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College Algebra—Winter 2005 Sections P.1&2 Section P.1--The Real Number System Our real numbers consist of several classes of numbers that can be grouped as in the following diagram. Notice that I have included all of the real numbers within the complex numbers. Natural numbers are those you learned before going to school: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...} Numerals 1 2 3 4 5 6 7 8 9 10 English one two three four five six seven eight nine ten Spanish uno dos tres cuatro cinco seis siete ocho nueve diez German eins zwei drei vier funf sechs sieben acht neun zehn French un deux trois quatre cinq six sept uit nouf dix Chinese yi er san si wu liu qi ba jiu shi Swahili moja mbili tatu nne tano sita saba nane tisa kumi Language? Whole numbers include all of the natural numbers and zero. Integers contain all of the natural numbers, zero and all of their negatives. Rational Numbers consist of fractions where the numerator and denominator are integers. The denominator is never zero! The decimal expansion of a rational number either 1 0. 5 or repeats like 13 0. 3333. . . terminates like 2 © W Clarke 1 1/4/2005 College Algebra—Winter 2005 Sections P.1&2 The set of Irrational Numbers is infinitely bigger than any of the above sets. Numbers like 2 , 3 , and are all irrational. Complex numbers include all of the above plus those numbers that involve i where i2 −1 . So 3 − 4i , log−2 , and sin−1 2 are all complex and not real. All real numbers can be place on a number line. We usually write the number line with 0 somewhere in the middle and positive integers spaced evenly to the right and negative integers spaced evenly to the left. We use the symbols: a b if " a is less than b " when a lies to the left of b on the number line. a b if " a is greater than b " when a lies to the right of b on the number line. Two numbers are called "opposites" if one number is the same as another except that they have differing signs. I personally dislike the term opposite when used to describe numbers and will not use it very often. The "absolute value" of a number is its positive distance from the origin 0 . |−5| 5 and |5| 5. Rules for algebraic operations We have attached names to various rules that we have made concerning multiplication and addition. These properties become very important as you move on to upper division algebra. I shall use them freely throughout the rest of the quarter. You may wish to refer back to this first lesson as a review. Expect one or more these properties to be on the next test! Commutative Property states that the order of multiplying or adding is not important: Example Multiplication Addition 23 32 23 32 Associative Property states how we multiply or add three numbers: Multiplication Addition Example 2 3 4 2 3 4 2 3 4 2 3 4 Identity Property Talks about multiplication by one and addition by zero. Multiplication Addition Example 1 3 3 1 3 0 3 3 0 3 © W Clarke 2 1/4/2005 College Algebra—Winter 2005 Sections P.1&2 Zero Property of Multiplications Talks about multiplication by zero. Multiplication Addition Example 0 3 3 0 0 Inverse Property Talks about multiplying to get an answer of one, or adding two numbers to get an answer of zero. Multiplication Example 1 3 3 3 1 3 Addition 1 −3 3 3 −3 0 Distributive Property States how we combine multiplication and addition in the same problem. Example: 2 3 4 2 3 2 4 If you wish to appear erudite, you may refer to these properties as the "field properties of the real numbers." Convention: Always do operations in the following order. 1. Perform operations inside grouping symbols first. These include parentheses (), ab brackets [], braces {}, and the fraction bar cd . 2. Perform exponential operations next 3. Then do multiplication and division from left to right 4. Finally do addition and subtraction from left to right. Some people like to remember this using the mnemonic: "Please excuse my dear aunt sally." Really we should write this in a table: 1 2 3 4 , , , fraction bar a 2 exponents , , − Please Excuse My Dear Aunt Sally Interval Notation We use the following notation very freely: Closed interval: 1, 2 x|1 ≤ x ≤ 2 Open interval: 1, 2 x|1 x 2 An combinations of these, e.g. 1, 2 1 x ≤ 2 and 2, x|x ≥ 2 © W Clarke 3 1/4/2005 College Algebra—Winter 2005 Sections P.1&2 Set theory Notation We will not place a great emphasis on set theory notation. However you should be familiar with the notation in the text. We do use the union ( ) symbol frequently: −, 2 3, 4 to represent all numbers less than two or between 3 and 4 inclusive. Question 9: Given that A −3, −2, −1, 0, 1, 2, 3 and B −2, 0, 2, 4, 6 , find AB A B −3, −2, −1, 0, 1, 2, 3, 4, 6 Question 33: Write without absolute value notation: |3| |−4| 3 4 12 Question 52: Graph the set: −4, 0 ∩ −2, 5 ) 0 [ ( -4 -2 ] 5 Since we are looking for the intersection, we just want the part of the graph between −2 and 0. [ -2 ) 0 Section P.2--Integer and Rational Exponents Review the properties of exponents. You should be very familiar with the basic laws from Intermediate Algebra or Algebra II. b m b n b m n bm b m −n bn b m n b m n ab n a n b n n a n ab n b n n n n n n Remember a b ≠ a b and a − b ≠ a − b © W Clarke 4 1/4/2005 College Algebra—Winter 2005 Sections P.1&2 From these you can quickly deduce that b 0 1 as long as b ≠ 0 b −n b1n b m n n bm Radicals are regarded as simplified if 1. the nth radical n x contains no perfect nth power of a number, 2. the radical does not contain a fraction, and 3. the radical does not occur in the denominator of a fraction. Furthermore don't leave negative exponents in an answer. 2 Not! Questions 1 & 2: Evaluate −5 2 and −5 −5 2 − 25 −5 2 25 2 −4x 2 y 3 2xy 2 Question 29: Simplify First distribute the exponents. 3 −4 2 x 4 y 6 2 3x 3y 6 16x 4 y 6 8x 3 y 6 Now do the arithmetic first with the numbers and then with each letter in turn. 2x Question 76: Simplify 3 −250 Notice that 5 3 125 and 250 125 2 so we can break up the radicand by 3 −125 2 and the cube root of −125 is −5 so 3 −250 −5 3 2 4 3 3 Question 85: Simplify 4 32y 3y 108y Notice that each term appears different so we can't just collect like terms. So simplify each radical: 4 3 32y 4 3y 3 108y 4 2y 3 4y 3y 3 3 4y 8y 3 4y 9y 3 4y © W Clarke 5 1/4/2005 College Algebra—Winter 2005 Sections P.1&2 17y 3 4y 6 2 5 2 Question 109: Simplify Here we multiply by the conjugate of the denominator: 2 5 − 2 6 2 5 2 6 2 5 2 2 5 −2 2 5 −2 6 2 5 −2 20−4 12 5 −1 16 3 5 −1 4 Notice no radical in the denominator, no perfect squares, and no fraction under the radical. © W Clarke 6 1/4/2005