Real Numbers and Algebraic Expressions Chapter 1 Real Numbers and Algebraic Expressions 1.2 1.3 1.4 1

Algebraic Expressions and Sets of Numbers 1.2 Algebraic Expressions and Sets of Numbers Objectives: Identify and evaluate algebraic expressions Identify natural numbers, whole numbers, integers, and rational & irrational numbers Find the absolute value of a number 2

Algebraic Expressions A variable is a letter used to represent any number. An algebraic expression is formed by numbers and variables connected by the operation of addition, subtraction, multiplication, division, raising to powers, and/or taking roots.

Evaluating Algebraic Expressions To evaluate an algebraic expression, substitute the numerical value for each variable into the expression and simplify the result. Example 1 Evaluate the expression for the given value. 5x – 2y when x = 8 and y = 2

Sets of Numbers Natural numbers: {1, 2, 3, 4, 5, 6 . . .} Whole numbers: {0, 1, 2, 3, 4 . . .} Integers: {. . . –3, –2, –1, 0, 1, 2, 3 . . .} The symbol, consisting of three dots, is called an ellipsis and means to continue in the same pattern.

Set Builder Notation A set can be written in set builder notation. For Example: { x | x is an even natural number less than 10} x is an even natural number less than 10 The set of all x such that Answer: {2, 4, 6, 8}

Set Builder Notation A set that contains no elements is called the empty set (or null set) symbolized by { } or {x|x is a month with 32 days} Answer: the empty set because no month has 32 days. The set has no elements. For Example:

Example 3 List the elements in each set. a. {x|x is a natural number greater than 200} b. {x|x is a whole number between 2 and 8} Solution a. {201, 202, 203, …} b. {3, 4, 5, 6, 7}

Identifying Numbers Real Numbers: {x|x corresponds to a point on the number line} Rational numbers: { |a and b are integers and b ≠ 0} Irrational numbers: {x|x is a real number and x is not a rational number} Can be written as a fraction Can NOT be written as a fraction

Absolute Value The absolute value of a real number a, denoted by |a|, is the distance between a and 0 on the number line. Distance of 4 |– 4| = 4 2 – 2 1 3 4 5 – 1 – 3 – 4 – 5 Distance of 5 |5| = 5 2 – 2 1 3 4 5 – 1 – 3 – 4 – 5

|3.4| = Example 6 Find each absolute value. a. = b. The negative sign outside the absolute value bars means “the opposite of the absolute value of 3.4” c.. This is read “the opposite of the absolute value of negative 6” |3.4| = 3.4 |6| = 6

Translating Phrases Beware of: subtracted from fewer than less than Addition (+) Subtraction (–) Multiplication (·) Division () sum plus added to more than increased by total difference of minus subtracted from less than decreased by less product times multiply twice of quotient divide into ratio Objective A Beware of: subtracted from fewer than less than

= a. 5 decreased by a number b. The quotient of a number and 12 Example 8 Translate each phrase to an algebraic expression. Use the variable x to represent each unknown number. a. 5 decreased by a number b. The quotient of a number and 12 = 5 – x = c. The sum of 4 and twice a number d. x less than the product of y and z = 4 + 2x = yz – x

Example 9 The symbol means an element of The symbol means not an element of. 5.2 of the integers 5 of the integers

Classification of Numbers Real Numbers (R) Rational (Q) Irrational (I) Can be written as fractions Ex. -7, 5, 5.2, 5½ , √16 Cannot be written as fractions Ex. Π, √20, ℮, -√24 Integers (Z) List which sets each is an element of: 5 √5 –2/5 List which sets each is an element of: 5 R, Q, Z, W, N √5 R, I –2/5 R, Q …-3,-2,-1,0,1,2,3,… Whole (W) 0,1,2,3,… Natural (N) 1,2,3,…

Algebraic Expressions and Sets of Numbers 1.2 summary Algebraic Expressions and Sets of Numbers Objectives: Identify and evaluate algebraic expressions Identify natural numbers, whole numbers, integers, and rational & irrational numbers Find the absolute value of a number 16

Operations on Real Numbers 1.3 Operations on Real Numbers Objectives: Add, subtract, multiply, & divide real numbers Evaluate expressions containing exponents Find roots of numbers Use the order of operations Evaluate algebraic expressions 17

Multiplying/Dividing Real Numbers Adding Real Numbers Adding two numbers with the same sign Add their absolute values. Use common sign as sign of sum. Adding two numbers with different signs Take difference of absolute values (smaller subtracted from larger). Use the sign of larger absolute value as sign of sum. Multiplying/Dividing Real Numbers Multiplying or dividing two real numbers with same sign Result is a positive number Multiplying or dividing two real numbers with different signs Result is a negative number

Example 1 15 + (3) = 9.1 + 1.2 = 4 + (15) = b. 4 + (9) = c. 15 + 21 = d. 9.1 + (2.3) = e. 2 – 11 = 19 f. 4 – (2) = g. 15  3 = h. 9.1  (1.2) = 4 + (2) = 2 5 6 18 11.4 10.3 2 + (11) = 9

Example 2 Multiply or Divide. a. (9)(2) b. 1.2(0.4) = 18 c. 0(12) d. (8)(2)(1)(2) = 0 e. f. g. h. = 0.48 = 32

Exponents Exponents that are natural numbers are shorthand notation for repeating factors. 34 = 3 · 3 · 3 · 3 3 is the base 4 is the exponent

Example 3 Evaluate each expression. a. b. c. d. 34 = 3 · 3 · 3 · 3 = 81 (–5)2 = (– 5)(–5) –62 = – (6)(6) = 25 = –36

Example 4 Find the value of each. a. b. c. d. e. f. = 2 = –5

2. Multiply or divide in order from left to right. Order of Operations Simplify expressions using the order that follows. If grouping symbols such as parentheses are present, simplify expressions within those first, starting with the innermost set. If fraction bars are present, simplify the numerator and denominator separately. 1. Evaluate exponential expressions, roots, or absolute values in order from left to right. 2. Multiply or divide in order from left to right. 3. Add or subtract in order from left to right. Objective D

Example 5 Simplify. a. b.

Example 6 Simplify:

Example 7 2 Simplify: Solution

Example 8 Evaluate each expression when x = 5 and y = 2. a. 4x – 5y b. 3y2 Solution

Add on: hw #16 and #23 help 2

Operations on Real Numbers 1.3 summary Operations on Real Numbers Objectives: Add, subtract, multiply, & divide real numbers Evaluate expressions containing exponents Find roots of numbers Use the order of operations Evaluate algebraic expressions 30

Properties of Real Numbers 1.4 Properties of Real Numbers Objectives: Use operation and order symbols to write mathematical sentences Identify identity numbers and inverses Identify and use the commutative, associative, and distributive properties Write algebraic expression Simplify algebraic expressions 31

Write each sentence as an equation The difference of x and 4 amounts to 15 x – 4 = 15 5 more than the product of 3 and b is 7. 3b + 5 = 7 The quotient of 9 and v is 2 more than v. 9/v = v + 2 3 added to one-half z is the same as 8 more than z 3 + ½ z = z + 8

Reciprocal of original # Complete the table Original Number Opposite of original # Reciprocal of original # 8 –2/3 – 8 1/8 2/3 – 3/2 1/0 is undefined

Properties Commutative Property: Associative Property: giving the same result irrespective of the order of arguments for example: 2 + 3 = 3 + 2 commutative property of addition 2•3 = 3•2 commutative property of multiplication but 5 – 2 ≠ 2 – 5 subtraction is NOT commutative Associative Property: having the property that bracketing of its arguments may be disregarded (in other words: everything remains in the same order the ()’s have been moved) for example: (2 + 3) + 5 = 2 + (3 + 5) associative property of addition (2•3)•5 = 2•(3•5) associative property of multiplication Distributive Property: for example: 5(2x – 3) = 10x – 15 (2x + 6y)0.2 = 0.4x + 1.2y

Use the commutative property to write an equivalent expression 5x + 3y m•n (2 + x) + 5 = 3y + 5x = n•m = (x + 2) + 5 or = 5 + (2 + x)

Use the associative property to write an equivalent expression a(b•c) (2 + x) + 5 = (5 + x) + 3 = (a•b)c = 2 + (x + 5)

Properties of Real Numbers 1.4 summary Properties of Real Numbers Objectives: Use operation and order symbols to write mathematical sentences Identify identity numbers and inverses Identify and use the commutative, associative, and distributive properties Write algebraic expression Simplify algebraic expressions 37