enclosed in parentheses is to be treated as one number. Key Point

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Thus if our intent was first to multiply 6 by 5 and then add 4, we would write… (6 × 5) + 4 © 2007 Herbert I. Gross next …but if we first wanted to add 5 and 4, we would have written… 6 × (5 + 4)

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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 next Note

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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 next Start with7 Multiply by 214 Add 317 Multiply by 468 Subtract 662 Divide by 231 next 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… © 2007 Herbert I. Gross next 7 × 2 + 3 × 4 – 6 ÷ 2 next or 23 () ( ) ( ) 14123 next

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And if the person had instead elected to replace 2 + 3 by 5 the expression would become… © 2007 Herbert I. Gross 7 × 5 × 4 – 6 ÷ 2 …and if we now performed the operations in the given order we would obtain.. 7 × 5 × 4 = 140140 – 6 = 134 134 ÷ 2 = 67 next 67 next answer

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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 © 2007 Herbert I. Gross next 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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Stated a bit differently, the output of each step is the input of the next step. That is… © 2007 Herbert I. Gross next Start with7 Multiply by 27 × 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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Trying to match the parentheses correctly is confusing. Therefore, we often agree to use grouping symbols other than parentheses. For Example © 2007 Herbert I. Gross next ((((7 × 2) + 3)) × 4) – 6) ÷ 2 Instead of writing… …we might write… { [(7 × 2) + 3] × 4} – 6 ÷ 2 next

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In this way, we look to match the symbols… When we see “(” © 2007 Herbert I. Gross next For Example we match it with“)”. When we see “[” we match it with“]”. When we see “{” we match it with“}”. etc.

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If you are intimidated by an expression such as… © 2007 Herbert I. Gross next Key Point { [(x × 2) + 3] × 4} – 6 ÷ 2 but not by the “recipe” Start withx Multiply by 2 Add 3 Multiply by 4 Subtract 6 Divide by 2y you have a language problem; not a math problem!!

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More specifically, to convert 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. © 2007 Herbert I. Gross next { [(x × 2) + 3] × 4} – 6 ÷ 2

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In summary… © 2007 Herbert I. Gross next Start withx Multiply by 2 Add 3 Multiply by 4 Subtract 6 Divide by 2y Starting after x, in terms of entering a sequence of key strokes on a calculator, the recipe would look like… ×+× –÷ …and y represents the answer. 23462

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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”. © 2007 Herbert I. Gross next Suppose we want to determine the value of x for which… { [(x × 2) + 3] × 4} – 6 ÷ 2 = 51 For Example

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From the previous problem, we can rewrite the question in the form… © 2007 Herbert I. Gross next Start withx Multiply by 2 Add 3 Multiply by 4 Subtract 6 Divide by 2 Or in the language of calculators, starting with x, the problem could be written as… For what value of x is it true that…? The answer51 next ×+× –÷23462 …and the answer would be 51.

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To find the value of x, w e could start with the answer (51) and successively undo each step of the recipe to obtain… © 2007 Herbert I. Gross next Start withx Multiply by 2 Add 3 Multiply by 4 Subtract 6 Divide by 2 The answer is51 12 24 27 108 102 51 The Indicated Process The “Unprocess” The answer is Unmultiply (÷) by 2 “Unadd” (-) 3 “Unmultiply” (÷) by 4 “Unsubtract” (+) 6 “Undivide” (×) by 2 Start with next

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© 2007 Herbert I. Gross next 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÷251 next x÷ 2-3÷4+6×251 becomes… 12 102108272412

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As a check we see that… © 2007 Herbert I. Gross Start with12 Multiply by 224 Add 327 Multiply by 4108 Subtract 6102 Divide by 251 The answer51 next

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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… © 2007 Herbert I. Gross next { [(x × 2) + 3] × 4} – 6 ÷ 2 = 51

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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… © 2007 Herbert I. Gross next { [(x × 2) + 3] × 4} – 6 = 102 { [(x × 2) + 3] × 4} – 6 ÷ 2 = 51

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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… © 2007 Herbert I. Gross next { [(x × 2) + 3] × 4} – 6 = 102

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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… © 2007 Herbert I. Gross next { [(x × 2) + 3] × 4} = 108 and since the braces can be omitted… [(x × 2) + 3] × 4 = 108 next { [(x × 2) + 3] × 4} – 6 = 102

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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… © 2007 Herbert I. Gross next [(x × 2) + 3] × 4 = 108 next (x × 2) + 3 = 27

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The last thing we did in the above equation was to add 3, so we now subtract 3 from both sides to obtain… © 2007 Herbert I. Gross next (x × 2) + 3 = 27 x × 2 = 24 And finally, we divide both sides by 2 to obtain… x = 12 next

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© 2007 Herbert I. Gross 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. Key Point next

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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. © 2007 Herbert I. Gross next Note

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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 (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 3 2 = 3 × 3 or 9. © 2007 Herbert I. Gross next A Brief Review of Exponents 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. next

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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: © 2007 Herbert I. Gross next and observe that for any positive whole number, n, 10 n in place value notation is a 1 followed by n zeroes. 10 1 = 10 10 2 = 10 ×10 = 100 10 3 = 10 × 10 × 10 =1,000 Thus, for example, 10 7 is an abbreviation for 1 0000000 or 10 million.,, 10 1 = 10 next 10 2 = 10 ×10 = 100 10 3 = 10 × 10 × 10 =1,000

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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. © 2007 Herbert I. Gross next For Example 2,345 becomes… 2 × 1,000 + 3 × 100 + 4 × 10 + 5 × 1 next

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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… © 2007 Herbert I. Gross next …which in terms of our exponent notation may be rewritten as… 2 × 1,000 + 3 × 100 + 4 × 10 + 5 next (2 × 1,000) + (3 × 100) + (4 × 10) + (5 × 1) (2 × 10³) + (3 × 10²) + (4 × 10¹) + (5)

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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. © 2007 Herbert I. Gross next Key Point 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

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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 next Rule

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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 © 2007 Herbert I. Gross next Caution

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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. © 2007 Herbert I. Gross next Note

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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. © 2007 Herbert I. Gross next

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As practice in using our PEMDAS agreement, what number is named by… 3 × 5 + 4 + 7 × 2 + 3 ? © 2007 Herbert I. Gross next Solving a Problem Answer: 36 next

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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 © 2007 Herbert I. Gross next

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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. © 2007 Herbert I. Gross next

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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. © 2007 Herbert I. Gross next Note

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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. © 2007 Herbert I. Gross next For Example

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Using the PEMDAS convention, we evaluate expressions as follows… © 2007 Herbert I. Gross next Summary 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

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Then perform all the multiplications and divisions, proceeding from left to right if there are ambiguities. © 2007 Herbert I. Gross next Finally, do all the additions and subtractions, again proceeding from left to right if there are ambiguities. next

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In evaluating expressions, we start with the innermost grouping symbols and work our way out. © 2007 Herbert I. Gross next However, in solving equations, we start with the outermost grouping symbols and work our way inward to isolate the “unknown”. next

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