Properties and Definitions

Transcription

1 Section 0.1 Contents: Operations Defined Multiplication as an Abbreviation Visualizing Multiplication Commutative Properties Parentheses Associative Properties Identities Zero Product Answers to Exercises Focus Exercises INTRODUCTION Algebra is all about symbols and abbreviations, using them and manipulating them. In fact, we use symbols for the four basic operations, addition (+), subtraction ( ), multiplication (x) and division ( ). The Greek mathematician Diophantus was one of the first to use symbols to abbreviate mathematical expressions and thought. Though his symbols were different from what we use today, he is credited with taking the discussion of mathematics and problem solving from sentence form into symbolic form. Could you imagine having to discuss everyday mathematics without the use of a plus sign or a times sign? Some of the symbols you ll be seeing in this course are variables, letters that represent numbers, like x, w, y, a, b, c, and so on; a new multiplication sign, a partially raised dot, ; instead of 3 x 5 we ll see 3 5 = 15; radicals, symbols that represent square roots and cube roots, like 2 3 and ; absolute value bars, symbols that represent the numerical value of a number, like - 5 = 5; exponents, symbols that represent an abbreviation for repeated multiplication, like 6 2 = 6 6; Letting symbols represent mathematical ideas allows us to abbreviate those ideas. A mathematical sentence written in words (English, not Greek) can be more easily understood if it is written symbolically, as long as you know what the symbols mean. For example, the simple English expression the sum of ten and six can be easily abbreviated as the numerical expression because sum means addition, and its symbol is the plus sign (+). ALGEBRA S SYMBOL FOR MULTIPLICATION This section and chapter are about preparing you, the student, for algebra. This section takes you through some of the important definitions and properties that form the foundation of algebra. Algebra is all about using symbols to represent numerical ideas. You will see that letters, called variables, are very big in algebra. Variables represent numbers, some of which we know and some of which we don t know. Sometimes it will be our job to find out which number the variable represents. One thing that may be new to you is the multiplication symbol we use in algebra. Because we use the letter x a lot and I mean a lot we shouldn t use the times sign that is most familiar to you (the x ). This could become too confusing. Instead, we use a raised dot that fits half-way between the numbers. It shouldn t be confused as a decimal point, nor should it be written like one. Robert H. Prior, 2004 page 0.1-1

2 The dot that represents multiplication looks like this: four times five is written 4 5 and it equals 20. We also sometimes use parentheses to represent multiplication; for example, (4)(5) also means four times five. Don t worry, you ll get used to it. Also, when multiplying two letters together, we sometimes don t put any sign between them at all; for example, if we were to multiply the letters w and y together, it could look like w y or just wy. OPERATIONS DEFINED There are four basic operations in mathematics, addition (+), subtraction ( ), multiplication ( ) and division ( ). The written form of an operation, such as 5 + 4, is called an expression. We also have words for the results of the operations as well. To get the result of an operation we must apply the operation. Also, once the operation is applied, the operation disappears! In all, we have 1) a sum is both the expression (written form) and the result of applying addition: the sum of 6 and 2 is written 6 + 2; when addition is applied, the sum of 6 and 2 is 8, and the plus sign disappears. 2) a difference is both the expression (written form) and the result of applying subtraction: the difference between 9 and 5 is written 9 5; when subtraction is applied, the difference between 9 and 5 is 4, and the minus sign disappears. 3) a product is both the expression (written form) and the result of applying multiplication: the product of 5 and 3 is written 5 3; when multiplication is applied, the product of 5 and 3 is 15. 4) a quotient is both the expression (written form) and the result of applying division: the quotient of 14 and 2 is written 14 2; when division is applied, the quotient of 14 and 2 is 7. Robert H. Prior, 2004 page 0.1-2

3 Here is a chart showing the different names of each part of each operation. (Especially important words are in bold. These are words that will be used throughout the text.) Operation Addition (sum) Subtraction (difference) Multiplication (product) Division (quotient) Names (with an example below each) Addend + addend = sum = 8 minuend subtrahend = difference 9 5 = 4 multiplier multiplicand = product 2 3 = 6 Also, factor factor = product dividend divisor = quotient 6 3 = 2 Example 1: Write both the expression and the result of the each of the following. a) the sum of 5 and 1 b) the difference between 10 and 6 c) the product of 4 and 7 d) the quotient of 20 and 4. Answer: Expression Result a) the sum of 5 and = 6 b) the difference between 10 and = 4 c) the product of 4 and = 28 d) the quotient of 20 and = 5 Exercise 1: Write both the expression and the result of the each of the following. Expression Result a) the sum of 9 and 3 = b) the difference of 12 and 4 = c) the product of 5 and 8 = d) the quotient of 30 and 5 = Robert H. Prior, 2004 page 0.1-3

4 Exercise 2: Determine whether the expression shown is a sum, difference, product or quotient. Expression Operation a) 10 6 b) 15 8 c) 27 9 d) To evaluate means to find the value of. When we find the sum of the expression and get 8, we are simply finding the value of 3 + 5; in other words, we are evaluating Exercise 3: Evaluate each of these expressions. a) = b) 9 2 = c) 7 4 = d) 12 3 = MULTIPLICATION AS AN ABBREVIATION Multiplication is an abbreviation for repeated addition. For example, 4 6 means The sum of four 6 s. Written out it is We know this sum to be 24, and we know that 4 6 is 24. The reason we know this so well is because we ve memorized this part in the multiplication table. Also, 6 4 means The sum of six 4 s, which is = 24. When we refer to 6 4 as The sum of six 4 s, we call the number 6 the multiplier of 4. Robert H. Prior, 2004 page 0.1-4

5 As 4 6 = 24, we can say that 24 is the product of 4 and 6. We can also say that 4 and 6 are factors of 24. Of course, 24 has other factors. For example, since 3 8 = 24, 3 and 8 are factors of 24. Also, you know that 2 12 = 24 and even 1 24 = 24. In a sense, we could say that factors are the pieces that make up a product. Of course, we also know that 24 6 = 4 and 24 8 = 3. This is to point out that the factors of 24 can divide evenly (without remainder) into 24. Here, therefore, is the definition of factor: If A, B and C are whole numbers, then (1) if A B = C, then both A and B are factors of C; and (2) if A is a factor of C, then A can divide evenly (without remainder) into C. Example 2: Think of a pair of numbers that are factors of the following. a) 14 b) 15 c) 17 Answer: Use your brain as much as you can to get these; a) 14 = 2 7 so, both 2 and 7 are factors of 14. b) 15 = 3 5 so, both 3 and 5 are factors of 15. c) 17 = 1 17 so, both 1 and 17 are factors of 17. Exercise 4: Think of a pair of numbers that are factors of the following. Use Example 2 as a guide. a) 21 = so, b) 31 = so, c) 35 = so, d) 63 = so, Robert H. Prior, 2004 page 0.1-5

6 VISUALIZING MULTIPLICATION Multiplication can also be represented visually, using geometry. Consider a row of four squares: This row has four squares. If we were to stack a total of 6 of these rows, how many squares would the entire stack have? 1. Each row has four squares Six 4 s is 24: = 24 There are 24 squares in this stack. 6. It s also true that four 6 s is 24: = 24. Shown graphically, we can think of four columns of 6 squares each: = = 24 squares Robert H. Prior, 2004 page 0.1-6

7 THE COMMUTATIVE PROPERTIES We can use the visual representation of multiplication to see a common, but important, property of mathematics called the commutative property. You can see that 6 4 = 24 and that 4 6 = 24. As you can see, when multiplying, it doesn t matter which factor is written first, the resulting product is still the same. The COMMUTATIVE Property of Multiplication If A and B are any numbers, then A B = B A. Example 3: Write each expression two different ways using the commutative property. a) The product of 7 and 3 can be written 7 3 or 3 7 b) The product of 5 and 4 can be written 5 4 or 4 5 Exercise 5: Write each expression two different ways using the commutative property. a) The product of 5 and 6 can be written or b) The product of 4 and 9 can be written or It probably won t surprise you that there is also a commutative property of addition, as well. It s easy to see it with numbers: = The COMMUTATIVE Property of Addition If A and B are any numbers, then A + B = B + A. Example 4: Write each expression two different ways using the commutative property. a) The sum of 7 and 3 can be written or b) The sum of 5 and 4 can be written or Robert H. Prior, 2004 page 0.1-7

8 Exercise 6: Write each expression two different ways using the commutative property. a) The sum of 5 and 6 can be written or b) The sum of 4 and 9 can be written or PARENTHESES Parentheses, both ( and ), are considered grouping symbols. Parentheses group different values together so that they can be treated as one quantity. A quantity is an expression that is considered to have one value. For example, (9 + 2) is considered to be a quantity; it has just one value: 11. In any expression that has parentheses, the quantity within those parentheses must be evaluated first. Example 5: Evaluate each expression a) 2 (6 + 3) b) (10 + 4) (8 1) Procedure: We must first evaluate the quantity. Then we can complete the evaluation. The process looks like this: Answer: a) 2 (6 + 3) 2 (6 + 3) the quantity (6 + 3) should be evaluated first; = 2 9 the value of is 9; there s no need to keep the parentheses, and = 18 since we have no more quantities to evaluate, it s safe to multiply 2 9. b) (10 + 4) (8 1) (10 + 4) (8 1) each quantity can be evaluated separately; = 14 7 the value of is 14, and the value of (8 1) is 7; = 2 to complete, we must apply division. Robert H. Prior, 2004 page 0.1-8

9 Exercise 7: Evaluate each of these completely. a) 6 + (20 5) b) 10 (3 2) c) 24 (9 5) d) 10 (7 + 1) e) (3 5) (8 2) f) (4 + 3) (6 4) g) (12 2) + (5 1) h) (20 + 1) (9 2) Parentheses are also sometimes used as separators to separate values from each other. Even though the quantity inside of parentheses must always be evaluated first, sometimes parentheses, in the end, make no difference. a) If the only operation in the expression is addition, then parentheses aren t really needed: Without parentheses With parentheses With parentheses (9 + 1) (1 + 5) = = = = 15 = 15 = 15 (a) They are all the same. The parentheses aren t really needed. b) If the only operation in the expression is multiplication, then parentheses aren t really needed: (2 3) 5 2 (3 5) = 6 5 = 6 5 = 2 15 = 30 = 30 = 30 (b) Again - they are all the same. The parentheses don t really make a difference. Robert H. Prior, 2004 page 0.1-9

10 Exercise 8: Evaluate each of these. Be sure to evaluate inside of any parentheses first. Notice the similarities between each set of problems. a) (10 + 3) + 7 c) (2 5) 3 b) 10 + (3 + 7) d) 2 (5 3) The exercises, above, lead to an important property about addition and multiplication. THE ASSOCIATIVE PROPERTIES In mathematics, if we wish to make a rule that applies to many (or all) numbers, then we write the rule using letters instead of the numbers. This way, the letters can stand for any number we wish. This next rule uses letters A, B and C to represent any numbers we wish to put in their place. The ASSOCIATIVE Properties of Addition and Multiplication If A, B, and C are any three numbers, then Addition Multiplication (A + B) + C = A + (B + C) = A + B + C (A B) C = A (B C) = A B C Example 6: Write each expression two different ways using parentheses; also, evaluate them. a) b) Answer: Use the Associative Property; notice that the order in which the numbers are written doesn t change. a) = (6 + 9) + 4 = 6 + (9 + 4) = 19 b) (2 5) 8 = 2 (5 8) = = 80 Robert H. Prior, 2004 page

11 Exercise 9: Write each expression two different ways using parentheses; also, evaluate them. Use Example 3 as a guide. a) = = = b) (4 + 9) + 5 = = = c) = = = d) 5 (3 4) = = = THE IDENTITIES Another important property of addition and multiplication is the notion of identity. The identity is a number that won t change the value of another number or quantity. For addition, the identity is 0 (zero), because a) = 6 b) = 8 c) = 3 d) = 15 Notice that adding 0 (zero) to a number doesn t change the value of the number. For multiplication, the identity is 1 (one), because a) 6 1 = 6 b) 8 1 = 8 c) 1 3 = 3 d) 1 15 = 15 Notice that multiplying 1 (one) to a number doesn t change the value of the number. Robert H. Prior, 2004 page

12 Exercise 10: Fill in the blank with the correct number. a) = b) 9 + = 9 c) 0 + = 12 d) = e) + 5 = 5 f) + 0 = 4 g) 7 1 = h) 23 = 23 i) 1 = 18 j) 1 8 = k) 16 = 16 l) 1 = 25 THE ZERO PRODUCT PRINCIPLE A common theme that will run throughout this course is this: With very few exceptions*, the rules of addition are different from the rules of multiplication. In other words, you should not apply the rules of multiplication to a sum (addition) and expect to get accurate results. *Two noted exceptions are the commutative and associative properties. Consider this: What s true for 0 (zero) in addition is not true for 0 in multiplication. We know that, because 0 is the identity for addition, = 3 and = 3. However, the product of 0 and a number, say 3, is 0: 3 0 = 0; that is, the sum of three 0 s is 0: = 0. It s also true, by the commutative property, that 0 3 = 0. This leads us to: The Zero Product Principle A 0 = 0 and 0 B = 0. Furthermore, if A B = 0, then one of the factors (A or B) must be 0: either A = 0 or B = 0. (It s even possible that both A and B could be 0 because 0 0 = 0.) Robert H. Prior, 2004 page

13 Answers to each Exercise Section 0.1 Exercise 1: a) the sum of 9 and 3 is = 12 b) the difference of 12 and 4 is 12 4 = 8 c) the product of 5 and 8 is 5 8 = 40 d) the quotient of 30 and 5 is 30 5 = 6 Exercise 2: a) product b) difference c) quotient d) sum Exercise 3: a) = 13 b) 9 2 = 7 c) 7 4 = 28 d) 12 3 = 4 Exercise 4: a) 21 = 3 7 so, both 3 and 7 are factors of 21. b) 31 = 1 31 so, both 1 and 31 are factors of 31. c) 35 = 5 7 so, both 5 and 7 are factors of 35. d) 63 = 7 9 so, both 7 and 9 are factors of 63. Exercise 5: a) The product of 5 and 6 can be written 5 6 or 6 5 b) The product of 4 and 9 can be written 4 9 or 9 4 Exercise 6: a) The sum of 5 and 6 can be written or b) The sum of 4 and 9 can be written or Exercise 7: a) 6 + (20 5) = = 10 b) 10 (3 2) = 10 6 = 4 c) 24 (9 5) = 24 4 = 6 d) 10 (7 + 1) = 10 8 = 80 e) (3 5) (8 2) = 15 4 = 11 f) (4 + 3) (6 4) = 7 2 = 14 g) (12 2) + (5 1) = = 11 h) (20 + 1) (9 2) = 21 7 = 3 Robert H. Prior, 2004 page

14 Exercise 8: a) (10 + 3) + 7 = = 20 c) (2 5) 3 = 10 3 = 30 b) 10 + (3 + 7) = = 20 d) 2 (5 3) = 2 15 = 30 Exercise 9: a) = (11 + 8) + 6 = 11 + (8 + 6) = 25 b) (4 + 9) + 5 = 4 + (9 + 5) = = 18 c) = (7 5) 2 = 7 (5 2) = 70 d) 5 (3 4) = (5 3) 4 = = 60 Exercise 10: a) = 4 b) = 9 c) = 12 d) = 6 e) = 5 f) = 4 g) 7 1 = 7 h) 23 1 = 23 i) 1 18 = 18 j) 1 8 = 8 k) 1 16 = 16 l) 25 1 = 25 Robert H. Prior, 2004 page

15 Section 0.1 Focus Exercises 1. In your own words, use an example to describe or define the following. a) Product b) Sum c) Quotient d) Difference 2. In your own words, describe the following. Use an example for both addition and multiplication. a) Associative Property for addition for multiplication b) Commutative Property for addition for multiplication Robert H. Prior, 2004 page

16 3. Evaluate each of these completely. a) 9 + (16 2) b) 12 (2 5) c) 36 (4 + 5) d) 6 (10 2) e) (8 + 9) (6 2) f) (18 6) (2 + 5) g) (3 + 15) (14 7) h) (5 6) + (24 3) Robert H. Prior, 2004 page