Natural Numbers Positive Integers. Rational Numbers

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1 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, -, - 2, -, 0,, 2,, Integers Note: is the German word for number. 4 5, 7, 29 Rational Numbers Note: This is the Hirst letter of. Circle One: a) 5 is a rational number TRUE FALSE b) π 4 is a rational number TRUE FALSE c) 6 5 is a rational number TRUE FALSE d) 7 5 is a rational number TRUE FALSE e) 0 is a rational number TRUE FALSE

2 De8inition: If the numbers a and b are both integers and b 0, then a is a rational number. b Decimal Expansions of Rational Numbers: Fact: The decimal expansions of rational numbers either or Examples 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 00, but how do 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 00x = b).2 Let x =

3 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

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 = 6 = 6 a = a = a a For each a, a 0 another g) 4 4 = 4 4 = element of denoted by a a a = a a = such that 4

5 Absolute Value 4 = 9 = Formal dehinition 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. This interpretation is important and useful because it extends to the complex numbers that we will study soon. Distance between two points:.) Find the distance between a) 4 and b) - and 5 c) 4 and -2 d) - and -4 e) and x f) x and y Notice that x y = Notice that =! whereas =!!! So, in general x + y 5

6 Chapter B - 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 = and x 5 = this is because by definition x =. 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 6

7 . Simplify a) ( 8) 2 b) 8 2 c) 0 Dividing is the same as d) y 6 e) a b! e) x x 2 f) y 4 y 5 g) 2a b 2 4a 4 b 2 h)

8 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. ( ) 4 4 Examples:! 8 = 8 = because =. ( )!!! 25 = because 5 =. In general if a n = b then and if This kind of statement can be condensed using the term This is often abbreviated or So in shorthand 8

9 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 ) = Also 8 2 = 2 So it would seem that 8 = In fact, more generally, it is true that x m n = Similarly, consider 64 = 64 2 = Whereas, 64 = 64 2 = So it would seem that 64 In general m n x 9

10 Our next goal is to define n a n. Let s do several examples to find a pattern. a) =!!!! b) 5 = c) ( 5) ( ) 5 5 =!!!! d) 2 = e) =!!!! f) 2 2 = g) ( 2) 4 ( ) =!!!! h) = n a n = Simplifying Radicals A radical expression is simplified when the following conditions hold:. 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. 2. Simplify 5 a) 2 b) ( 26) c) 9 2 d) 64 0

11 6 e) ( ) 6 f) 4 6x 4 g) x 5 h) 648x 4 y 6 8 i) 6x 4 y 2 j) 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 in a helpful form.. Simplify a) 2!!!!!! b) 2

12 c) 4x!!!!!! d) ( 9x) y 4 50x 8 y 5 Notice: ( x + ) ( x ) =. To simplify x 5 we multiply it by in the form of. So x 5 = and = 2

13 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) 6x 4 x 8x x 250x c) 27 Please notice =! Whereas 25 = In other words In general

14 Chapter C - Polynomials Which of these are examples of polynomials? Polynomial Not a polynomial Polynomials should not have x 2 + 2x + x 7 + π ( 2) x 2 + 2x + 4 x x + x2 ( 5x + x 2 2) 4 ( 5x + x 2 2) 2 x 2 + sin x 5 Polynomials are expressions of the form a n x n + a n x n + + a 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 4

15 Techniques for Factoring Polynomials Common Factors: 2x 4 + 4x 2 + 2x Factor by Grouping: x + x 2 2x 4 Factor using special patterns: Look what happens when you multiply (x )(x + ) = (4 y)(4 + y) = Difference of Squares a 2 b 2 = Look what happens when you expand (x + 5) 2 = (y z) 2 = Square of a Binomial a 2 + 2ab + b 2 = a 2 2ab + b 2 Look what happens when you multiply (a + b)(a 2 ab + b 2 ) Sum of Cubes a + b Difference of Cubes a b 5

16 . Factor a) x 2 44 b) 6x 4 y 6 c) 27 y d) x e) a 2 +0a + 25 f) x 4 + 8x 2 +6 Factoring quadratics, lead coefficient : 2. Factor : x 2 +0x +6 = 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 Factoring quadratics with lead coefficient is not Use the Blankety-Blank Method! 4. Factor: 0x 2 +x + = Find 2 numbers whose product is and whose sum is 5. Factor: x 2 4x 5 = Find 2 numbers whose product is and whose sum is 6

17 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. 2x +x 2 + 7x 20 x x 5 x + x x 2 + 2x 7

18 Chapter D - Rational Expressions A rational expression is A rational expression is undefined when its For what values of x is the rational expression x + x 2 defined? Simplifying rational expressions means reducing it to lowest terms. This is achieved by canceling factors common to the numerator and the denominator.. x +x 2 + 0x x 2 +0x i y2 y 2 y + 2 y 2. v 2 4v 2 (v 2 + 7v +2) v + 4 8

19 4. x x + x x 2 + 2x + 5. x x 6. u 4 u + u2 8u +6 u 2 9

20 A compound or complex fraction is an expression containing fractions with 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 t + 4 2t 5 t 8. ( x + h) 2 x 2 h 20

21 Chapter E - Complex Numbers 6 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 =, =, so 6 = i is not used in ordinary life, and humankind existed for 000 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 7 5i are called More formally is the set of all numbers When a complex number has been simplified into this form, it is called. Put the following complex numbers into standard form: a) ( 6 + 9) + ( 00) = b) ( a + bi) + (c + di) = c) ( 2 + i)(4 + 5i) = d) ( 5 + 4i)( 2i) = e) ( )( ) = 2

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.

22 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 C i E 4 2 B 2 + i D i F 2i 22

23 Absolute Value of a Complex Number: If z then z is defined as its Calculate + 4i Consider a + bi, an arbitrary element of. What is a + bi? In general z =. Calculate 5 + i 2

24 Complex Conjugate If z = + 4i, then z =. If z = 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 = Finally notice that ( + 4i) ( 4i) = More generally ( a + bi) ( a bi) = in other words ziz = 24

25 Distance Between Two Complex Numbers Plot and label two points in the complex plane z = + 5i and z 2 = + 2i The distance between z and z 2 is What if the 2 points were arbitrary? z = a + bi and z 2 = c + di The distance between z and z 2 is Notice that z z 2 = So z z 2 = So we have shown that the distance between two points z and z 2 is 25

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

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