Chapter 1: Fundamentals of Algebra Lecture notes Math 1010

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1 Section 1.1: The Real Number System Definition of set and subset A set is a collection of objects and its objects are called members. If all the members of a set A are also members of a set B, then A is a subset of B. Ex.1 Days of the week. Ex.2 Natural numbers. Ex.3 Integers. Notice that the set of natural numbers is a subset of the set of integers. 1

2 Ex.4 Numbers. Set of natural numbers: Set of whole numbers: Set of integers: Set of rational numbers: the rational numbers are of the form x y, where x and y are integers and y 0. For example, 1 2, 3 5, 9 4. When expressed in decimal forms, rational numbers are either terminating decimals with a finite number of digits (for example, 1 4 = 0.25) or repeating decimals in which a pattern repeats over and over (for example, 1 3 = = 0. 3). Set of irrational numbers: the irrational numbers are numbers that cannot be expressed in the form x y, where x and y are integers and y 0. When written as decimals, irrational numbers neither terminate nor have a repeating pattern. For example, π, 2: π = , 2 = Set of real numbers: the set of real numbers consists of both rational and irrational numbers. 2

3 Rounding The basic process of rounding numbers takes two steps: Step 1: Decide which decimal place (for example, tens, ones, tenths or hundredths) is the smallest that should be kept. Step 2: Look at the number in the nearest place to the right (for example, if rounding the tenths, look at hundredths). If the value in the next place is less than 5 round down, if it is 5 or greater than 5, round up. Ex rounded to the nearest thousandth is rounded to the nearest hundredth is rounded to the nearest tenth is rounded to the nearest one is rounded to the nearest ten is rounded to the nearest hundred is 400. Ex.6 Round π to four decimal places. Ex.7 Classifying real numbers Which of the numbers in the set are (1) natural numbers (2) integers (3) rational numbers (4) irrational numbers {3, 2, 4, π, 0, 4} 9 Order on the real number line If the real number a lies to the left of the real number b on the real number line, then a is less than b, which is written a < b. This relationship can also be described by saying that b is greater than a and writing b > a. The expression a b means that a is less than or equal to b, and the expression b a means that b is greater than or equal to a. The symbols <, >,, are called inequality symbols. When asked to order two numbers, you are simply being asked to say which of the two numbers is greater. Law of Tricotomy For any two real numbers a and b, exactly one of the following orders must be true: a < b, a = b, a > b. 3

4 Ex.8 Ordering real numbers (1) 2 < 1 (2) 0 < 5 5 (3) 12 > 9 23 (4) 1 4 > 1 2 Distance between two real numbers If a and b are two real numbers such that a b, then the distance between a and b is given by b a. Note that if a = b the distance between a and b is 0. If a b, the distance between a and b is always positive. Ex.9 Find the distance between each pair of real numbers. (1) 2 and 3 (2) 0 and 5 (3) 4 and 1 (4) 1 and 1 2 Opposites and additive inverses Let a be a real number. (1) a is the opposite of a. (2) The opposite of a negative number is called double negative: ( a) = a. (3) Opposite numbers are also referred to as additive inverses because their sum is 0: a + ( a) = 0. Definition of absolute value The distance between a real number a and 0 is called absolute value of a and it is { a, if a 0 a = a, if a < 0 Ex.10 (1) 2 = 2 (2) 9 14 = 9 14 (3) = (4) 3 = (3) = 3 4

5 Ex.11 Place the correct symbol (<, >, or =) for each pair of real numbers. (1) 2, 1 (2) 3, 3 (3) 11, 14 (4) 4, 4 Distance between two real numbers If a and b are two real numbers, then the distance between a and b is given by b a = a b. Ex.12 Find the distance for each pair of real numbers. (1) 2 and 3 (2) 1 and 5 Section 1.2: Operations with Real Numbers Addition and subtraction of two real numbers To add two real numbers with like signs, add their absolute values and attach the common sign to the result. To add two real numbers with unlike signs, subtract the smaller absolute value from the greater absolute value and attach the sign of the number with the greater absolute value. The result of adding two real numbers is the sum of the two numbers, and the two real numbers are the terms of the sum. To subtract the real number b from the real number a: a b = a + ( b) The result of subtracting two real numbers is the difference of the two numbers. Ex.1 (1) (2) 4 + ( 5.2) (3) 9 21 (4) 2.1 ( 3.4) 5

6 Ex.2 Evaluate ( 2) Addition and subtraction of two fractions (1) Like denominators: a c + b c = a + b c a c b c = a b c (2) Unlike denominators: a b + c ad + bc a = d bd b c ad bc = d bd Then simplify the result. Another way is to find the least common multiple of the denominators (see examples). Ex.3 (1) (2) Multiplication and division of two real numbers To multiply two real numbers with like signs, multiply their absolute values. To multiply two real numbers with unlike signs, multiply their absolute values and attach a minus sign to the result. The product of zero and any other number is zero. The result of adding two real numbers is the product of the two numbers, and the two real numbers are the factors of the product. To divide the real number a by the real number b 0: a b = a 1 b = a b The result of dividing two real numbers is the quotient of the two numbers. The number a is the dividend and the number b is the divisor. 6

7 Ex.4 (1) 3 5 (2) ( 7)( 2) (3) 9 ( 3) (4) 7 21 Multiplication and division of two fractions (1) Multiplication: (2) Division: Then simplify the result. a b c d = ac bd a b c d = ad bc b, d 0 b, d 0 Ex.5 (1) (2)

8 Exponential notation Let n be a positive integer and let a be a real number. Then the product of n factors of a is given by a n = a a a }{{} n times In the exponential form a n, a is the base and n is the exponent. Writing the exponential form a n is called raising a to the nth power. When a number is raised to the first power we will just write the number (for example 3 1 = 3). Raising a number to the second power is called squaring the number. Raising a number to the third power is called cubing the number. Ex.6 Evaluate the following expressions. (1) ( 5) 4 (2) 5 4 (3) ( 2 3 ) 3 (4) ( 5) 3 (5) 2 4 Order of operations To evaluate an expression involving more than one operation, use the following order. (1) First do operations that occur within symbols of grouping. (2) Then evaluate powers. (3) Then do multiplications and divisions from left to right. (4) Finally do additions and subtractions from left to right. Then simplify the result. Another way is to find the least common multiple of the denominators (see examples). Ex.7 Evaluate the following expressions. (1) 20 ( 5) (2) (3) 1 [4 (3 4 6)] (4) (2 3) 2 (5)

9 Section 1.3: Properties of Real Numbers Basic properties of real numbers Let a, b, c represent real numbers, variables or algebraic expressions. Commutative Property of Addition: For example, = Commutative Property of Multiplication: For example, 2 5 = 5 2. Associative Property of Addition: For example, (1 + 3) + 11 = 1 + (3 + 11). Associative Property of Multiplication: a + b = b + a ab = ba (a + b) + c = a + (b + c) (a b) c = a (b c) For example, (2 5) 3 = 2 (5 3). Distributive Properties: (1) a(b + c) = ab + ac For example, 2(3 + 4) = (2) (a + b)c = ac + bc For example, (2 + 4)3 = (3) a(b c) = ab ac For example, 2(6 3) = (4) (a b)c = ac bc For example, (5 3)4 = Additive Identity Property: a + 0 = 0 + a = a For example, = = 5. Multiplicative Identity Property: a 1 = 1 a = a For example, 34 1 = 1 34 = 34. Additive Inverse Property: a + ( a) = 0 For example, 3 + ( 3) = 0. Multiplicative Inverse Property: a 1 a = 1, a 0 For example, = 1. 9

10 Additional properties of real numbers Let a, b, c represent real numbers, variables or algebraic expressions. Properties of Equality Addition Property of Equality: Multiplication Property of Equality: Cancellation Property of Addition: Cancellation Property of Multiplication: Properties of Zero Multiplication Property of zero: Division Property of Zero: Division by Zero is Undefined: Properties of Negation Multiplication by 1: Placement of Negative Signs: Product of Two Opposites: If a = b, then a + c = b + c If a = b, then ac = bc If a + c = b + c, then a = b If ac = bc, then a = b ( 1) a = a, 0 a = 0 0 a = 0, a 0 a is undefined 0 ( 1)( a) = a ( a)b = (ab) = a( b) ( a)( b) = ab Ex.1 Prove that if a + c = b + c, then a = b. 10

11 Ex.2 In the solution of the equation 3x + 4 = 6 identify the property of real numbers that justifies each step. 11

12 Section 1.4: Algebraic Expressions Algebraic expression A collection of letters, called variables and real numbers, called constants combined using the operations of addition, subtraction, multiplication, or division is called an algebraic expression. The terms of an algebraic expression are those parts that are separated by addition. The terms with variables are called variables terms and the terms with only constants are called constant terms. The numerical factor of a term is called the coefficient. Ex.1 Identify the terms and the coefficients in the following algebraic expressions. (1) 5x 1 3 (2) 2y + 6x 4 (3) 2 x + 3x2 3y Simplifying algebraic expression Two terms are said to be like terms if they are both constant terms or if they have the same variable factor. One way to simplify an algebraic expression is to combine like terms. Another way is to remove symbols of grouping and then combine like terms. 12

13 Ex.2 Simplify the following algebraic expressions. (1) 2x + 3x 3 (2) y 2y (3) 3xy 2 + 5x 7xy 2 + 4x (4) 4(x 2 + 2) + x(x 3) (5) 5x 2x[3 + 2(x 7)] (6) 3x 2 (5x 3 ) + 9x 5 13

14 Evaluating algebraic expression To evaluate an algebraic expression, substitute numerical values for each of the variables in the expression. Ex.3 Evaluate each algebraic expression when x = 3 and y = 1. (1) 2x 3 (2) x (3) 3xy 2 + 4x (4) y x (5) x 2 4x + 5y (6) 2xy x+1 14

15 Section 1.5: Constructing Algebraic Expressions Constructing algebraic expressions (1) Addition: Key words are: sum, plus, greater than, increased by, more than, exceeds, total of. For example, the sum of 3 and y is 3 + y. (2) Subtraction: Key words are: difference, minus, less than, decreased by, subtracted from, reduced by, the remainder. For example, four less than x is x 4. (3) Multiplication: Key words are: product, multiplied by, twice, times, percent of. For example, five times x is 5x. (4) Division: Key words are: quotient, divided by, ratio, per. For example, x divided by two is x 2. 15

16 Ex.1 Translate the following verbal phrases. (1) Four less than the product of two and x. (2) The difference of a and seven, all divided by four. (3) The sum of two and a number. (4) Seven less than twice the sum of a number and 5. 16

17 Ex.2 Without using a variable, write a verbal description for each expression. (1) 3x 5 (2) 3+x 4 (3) 2(5x + 3) Constructing mathematical models To construct a mathematical model you need to do three steps: (1) Construct a verbal model that represents the problem situation. (2) Assign labels to all quantities in the verbal model. (3) Construct an algebraic expression. Ex.3 A cash register contains n nickels and d dimes. Write an algebraic expression for this amount of money in cents. 17

18 Ex.4 A person who is riding a bicycle travels at a constant rate of 12 miles per hour. Write an algebraic expression showing how far the person can ride in t hours. Ex.5 A person adds k liters of fluid containing 55% antifreeze to a car radiator. Write an algebraic expression that indicates how much antifreeze was added. Ex.6 A person s annual salary is y dollars. Write an expression for the person s monthly salary. 18

19 Labels for integers Two integers are consecutive if they differ by 1. Let n represents an integer. Then even integers, odd integers, and consecutive integers can be represented as follows. (1) 2n denotes an even integer. For example, 2 = 2 1, 8 = 2 4. (2) 2n 1 and 2n + 1 denote odd integers. For example, 5 = , 9 = 2 5 1, 11 = (3) {n, n + 1, n + 2,...} denotes the set of consecutive integers. Ex.7 Write an expression for the following phrase: The sum of two consecutive integers, first of which is n. Ex.8 Write expressions for the perimeter and the area of the rectangle with sides of length (w + 12) in. and 2w in. 19