How To Understand The Laws Of Algebra

Transcription

1 Welcome to Math Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move step-by-step and at your own pace through these notes. To do so, use the arrow keys or the mouse to move from slide to slide, forwards or backwards. Also use the index dots at the top of this slide (or the index at the left, accessible from the Adobe Acrobat Toolbar) to access the different sections of this document. To prepare for the Chapter 0 Quiz (September 9th at the start of class), please Read all the following material carefully, especially the included Examples. Memorize and understand all included Definitions and Procedures. Work out the Exercises section, which explains how to check your answers. Do the Quiz Review and check your answers by referring back to the Examples.

2 A basic real-life problem The area of a square with side length s is s 2. If we call the area A, we can write a formula for A in terms of s: A = s 2 This simple formula is described in many ways. A depends on s. s is the independent variable and A is the dependent variable. s is the input and s 2 is the output. A is a function of s. Obvious basic questions about squares include: If the side length of a square is 12, what is its area? Solution: A = s 2. Since s = 12, then A = s 2 = 12 2 = 144. Answer: The area is 144. If the area of a square is 100, what is its side length? Solution: A = s 2. Since A = 100, then 100 = s 2 and so s = 10. Answer: The side length is 10.

3 Some things to be think about: If the question uses words, the answer must use those words. In science classes, you must specify units. In math class, we don t use units unless the problem includes them. For example, we are willing to say that a square with side length 12 has area 12 2 = 144. If this bothers you, insert the word unit and say: A square with side length 12 units has area 144 square units. Now let s think about a square that is growing. Suppose at a starting time t = 0 the side length of the square is 10 units. Also suppose that the side length grows at a rate of 3 units per second. Then the formula for the side is easy to figure out: s = t In this formula, s depends on t. t is the independent variable and s is the dependent variable. t is the input and t is the output. s is a function of t.

4 Let s put everything together. Let t be time, s the side length of a square, and A the square s area. Then A = s 2 and s = t Let s focus on the algebra: forget the real-life problem. Now A depends on s, which in turn depends on t. This double dependence generates new questions: Find a formula showing how A depends on t. Solution: Substitute (10 + 3t) for s in A = s 2. Answer: A = (10 + 3t) 2. Stop here. You weren t asked to simplify. When t = 3, find A. Solution: since t = 3 and A = (10 + 3t) 2, we get A = ( ) 2 = 19 2 = 361. Answer: when t = 3, then A = 361. When A = 256, find t. Solution:(10 + 3t) 2 = 256. Therefore t = ± 256 = ±16. This may come as a surprise. If t = 16, then t = 2. If t = 16, then t = 26/3. Answer: t = 2 or t = 26/3. This may come as a bigger surprise.

5 The difference between the last two examples: The first example asked for the side length of a square. Although s 2 = 100 has two solutions s = 10 and s = 10, we rejected the negative answer (without saying so at the time) because the side length of a square is a positive number. The second example (check the wording of the question) talked about A, s, and t, not about area, side, and time. Therefore the question must be answered using algebra, without reference to the underlying physical problem. Question: Why do these notes consistently use the phrase side length? What s wrong with just saying side? This preliminary section of online notes deals with the basics of algebra. An important goal is to develop machinery for analyzing real-life problems including, but typically more complicated than, the one discussed above. Please read carefully, since this material is not always explained in high school classes or in textbooks.

6 A possibly surprising ingredient of student success in mathematics Success in mathematics requires the combined focus of intellect, logic, and memory. Students in this course will be required not only to do algebra problems, but also to reproduce on each examination a selection of definitions and procedures in these notes. The understanding of mathematics requires memorization of definitions. The efficient application of mathematics requires memorization of procedures. In recent years, memorization has been denigrated and undervalued. Today s students and educators would do well to read Chapter 60 in historian Daniel Boorstin s remarkable cultural history entitled The Discoverers (Random House, 1983). Here is the opening paragraph: Before the printed book, Memory ruled daily life and the occult learning, and fully deserved the name later applied to printing, the art preservative of all arts. The Memory of individuals and of communities carried knowledge through time and space. For millennia personal Memory reigned over entertainment and information, over conduct of professions. By Memory and in Memory the fruits of education were garnered, preserved, and stored. Memory was an awesome faculty everyone had to cultivate, in ways and for reasons we have long since forgotten. In these last five hundred years we see only pitiful relics of the empire and the power of Memory.

7 Introduction: Algebra Review This introduction for Math contains important review material. The main goal is to help you understand how and why we use algebra formulas. Whole numbers have certain obvious properties such as = and = 3. These basic properties, shared by all numbers, are called the Laws of Algebra. Here is a familiar example. You know that = = = To convey the idea that the order of adding numbers doesn t matter, we use letters to represent arbitrary numbers, and say that A + B = B + A for all real numbers A and B. This is the commutative law for addition. All of algebra is summarized in nine (just 9!) Algebra Laws. In the list below, A,B, and C stand for any real numbers. We use the symbol for multiplication.

8 Four Laws for Addition A + B = B + A (Addition is commutative) A + (B + C) = (A + B) + C(Addition is associative) A + 0 = A = 0 + A (0 is the identity for Addition) Every number A has a negative, written A, satisfying A + A = 0 Four Laws for Multiplication A B = B A (Multiplication is commutative) A (B C) = (A B) C(Multiplication is associative) A 1 = A = 1 A (1 is the identity for multiplication) Every nonzero number A has a reciprocal, written A 1, satisfying AA 1 = 1 One Distributive Law A (B + C) = A B + A C

9 We use some special notation and abbreviations, as follows: The fraction 1 A is another way of writing A 1. A 1 and A/B, read A divided by B, are abbreviations for the product A B B. We will not use the symbol for division. A power such as A 5 is just an abbreviation for a product, in this case A 5 = A A A A A. A n 1 is an abbreviation for A n. ( ) n A (the n th root of A ) satisfies n n A = A. When n = 2, we write just A, which we call the square root of A.

10 Using substitution to rewrite expressions The polynomial expression t 2 arises in physics as follows. If you drop a rock from a height of 100 feet, the height of the rock above the ground t seconds later is t 2, feet, provided 0 t 2.5. Since t can vary within the stated range of values, it is common to refer to t 2 as a polynomial in the variable t. However, we will often refer to t simply as a letter (that s what it is, after all). Definition An algebra expression contains letters and numbers, and combines them by using parentheses and symbols for algebra operations. Definition A numerical expression contains numbers, and combines them by using parentheses and symbols for algebra operations.

11 We will use the word expression for algebra expression. Note that an expression without any letters is a numerical expression. Here is a precise statement of how to construct expressions: Any letter is an expression. Any whole number is an expression. If E is an expression, then (E) is an expression. If E and F are expressions, then so are E + F, E F, E F, E/F, and EˆF. We use a dot, not the calculator symbol *, for multiplication. In practice, most multiply signs are omitted to save space. For example, the usual way of writing 3 A + 2 B C (C + 2) (C + 3) is 3A + 2BC(C + 2)(C + 3). Finally, we often write E/F as E F and usually write EF for E ˆF. Very often, we modify a starting expression by replacing one or more of its letters by other expressions. This important process is called substitution.

12 Definition of substitution To substitute an expression E for a letter in a formula, replace every occurence of the letter in the formula by (E). The parentheses are crucial! Example 1: In the expression A 2 B 2 substitute x + 3 for A and y 2 for B. Answer : (x + 3) 2 (y 2) 2 Please note the following: both statements are important. The substituted expression must be enclosed in parentheses. Exception: if the substituted expression is a letter, the parentheses are not needed. (x + 3) 2 (y 2) 2 is the only correct answer. The instruction says just substitute. Do not simplify or otherwise modify an expression unless you are explicitly asked to do so.

13 Example 2: In the expression A 2 B 2 substitute A + 3 for A and B 2 for B. Answer: (A + 3) 2 (B 2) 2 In this example, the statement Substitute A + 3 for A cannot sensibly be rephrased as Let A = A + 3. On the other hand, you may say: In expression (A + 3) 2 (B 2) 2, let A = 3 and B = x + 2. But it is best to say In expression (A + 3) 2 (B 2) 2, substitute 3 for A and x + 2 for B. Example 3: In the statement f(x) = 3 7x 2x 3 substitute x + h for x. Answer: f((x + h)) = 3 7(x + h) 2(x + h) 3. This looks a little funny. Double parentheses ((...)) should always be replaced by single parentheses and so we can write this solution more neatly as f(x + h) = 3 7(x + h) 2(x + h) 3. Again, don t multiply out unless you are asked to do so.

14 Based on your previous math course, you might think that the symbol f(x) in Example 3 means a function of x. That doesn t have to be the case. Unless you are informed that a function is involved, f(x) might mean the product of letters f and x. But it really doesn t matter: the stated solution is correct in both cases, whether f(x) indicates a function of x or just a product of variables. Substitution is a purely mechanical operation that can be done by doing a find and replace operation with a word processor: find every occurrence of x and replace it by (x + h). When you substitute an expression for a variable in a formula, you do not need to, and perhaps you should not, think about what the algebra operations mean. Rather, it would be wise to consider the 1911 advice of mathematician and philosopher Alfred North Whitehead: It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking about what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in battle they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.

15 Correct use of parentheses is crucial for student success Procedure: three rules for parentheses 1. When you substitute an expression for a letter, replace the letter by the expression enclosed in parentheses. 2. Insert parentheses whenever you expand an expression. Specifically, each time you multiply or take a power of a parenthesized expression, put parentheses around the result. 3. When you substitute an expression for a function value f(x), put parentheses around the result. Example 4: Rule 1: Substitute x + h for x in 3 x 2. Answer: 3 (x + h) 2, not 3 x 2 + h or 3 x + h 2. Rule 2: Simplify 3 3(x + h). Answer: 3 3(x + h) = 3 (3x + 3h) = 3 3x 3h, not 3 3x + 3h.

16 Rule 3: ( to be discussed in Section 2.1) If f(x) = x 2 then f(x + h) = (x + h) 2, not x 2 + h and not x + h 2. Definition An identity, also called an algebra law, is one of the nine Laws of Algebra listed above, or a new algebra law obtained by substituting expressions for the letters in a known algebra law. Each of the nine Laws of Algebra is a statement that two expressions are equal. New identities are produced by substitution in previously known identities. Therefore every identity is a statement that two expressions are equal. Example 5: Substituting x + y for A and z for B in (A + B)(A B) = A 2 B 2 yields the new identity ((x + y) + z)((x + y) z) = (x + y) 2 z 2 which can be rewritten (x + y + z)(x + y z) = (x + y) 2 z 2 In this example, omitting parentheses when you substitute is a mistake that (luckily) happens not to affect the answer. But it is still a mistake.

17 Example 6. Substituting z for A and x + y for B in (A + B)(A B) = A 2 B 2 yields the new identity (z + (x + y))(z (x + y)) = z 2 (x + y) 2, which can be rewritten (z + x + y)(z x y) = z 2 (x + y) 2. In this example, omitting parentheses when you substitute is a mistake that yields an incorrect answer. Every identity is written as a statement that two expressions are equal. Here is an example. Assume you already know the FOIL identity (A + B)(C + D) = AC + AD + BC + BD. Then the familiar identity (A + B)(A + B) = A 2 + 2AB + B 2 can be obtained by substituting A for C and B for D in the FOIL identity, as follows: Start with (A + B)(C + D) = AC + AD + BC + BD Substitute A for C and B for D: (A + B)(A + B) = AA + AB + BA + BB Use the commutative law AB = BA: (A + B)(A + B) = AA + AB + AB + BB Rewrite using powers and collect terms: (A + B)(A + B) = A 2 + 2AB + B 2

18 If you think that the above discussion is too detailed, I agree completely. The identity (A + B)(A + B) = A 2 + 2AB + B 2 should be memorized. Similarly, the identity (A + B)(A B) = A 2 B 2 should be memorized. Every identity is written using letters but it is equally true when any expressions are substituted for those letters. In particular, identities produce true statements when numbers are substituted for letters. Example 7: Start with the Distributive Law A(B + C) = AB + AC. Substitute x for A, x + 1 for B, and x + 1 for C to get ((x))((x + 2) + (x + 1)) = (x)(x + 1) + (x)(x + 2) Removing doubled parentheses yields (x)((x + 2) + (x + 1)) = (x)(x + 1) + (x)(x + 2). This is the answer to the question: you were not asked to simplify the result of substitution.

19 Example 8: Start with the identity: (A + B)(A B) = A 2 B 2 Substitute 3x for A and 5 for B: ((3x) + (5)) ((3x) (5)) = (3x) 2 (5) 2 Remove unnecessary parentheses: (3x + 5) (3x 5) = (3x) We will discuss in detail in the next section why (3x) + 5 can be rewritten without parentheses as 3x + 5. In contrast, the parentheses in (3x) 2 cannot be omitted, since (3x) 2 = (3x)(3x) = 9x 2 is not the same as 3x 2. Example 9: Check that the new identities obtained in Examples 7 and 8 yield true statements when you substitute 3 for x. 7 for x

20 What does the word simplify mean? The identity x 2 + 3x + 2 = (x + 1)(x + 2) does not say whether one expression is simpler than the other. When you start with (x + 1)(x + 2) and then multiply out and collect terms to get x 2 + 3x + 2, you are expanding (multiplying out) the product and rewriting it as a simplified sum. When you start with x 2 + 3x + 2 and then factor it to get (x + 1)(x + 2), you are rewriting the sum as a simplified product. Every simplify instruction should say which kind of simplified form is desired. Some books ignore this advice. In this course, we say either Rewrite as a simplified sum, which means: collect like terms or Rewrite as a simplified product, which means: the answer should be completely factored.

21 Parentheses and PEMDAS: order of operations When you write a story you organize your narrative by using paragraphs. When you write mathematics, you organize your computations by using parentheses. In the last example above, we said that ((3x) + 5) could be rewritten more simply as (3x + 5). Why are we allowed to remove the parentheses around the 3x? The problem is that the expression is ambiguous: it doesn t indicate whether the result should be (3 4) + 5 = = 17 or 3 (4 + 5) = 3 9 = 27. It seems that parentheses must be used to make clear what is intended. The difficulty with this approach is that using a lot of parentheses produces unreadable expressions. That s why mathematicians and computer scientists formulated the PEMDAS rule, which specifies the order of operations. The real purpose of the PEMDAS rule is to reduce the number of parentheses required needed to write complicated expressions.

22 Here s an easy example. In the expression , there are many ways to insert parentheses. However, the associative law guarantees that all of these methods give the same answer. This is obvious from everyday experience. After all, when you pour containers of 3, 4, 5, 6, and 7 gallons (in the order listed) into intermediate containers and then into a single barrel, the barrel will contain 25 gallons, regardless of which intermediate containers you used. Similarly, no matter how you insert parentheses in the expression , you will get the same answer. For example, check that (3 + 4) + (5 + (6 + 7)) = 3 + (((4 + (5 + 6)) + 7)). However, the above example is an exception: the value of many (most) expressions does change, depending on how parentheses are inserted. Here are a few examples: =? Two ways: (8 5) + 2 = = 5 but 8 (5 + 2) = 8 7 = =? Two ways: (8 5) + 2 = = 42 but 8 (5 + 2) = 8 7 = =? Two ways: 3 (4 2 ) = 3 16 = 48 but (3 4) 2 = 12 2 = 144 To resolve these ambiguities, we need to specify the order of applying arithmetic operation symbols: ˆ + and \.

23 The PEMDAS rule explains the order of operations intended when parentheses are missing. The letter P tells you that the general goal is to get rid of parentheses. The letters EMDAS, which probably should be written as E(M or D)(A or S), tell you what is the proper order of operations. More precisely, EMDAS tells you that at any stage of a simplification process, the first operation to apply is chosen as follows. Definition: Order of operations E: (Exponent): If there is a power symbol ˆ, apply the one furthest to the right. MD: (Multiply or Divide): Otherwise, if there is a or / symbol, apply the one furthest to the left. AS: (Add or Subtract) Otherwise, if there is a + or symbol, apply the one furthest to the left. After you apply that operation, you get a new expression. Apply these rules to the new expression, and continue applying the rules until the result has no parentheses.

24 In practice, you should remember the following special cases: Multiply before you Add or Subtract: = 2 + (3 4) = = 14, not (2 + 3) 4 = 5 4 = 20 Exponentiate before you Multiply: 3 2ˆ2 = 3 (2ˆ2) = 3 4 = 12, not (2 3)ˆ2 = 6 ˆ 2 = 36. Similarly: = 3 (2 2 ) = 3 4 = 12, not (2 3) 2 = 6 2 = 36 Add and/or Subtract from left to right: = (100 2) + 1 = = 99, not 100 (2 + 1) = = 97. Multiply and/or Divide from left to right: 8/2 4 = (8/2) 4 = 4 4 = 16, not 8/(2 4) = 8/8 = 1 In practice, this last statement is not well known. It is safer to write (8/2) 4 or 8/(2 4), whichever is intended.

25 Special facts about modern math notation Most multiply signs are omitted: 3 x y (x + 5) (2 3) = 3xy(x + 5)(2 3). Fraction lines are usually written horizontally: A/B = A B. Substituting x + 2 for A and x + 3 for B gives (x + 2) (x + 2)/(x + 3) = (x + 3) = x + 2 x + 3. Using a horizontal fraction line takes more space but is clearer, since you can omit the parentheses required when you use slash notation. Use superscript notation: write 3 4 for 3ˆ4. If the first symbol in an expression is a minus sign, it means negate, i.e., multiply by 1. Thus 3 2 = ( 1)(3 2 ) = ( 1)(9) = 9, not ( 3) 2 (which equals +9). We do not follow the practice of some K-12 algebra books that use two different minus signs, one to indicate subtract, and the other to indicate the sign of a negative number.

26 Example 10: Apply PEMDAS to find the value of ( ) 2 + (8/2 4 3) 3. Here find the value means: rewrite as a single number. Step 1: = = 11, since we multiply before we add. Step 2: 8/2 4 3= 4 4 3, since we multiply/divide from left to right = 16 3 = 13, since we multiply before we subtract. Using the results of Steps 1 and 2, we get ( ) 2 + (8/2 4 3) 3 = (11) 2 + (13) 3 = Remove unnecessary parentheses. = Exponentiate before you Add. = All of the above examples involved numerical expressions only. We are also interested in using PEMDAS as a guide to rewriting algebra expressions with letters.

27 Example 11: Rewrite (x + 2)(x + 3) x 2 as a simplified sum. Solution: (x + 2)(x + 3) x 2 is given. Note that x 2 can t simplify further. = x 2 + 5x + 6 x 2 Multiply before you subtract. = 5x + 6. This example is very misleading, because we got the right answer somewhat by accident. We didn t follow Parenthesis Rule 2: Whenever you Multiply or Exponentiate, put parentheses around the result. In the following problem, omitting parentheses will produce a wrong answer and will earn zero part credit. Example 12: Rewrite x 2 (x + 2)(x + 3) as a simplified sum. Solution: x 2 (x + 2)(x + 3) is given. = x 2 (x 2 + 5x + 6) Multiply before you subtract. Insert parentheses! = x 2 x 2 5x 6 Remove parentheses by distributing the minus sign. = 5x 6. The answer would have been incorrect had we forgetten to use parentheses.

28 Example 13: Rewrite x 2 (x + 4) (x + 2) 2 as a simplified sum. In the following solution, we apply PEMDAS by Exponentiating first to expand (x + 2) 2. Start: x 2 (x + 4) (x + 2) 2 Exponentiate first: Result in parentheses! x 2 (x + 4) (x 2 + 4x + 4) Note that x 2 is already simplified, so we ignore it: Multiply before you subtract: (x 3 + 4x 2 ) (x 2 + 4x + 4) The first set of parentheses is not needed: x 3 + 4x 2 (x 2 + 4x + 4) The second set is needed! Distribute the minus sign: (x 3 + 4x 2 ) x 2 4x 4 Collect terms: x 3 + 4x 2 x 2 4x 4 x 3 + 3x 2 4x 4

29 You can perform a partial check by substituting a number (say 1) for x. However, you run the risk that you will make an algebra slip in your check even though your original work was correct. x 2 (x + 4) (x + 2) 2 =? x 3 + 3x 2 4x 4 Substitute 1 for x. 1 2 (1 + 4) (1 + 2) 2 =? (5) 3 2 =? =? =? 0 4 YES! Again, this is only a partial check. Advanced math fact: The general rule is that polynomials with highest power x 3 are identical if they agree at 4 different values of x. Therefore you would need to check your answer for 3 additional values of x to be sure it is correct.

30 Worked examples 3 4(x + 2) = 3 (4x + 8) 14) = 3 4x 8 = 4x 5 16) x 2 x(x + 2) = x 2 (x 2 + 2x) = x 2 x 2 2x = 0 2x = 2x (x)(x + 4) (x + 7) 2 = (x 15) 2 + 4x) (x x + 49) = x 2 + 4x x 2 14x 49 = 10x 49 17) (x + 3)(x + 4) 3(x + 7) 2 = (x 2 + 7x + 12) 3(x x + 49) = x 2 + 7x x 2 42x 147 = x 2 3x 2 + 7x 42x = 2x 2 35x 135

31 Exercises Click on Wolfram Calculator to find an answer checker. For example, type in simplify (2 x)ˆ2. Click on Wolfram Algebra Examples to see how to check various types of algebra problems. Click on WA to work on WebAssign HW problems. 1. Rewrite each of the following as a simplified sum: a) (3 x) 2 (4 x) 2 b) (x + 2) 2 (x + 2) c) 5x (3x + 1) 2 d) (5x + 2)(2x 1) (3x + 1) 2 2. Substitute x + 2 for x in each of the following and rewrite the answer as a simplified sum. a) 4(3 x) b) 2 x 2 c) (2 x) 2 d) (x + 1) 2 x 2 3. Substitute x h for x in each of the following and rewrite the answer as a simplified sum. a) 4(3 x) b) 2 x 2 c) (2 x) 2 d) (x + 1) 2 x 2

32 4. Substitute 2 for h in each of your answers in question 3 above. Your answers should match exactly your answers to question In the function definition f(x) = x Find f(x + h) by substituting x + h for x. Simplify your answer. Find f(x h) by substituting x h for x. Simplify your answer. Use the last two results to simplify f(x+h) f(x h) h. Simplify your answer. 6. Redo problem 5, this time using f(x) = 3 7x 2x Redo problem 5, this time using f(x) = x 3.

33 Review exercises for Chapter 0 Example 1: In the expression A 2 B 2 substitute x + 3 for A and y 2 for B. Example 2: In the expression A 2 B 2 substitute A + 3 for A and B 2 for B. Example 3: In the statement f(x) = 3 7x 2x 3 substitute x + h for x. Example 4: Substitute x + h for x in 3 x 2. Example 5: Substitute x + y for A and z for B in (A + B)(A B) = A 2 B 2 Example 6. Substitute z for A and x + y for B in (A + B)(A B) = A 2 B 2 Example 7: Substitute x for A, x + 1 for B, and x + 1 for C in A(B + C) = AB + AC. Example 8: Substitute 3x for A and 5 for B in the identity: (A + B)(A B) = A 2 B 2. Then remove unnecessary parentheses: Example 9: Check that the new identities obtained in Examples 7 and 8 yield true statements when you substitute a) 5 for x b) 7 for x.

34 Example 10:Find each of the following: a) b) 3 2ˆ2 c) d) e) 8/2 4 e) ( ) 2 + (8/2 4 3) 3. Rewrite each of the following as a simplified sum: Example 11: (x + 2)(x + 3) x 2 Example 12: x 2 (x + 2)(x + 3) Example 13: x 2 (x + 4) (x + 2) 2 Example 14: 3 4(x + 2) Example 15: (x)(x + 4) (x + 7) 2 Example 16: x 2 x(x + 2) Example 17: (x + 3)(x + 4) 3(x + 7) 2