Rational Numbers. a) 5 is a rational number TRUE FALSE. is a rational number TRUE FALSE

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1 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 1 Chapter 1A - - Real Numbers Types of Real Numbers Name(s) for the set 1, 2,, 4, Natural Numbers Positive Integers Symbol(s) for the set, -, - 2, - 1 Negative integers 0, 1, 2,, 4, Non- negative Integers Whole Numbers, -, - 2, - 1, 0, 1, 2,, Integers Note: is the German word for number. 4 5, 17 1, 1 29 Rational Numbers Note: This is the Iirst letter of 1. Circle One: a) 5 is a rational number TRUE FALSE b) π 4 is a rational number TRUE FALSE c) 16 5 is a rational number TRUE FALSE d) 17 5 is a rational number TRUE FALSE e) 0 is a rational number TRUE FALSE

2 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 2 De8inition: If a, b! and b 0, then a b!. Translation: Decimal Expansions of Rational Numbers: Fact: The decimal expansions of rational numbers either or Examples 1 4 = ( This is a terminating decimal.) 4 9 = (This is a repeating decimal.) You know how to express a terminating decimal as a fraction. For example.25 = 25, but how do 100 you express a repeating decimal as a fraction? 2. Express these repeating decimals as the quotient of two integers: a).777 or.7 Let x = Then 100x = b).12 Let x = Then

3 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! There are a lot of rational numbers, but there are even more irrational numbers. Irrational numbers cannot be expressed as the quotient of two integers. Their decimal expansions never terminate nor do they repeat. Examples of irrational numbers are The union of the rational numbers and the irrational numbers is called the or simply the. It is denoted by. Math Symbols: iff or

4 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 4 Properties of Real Numbers 4. What properties are being used? Name of Property a) ( 40) + 8(x + 5) = 8(x + 5) + ( 40) a + b = b + a a,b! b) 8(x + 5) + ( 40) = 8x ( 40) a(b + c) = ab + ac a,b,c! c) 8x ( 40) = 8x + (40 + ( 40)) (a + b) + c = a + (b + c) a,b,c! d) = = a = a + 0 = a a! e) 7 + ( 7) = ( 7) + 7 = 0 For each a!, another element of! denoted by a such that a + ( a) = ( a) + a = 0 f) 6 1 = 1 6 = 6 1 a = a 1 = a a! For each a!, a 0 another g) = = 1 element of! denoted by 1 a a 1 a = 1 a a = 1 such that

5 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 5 When do we use the commutative property of addition? Consider The rules for order of operation tell us to add left to right, but the commutative property of addition allows us to change the order and make our life easier = When do we use the associative property of multiplication? Consider = Is subtraction commutative? Example: Is division commutative? Example: Is subtraction associative? Example: Is division associative? Example: Absolute Value 4 = 9 = Formal deiinition for absolute value x = It will help if you learn to think of absolute value as (Notice that -4 is 4 units away from the origin.)

6 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 6 Distance between two points:.) Find the distance between a) 4 and 1 b) -1 and 5 c) 4 and -2 d) - and -4 e) 1 and x f) x and y Notice that! 4 1 = 1 4! and! 2 = 2 ( ). In general x y = Notice that! =! whereas = So, in general x + y Suggested problems: Text: 1-9, 22, 2 My Previous Exams: Spring 201 1A:, 4, Spring A: 1b, c, d, e Dr. Scarborough s Previous Exams Fall 201 I: p2:2, Fall 2012 I: p2:1 Dr. Scarborough s Fall 201 WIR 1: 1, 2, 4, 6, 21 Dr. Kim s Fall 2014 WIR:

7 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 7 Chapter 1B - Exponents and Radicals Multiplication is shorthand notation for repeated addition = 5 4 Exponents are shorthand notation for repeated multiplication. So = xixixix = You probably know that 1 = and x 5 = this is because by definition x 1 =. In general if m Ν, then x m = Properties of exponents Example In general a 4 a x m x n b 8 b 5 x m x n y y x 0 for x 0 Note: and is an More Properties of Exponents Example In general (xy) x 2 y (xy) m x m y

8 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 8 1. Simplify a) ( 8) 2 b) 8 2 c) 0 Dividing is the same as d) 1 y 6 e) a b 1! e) x x 2 f) y 4 y 5 1 g) 12a b 2 4a 4 b 1 2 h)

9 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 9 Roots or Radicals TRUE or FALSE: 4 = ±2. Important distinction: x 2 = 25 has solutions: x = and x = 25 represents exactly number. 25 =. Recall that 2 5 = 2. Now suppose you have x 5 = 2 with the instructions: Solve for x. How do you express x in terms of 2? The vocabulary word is In this case x equals the. The notation is x = or x = This second notation is called a fractional exponent. ( ) Examples:! 81 = 81 = because =. ( ) 1!!! 125 = because 5 =. In general if a n = b then and if So in shorthand

10 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 10 Properties of Roots (just like properties of exponents!) Example In general 27i8 m xy m m x y 2 Now consider 8. Notice that ( 8 2 ) 1 = Also = 2 So it would seem that 8 = In fact, more generally, it is true that x m n = Similarly, consider 64 = = Whereas, 64 = = So it would seem that 64 In general m n x

11 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 11 Simplify n a n for all n!. In recitation you graphed y = x. Hopefully you found that its graph looked just like the graph of y = x. In other words, you saw that x = You also graphed y = 2 x 2. Hopefully you found that its graph looked just like the graph of y = x. In other words, you saw that 2 x 2 = Most people believe that = and agree easily that n 2 n = n But ( 2) 4 ( ) = while 2 = So hopefully you see believe n ( 2) n = 2 whenever n is and n ( 2) n = 2 whenever n is So to simplify n a n for any a! and any n! we have n a n = Simplifying Radicals A radical expression is simplified when the following conditions hold: 1. All possible factors ( perfect roots ) have been removed from the radical. 2. The index of the radical is as small as possible.. No radicals appear in the denominator.

12 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! Simplify 5 a) 2 b) ( 216) c) 9 d) 64 6 e) ( 1) 6 f) 4 16x 4 g) x 5 h) 648x 4 y 6 8 i) 16x 4 y 12 j)

13 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 1 Rationalizing the Denominator Rationalizing the denominator is the term given to the techniques used for eliminating radicals from the denominator of an expression without changing the value of the expression. It involves multiplying the expression by a 1 in a helpful form.. Simplify a) 1 2!!!!!! b) 1 2 c) 1 4x!!!!!!! d) ( 9x) y 4 50x 8 y 5 Notice: ( x + ) ( x ) =. To simplify we multiply it by 1 in the form of.

14 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 14 So = and = Adding and Subtracting Radical Expressions Terms must be alike to combine them with addition or subtraction. Radical terms are alike if they have the same index and the same radicand. (The radicand is the expression under the radical sign.) 4. Simplify a) b) 16x 4 x 18x x 250x c)

15 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 15 Please notice =! Whereas 25 = In other words In general Suggested Problems: Text: 1-2 My Previous Exams:! Spring A: 11, 14, Fall 201 1A: 1, 2,!!!! Spring 201 1A: 1, 2,! Fall A: 1 a, b, d, Dr. Scarborough s Previous Exams Fall 201 I: p2:2, Fall 2012 I: p2:1 Dr. Scarborough s Fall 201! WIR 1:, 5, 7-10, 12-1, 15-16, 19-20,!!!!! WIR 2: 2,!!!!! WIR : 12,1, 21, 22, 24 Dr. Kim s Fall 2014 WIR:

16 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 16 Chapter 1C - Polynomials Which of these are examples of polynomials? Polynomial Not a polynomial Polynomials should not have x 2 + 2x +1 x 7 + π ( 2) x 2 + 2x +1 4 x 1 1 x + x2 ( 5x + x 2 2) 1 4 ( 5x + x 2 2) 2 x 2 + sin x 5 Polynomials are expressions of the form a n x n + a n 1 x n 1 +!+ a 1 x + a 0 where a i! and n! n is called a n is called a 0 is called If n = 2, the polynomial is called a If n =, the polynomial is called a A polynomial with 2 terms is called a A polynomial with terms is called a Factoring polynomials will help us in later chapters when we need to find the roots of polynomials.

17 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 17 Techniques for Factoring Polynomials Common Factors: 12x 4 + 4x 2 + 2x = Factor by Grouping: x + x 2 12x 4 (Factoring by grouping is your best bet if you have a long cubic!) Factor using special patterns: Look what happens when you multiply and simplify: (x )(x + ) = (4 y)(4 + y) = Difference of Squares a 2 b 2 = Look what happens when you expand (x + 5) 2 = (x 2) 2 = Square of a Binomial a 2 + 2ab + b 2 = a 2 2ab + b 2 = a) x b) 16x 4 y 6

18 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 18 c) a 2 +10a + 25 d) x 4 + 8x Look what happens when you multiply (a + b)(a 2 ab + b 2 ) Sum of Cubes a + b Difference of Cubes a b 1. Factor a) 27 y b) x c) 27x d) x 5 4x x e) x + 6x 2 12x 24 f) 16x 4 25x 2 Factoring quadratics, lead coefficient 1: 2. Factor : x 2 +10x +16 = Find 2 numbers whose product is and whose sum is. Factor: x 2 2x 48 = Find 2 numbers whose product is and whose sum is

19 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 19 Factoring quadratics with lead coefficient is not 1 Use the Blankety-Blank Method! 4. Factor: 10x 2 +11x + = Find 2 numbers whose product is and whose sum is 5. Factor: x 2 14x 5 = Find 2 numbers whose product is and whose sum is Important Fact: a 2 + b 2 does not factor over the real numbers. So x cannot be factored nor can y Dividing Polynomials (This is a lot like long division of integers!) 6. y = 4x x 2 5x + 2 x 1 =!

20 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! y = x5 x + x x 2 + 2x Suggested Problems: Text: 1, 2, 16, 17, 18, 21-8 My Previous Exams:! Spring 2014: 8, 10,! Fall 201 1A:, 4, 12!!!! Spring 201 1A: 5,! Fall A: 1c, 2 Dr. Scarborough s Previous Exams Fall 201 I: p2:2, Fall 2012 I: p2:1 Dr. Scarborough s Fall 201 WIR 1: 11, 17, 18,!! WIR : 17, 28 Dr. Kim s Fall 2014 WIR

21 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 21 Chapter 1D - Rational Expressions A rational expression is the quotient of two Division by is not defined for real numbers, so sometimes restrictions must be placed on the variables of rational expressions. Simplifying Rational Expressions: 1. Simplify the rational expressions and list any restrictions that exist for x. x 7 x 4 x 5 x = Restrictions on x : 2. x + 2 x = Restrictions on x : Addition and Subtraction of Rational Expressions. x x = Restrictions on x :

22 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! x 4 x 2 2x + 4 x2 8x +16 = x + 8 Restrictions on x : Multiplication and Division of Rational Expressions 5. 8 x + 2 i x 2 x 2 x 2 1 = Restrictions on x : 6. x 2 4x 21 (x 2 + 7x +12) = x + 4 Restrictions on x :

23 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 2 7. x 1 x +1 x 1 =!!! x 2 + 2x +1 Restrictions on x : Compound Fractions A compound (or complex) fraction is an expression containing fractions within the numerator and/or the denominator. To simplify a compound fraction, first simplify the numerator, then simplify the denominator, and then perform the necessary division. 8. x + 1 2x 7 5 x +1 Restrictions on x :

24 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! ( x + h) 2 x 2 h = Restrictions on x : Restrictions on h : Suggested Problems: Text: 1-25 My Previous Exams:! Fall A:, 4 Dr. Scarborough s Previous Exams: Fall 201 I: p8:10, Fall 2012 I: p2:4, p8:10 Dr. Scarborough s Fall 201 WIR 2:, 5!!! WIR : 6, 16, 25 Dr. Kim s Fall 2014 WIR:

25 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 25 Chapter 1E - Complex Numbers 16 exists! So far the largest (most inclusive) number set we have discussed and the one we have the most experience with has been named the real numbers. And x!, x 2 0 But there exists a number (that is not an element of! ) named i and i 2 = Since i 2 =, 1 =, so 16 = i is not used in ordinary life, and humankind existed for 1000 s of years without considering i. i is, however, a legitimate number. i, products of i, and numbers like 2 + i are solutions to many problems in engineering. So it is unfortunate that it was termed imaginary! i and numbers like 4i and i are called Numbers like 2 + i and 17 5i are called More formally! is the set of all numbers When a complex number has been simplified into this form, it is called 1. Put the following complex numbers into standard form: a) ( ) + ( 1 100) = b) ( a + bi) + (c + di) = c) ( 2 + i)(4 + 5i) = d) ( 5 + 4i)( 2i) =

26 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 26 e) i = f) i 4 = g) i 5 = Graphing Complex Numbers When we graph elements of!, we use When we graph elements of!, we use the complex plane which will seem a lot like the Cartesian plane. In the Cartesian plane, a point represents In the complex plane, a point represents The horizontal axis is the The vertical axis is the 2. Graph and label the following points on the complex plane: A + 4i B 2 + i C 2 i D i E 4 F 1 2i

27 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 27 Absolute Value of a Complex Number: If z! then z is defined as its Calculate + 4i (How far is + 4i from the origin?) Consider a + bi, an arbitrary element of!.!! What is a + bi? (How far is a + bi from the origin?) In general z = Notice that z is a real number. It is z s distance from the origin. We did not use i to calculate z.. Calculate 5 + i. (How far is 5 + i from the origin?) 5 + i =

28 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 28 Complex Conjugate If z = + 4i, then z =. If z = 1 2i, then z =. Graphically the complex conjugate is the of the number through the More generally if z = a + bi, then z = 4. Put the following complex numbers into standard form. a) 2 i =!!! b) 4i = c) 5 =!!! d) i 2 = Finally notice that ( + 4i) ( 4i) = More generally ( a + bi) ( a bi) = in other words ziz =

29 Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 29 Distance Between Two Complex Numbers Plot and label two points in the complex plane z 1 = + 5i and z 2 = 1+ 2i The distance between z 1 and z 2 is Just for fun, calculate z 1 z 2 = And z 1 z 2 = What we have seen is that for this particular z 1 and z 2, the distance between these points is equal to. But this is actually true for all complex numbers. The distance between two general points z 1 and z 2 is Suggested Problems: Text: 1-12 My Previous Exams:! Fall 201 1A: 5,! Spring 201 1A: 7,!! Fall A: 5 Dr. Scarborough s Previous Exams:! Fall 201 I: p6:,!! Fall 2012 I: p:7, p6:2 Dr. Scarborough s Fall 201 WIR 2: 4, 6, 7, 8 WIR : 8, 19 Dr. Kim s Fall 2014 WIR:

Natural Numbers Positive Integers. Rational Numbers

Chapter A - - Real Numbers Types of Real Numbers, 2,, 4, Name(s) for the set Natural Numbers Positive Integers Symbol(s) for the set, -, - 2, - Negative integers 0,, 2,, 4, Non- negative integers, -, -