1. The Real Number System Section P.1

2. Set, Unions, and Intersections Part 1

3. The Real Number System In mathematics it is useful to place numbers with similar characteristics into sets. Integers – {…, -3, -2, -1, 0, 1, 2, 3, …} Rational Numbers – All terminating or repeating decimals. – All integers are rational numbers. Irrational Numbers – All nonterminating, nonrepeating decimals. Real Numbers – All rational and irrational numbers. Prime Number – A positive integer other than one that has no positive integer factors other than itself and 1. Composite Number – A positive integer greater than 1 that is not a prime number.

4. The Real Number System Example: Classify each of the following as integer, rational, irrational, real, prime, or composite. a)-0.2 b)0 c)0.333333…  e)6 f)7 g)41 h)51 i)0.71771777177771…

5. The Real Number System Each number of a set is called an element of the set. – For instance, if C = {2, 3, 5}, then the elements of C are 2, 3, and 5. – The notation 2 Є C is read “2 is an element of C.” Set A is a subset of set B if every element of A is also and element of B. – For instance the set of negative integers {-1, -2, -3, -4,…} is a subset of the set of integers. – The set of positive integers {1, 2, 3, 4,…}, also known as natural numbers, is also a subset of the set of integers.

6. The Real Number System The empty set, or null set, is the set that contains not elements. – For example, the set of people who have run a two- minute mile is the empty set. A finite set is characterized as a set in which all elements of the set can be listed. – For example, the set of natural numbers less than 6 is {1, 2, 3, 4, 5}. All elements can be listed. In an infinite set it would be impossible to list all elements. – The set of all natural numbers is an infinite set because not all elements can be listed.

7. The Real Number System Sets are often written using set-builder notation. Set-builder notation can be used to describe almost any set, but it is especially useful when writing infinite sets. – The set of all real numbers greater than 2 is written: {2n|n Є natural numbers} – Read as, “the set of elements 2n such that n is a natural number.” {x|x > 2, x Є real numbers} – Read as, “the set of x such that x is greater than 2 and x is an element of the real numbers.”

8. The Real Number System Example: List the four smallest elements in: {n 3 |n Є natural numbers}

9. Union and Intersection of Sets The union of two sets, written A U B, is the set of all elements that belong to either A or B. A U B = {x|x Є A and x Є B} Example: Given A = {2, 3, 4} and B = {0, 1, 2, 3} then A U B = {0, 1, 2, 3, 4} The intersection of two sets, written A ∩ B, is the set of all elements that are common to both A and B. A ∩ B = {x|x Є A and x Є B}

10. Union and Intersection of Sets Example: Find each intersection or union given: A = {0, 1, 2, 3, 4, 5, 6, 7, 10, 12} B = {0, 3, 6, 12, 15} C = {1, 2, 3, 4, 5, 6, 7} – A U C – B ∩ C – A ∩ (B U C) – B U (A ∩ C)

11. Absolute Value (Distance), Interval Notation, and Order of Operations Part 2

12. Absolute Value and Distance Absolute value denotes the distance between 0 and a point on the real number line. |3| = 3|-3| = 3 Distance between two points on a number line is given by: d(a, b) = |a – b| d(-2, 5) = |-2 – 5| = |-7| = 7

13. Absolute Value and Distance Example: Express the distance between a and -3 on the number line using absolute value.

14. Interval Notation (a, b) – Represents all real number between a and b, not including a and b. Open interval [a, b] – Represents all real numbers between a and b, including a and b. Closed interval (a, b] or [a, b) – Represents all real numbers between a and b, not including a but including b, or vice versa. Half-open interval

15. Interval Notation (-∞, a) – Represents all real numbers less than a. (b, ∞) – Represents all real numbers greater than b. (-∞, a] – Represents all real numbers less than or equal to a. [b, ∞) – Represents all real numbers greater than or equal to b. (-∞, ∞) – Represents all real numbers.

16. Graph a Set in Interval Notation Example: Graph (-∞, 3]. Write the interval in set- builder notation.

17. Graph Intervals Example: Graph the following. Write a) and b) using interval notation. Write c) and d) using set-builder notation. a){x|x ≤ -1} U {x|x ≥ 2} b){x|x ≥ -1} ∩ {x|x < 5} c)(-∞, 0) U [1, 3] d)[-1, 3] ∩ (1, 5)

18. Order of Operations If grouping symbols are present, evaluate by performing the operations within the grouping symbols, innermost grouping symbols first, while observing the order given in steps 1 to 3. Step 1 – Evaluation exponential expressions. Step 2 – Do multiplication and division as they occur from left to right. Step 3 – Do addition and subtraction as they occur from left to right.

19. Variable Expressions 3x 2 – 4xy + 5x – y – 7 Terms – Variable Terms – Constant Terms – Numerical Coefficient Evaluate – Arrive at a single numerical answer/solution.

20. Evaluating a Variable Expression Example: Evaluate x 3 – y 3 x 2 + xy + y 2 when x = 2 and y = -3

21. Evaluating a Variable Expression Example: Evaluate (x + 2y) 2 – 4z when x = 3, y = -2, and z = -4

22. Simplifying Variable Expressions and Properties of Real Numbers Part 3

23. Simplifying Variable Expressions Multiplication – Product Division – Quotient Addition – Sum Subtraction – Difference

24. Properties of Real Numbers Addition Properties Closurea + b is a unique real number Commutative a + b = b + a Associative (a + b) + c = a + (b + c) Identity There exists a real number 0 such that a + 0 = 0 + a = a Inverse For each real number a, there is a unique real number -a such that a + (-a) = (-a) + a = 0 Distributivea(b + c) = ab + ac

25. Properties of Real Numbers Multiplication Properties Closureab is a unique real number Commutative ab = ba Associative (ab)c = a(bc) Identity There exists a real number 0 such that a × 1 = 1 × a = a Inverse For each nonzero real number a, there is a unique real number 1/a such that a × 1/a = 1/a) × a = 1 Distributivea(b + c) = ab + ac

26. Properties of Real Numbers Example: Identify the property of real numbers. a)(2a)b = 2(ab) b)(⅕)11 is a real number c)4(x + 3) = 4x + 12 d)(a + 5b) + 7c = (5b + a) + 7c e)(½ ∙ 2)a = 1 ∙ a f)1 ∙ a = a

27. Simplifying Variable Expressions Question: Are the terms 2x 2 and 3x like terms? Example: Simplify the following expressions. a)5 + 3(2x – 6) b)4x – 2[7 – 5(2x – 3)]

28. Properties of Equality Let a, b, and c be real numbers – Reflexive a = a – Symmetric If a = b, then b = a. – Transitive If a = b and b = c, then a = c. – Substitution If a = b, then a may be replaced by b in any expression that involves a.

29. Properties of Equality Example: Identify the property of equality illustrated in each statement. a)If 3a + b = c, then c = 3a + b. b)5(x + y) = 5(x + y) c)If 4a – 1 = 7b and 7b = 5c + 2, then 4a – 1 = 5c + 2. d)If a = 5 and b(a + c) = 72, then b(5 + c) = 72.