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Math Analysis Notes Section 1.1 Page 1 of 3 Section 1.1: Real Numbers Big Idea: ALGEBRA IS A SYSTEM FOR SIMPLIFYING AND UNDOING MULTI-STEP ARITHMETIC CALCULATIONS! This section is a review of how to represent the properties of real numbers, negative numbers, fractions, sets, and absolute values in the (terse) language of mathematics. Big Skill: You should be able to represent intervals of real numbers using set-builder notation, interval notation, and graphical methods. Subsets of the real number system: Natural numbers: 1, 2,3, 4, Integers: , 3, 3, 1, ,1, 2,3, m r r , where m, n Some rational numbers have a finite number of digits in n 1 their decimal representation (½ = 0.5), while others have an infinite number of repeating digits 0.142857 . 7 Rational numbers: Practice: Convert 0.87 to a fractional form. Irrational numbers: Any number that can not be written as a rational number. An irrational number has an infinite number of non-repeating digits in its decimal representation. We can only represent an irrational number exactly by its description, such as 2 the positive solution to the equation x2 = 2, or = the ratio of the circumference to the diameter of any circle. Real numbers: = the union of the sets of all rational and irrational numbers. Properties of Real Numbers: Commutative Property of Addition: a b b a Commutative Property of Multiplication: ab ba Associative Property of Addition: a b c a b c Associative Property of Multiplication: ab c a bc Distributive Property: a b c ab ac , b c a ba ca I state these properties to remind you that any algebraic expression, no matter how complicated, boils down to doing repeated arithmetic operations on a pair of numbers, and because you are going to encounter mathematical objects in higher mathematics classes where these properties don’t always apply. Math Analysis Notes Section 1.1 Page 2 of 3 Practice: State the properties of real numbers used in the following simplification: 2 x 5 3x 2 x 2 5 3x 2 x 10 3 x 10 2 x 3 x 10 2 x 3 x 10 2 3 x 10 5 x More Properties of Addition: Additive Identity: The number 0; 0 leaves a number unchanged after addition: a + 0 = a. Additive Inverse: The negative of a number; a number plus its additive inverse equals the additive identity: a + -a = 0. Inverse of the Addition Operation: Subtraction “undoes” addition. Subtraction is defined as the addition of the negative of a number: a – b = a + (-b). Properties of Negatives: These follow from the properties of addition. 1. (-1)a = -a 2. –(-a) = a 3. (-a)b = a(-b) = -(ab) 4. –(a + b) = -a – b 5. –(a – b) = b – a More Properties of Multiplication: Multiplicative Identity: The number 1; 1 leaves a number unchanged after multiplication: a·1 = a. Multiplicative Inverse: The reciprocal of a number; a number times its multiplicative inverse equals the multiplicative identity: a·(1/a) = 1. Inverse of the Multiplication Operation: Division “undoes” multiplication. Division is defined as multiplying by the reciprocal of a number: a b = a·(1/b). Properties of Fractions: These follow from the properties of multiplication. a c ac 1. b d bd a c a d 2. b d b c a b ab 3. c c c a c ad bc 4. b d bd ac a 5. bc b a c 6. If , then ad bc b d Math Analysis Notes Section 1.1 Page 3 of 3 Ways to represent sets of real numbers: 1. Listing e.g. The set A of all integers whose absolute value is less than 2 can be written as A = {-1, 0, 1} 2. Set-builder notation (same example): A = {x | x is an integer and |x| < 2}, or A = {x | x and |x| < 2} 3. Interval notation (combined with set builder notation) A = {x | x (-2, 2) and x Z} 4. Interval notation for real numbers: a. [ or ] means the endpoint is in the set, b. ( or ) means the endpoint is not in the set c. [1, 5) means all numbers between 1 and five, including 1 but not including 5 d. (-1, ) means all numbers greater than -1 5. Combining sets using Union ( ): Forms a new set by combining all elements from both sets. {1, 2, 3, 4, 5} {4, 5, 6, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8} Practice: [-1, 1) (0, 2] = 6. Combining sets using Intersection ( ): Forms a new set by combining only common elements from both sets. {1, 2, 3, 4, 5} {4, 5, 6, 7, 8} = {4, 5} Practice: [-1, 1) (0, 2] = Absolute value: a if a 0 Definition: The absolute value of a number a is a . a if a 0 Properties of the Absolute Value: 1. a 0 2. a a 3. ab a b 4. a a b b Linear Distance between points: If a and b are points on the real line, then the distance between them is d a, b b a . Technology Practice: Use Microsoft Excel to create a filled-in table for problem #80 from section 1.1