Chapter 1: Fundamentals


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1 Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives, nd 0. Rtionl Numbers: { m n n, m re integers, n 0}; These re rtios of integers (frctions). All numbers with repeting decimls cn be written s frctions. Irrtionl Numbers: These re the nonrepeting, nonterminting numbers. Ex: π, 2, 3 4,... Rel Numbers: All of the bove put together. Exmple: Consider the list of numbers 12, 1.987, 5 2, π 4, 4, 7 5, 0, List the numbers tht re: Nturl numbers Integers Rtionl numbers Irrtionl numbers Rel numbers cn be represented by points on the rel line, or number line. If number is further to the left thn b on the number line, then < b. If is to the right of b, then > b. Properties of Rel Numbers Commuttive Property: + b = b +, b = b Associtive Property: + (b + c) = ( + b) + c, (bc) = (b)c Distributive Property: (b + c) = b + c, ( + b)c = c + bc Exmple: Write 8x(3 2y) without prentheses. 1
2 Some Properties of Frctions b c d = c bd b c d = b d c (Multiply by the reciprocl.) c + b c = + b c Note: You cn only do this when the frctions hve the sme denomintor: b + c b + c 4. If b = c, then d = bc (Cross multiplying.) d 5. c cb = b (Cncel common fctors.) To dd frctions with different denomintors, we hve to use the Lest Common Denomintor. 5 Exmple: A set is collection of objects, which re clled elements. Sets of rel numbers tht mke up portions of the number line cn be represented by intervls. A closed intervl, [, b] = {x x b}; the set of ll number between nd b, including the endpoints. An open intervl, (, b) = {x < x < b}; the set of ll numbers between nd b, not including the endpoints. An intervl cn lso be hlfopen, hlfclosed: such s (, b] or [, b). Prentheses indicte tht the endpoint IS NOT included. Squre brckets indicte tht the endpoint IS included. 2
3 The union of two sets or intervls A nd B, denoted A B, is the set of elements tht: The intersection of two sets or intervls A nd B, denoted by A B, is the set of elements tht: Exmple: Grph the set (, 7] [2, 8). Exmple: Grph the set [ 4, 5] (1, ) The bsolute vlue of number x, denoted x, is how fr x is wy from the origin. It is mesure of distnce, so it must lwys be greter thn or equl to 0. x = Exmple: 2 12 = π 4 = The distnce between two points nd b is: d(, b) = b or equivlently b. Exmple: Find the distnce between the numbers 4 nd
4 1.2 Exponents nd Rdicls Exponent Rules 1. n = = 1, for 0 3. n = 1 n, for 0 4. m n = m+n 5. m n = m n 6. ( m ) n = mn 7. (b) n = n b n Note: This only works with multipliction. ( + b) n n + b n 8. ( b )n = n b n 9. ( b ) n = ( b )n 10. n b m = bm n Exmples: Simplify nd write without negtive exponents. ( ) x 3 (4x 4 y) 2 3y 2 ( xy 2 z 3 x 2 y 3 z 4 ) 2 4
5 Rdicls The symbol mens the positive squre root of. To find, find the number b such tht b 2 =. In ll cses (except when = 0), there re two numbers b tht mke this true. The nswer is the POSITIVE one. Similrly, to find 3, the cube root (or third root) of, find the number b such tht = b 3. In generl, to find n, find the number b such tht b n =. If n is EVEN, we must hve 0 nd b 0. If n is odd, there re no restrictions. Does 2 lwys equl? Does 3 3 lwys equl? So, we define n n = Simplify the following expressions: 6 ( 7) 18 x 6 y 12 5 ( 4) 15 r 10 s 25 5
6 Properties of Rdicls 1. n b = n n b n 2. b = n 3. n b m n = mn Exmples: Simplify these expressions Note: It is NOT true tht 2 + b 2 = + b. All these properties follow from properties of exponents when we use the following: n = 1/n n m = m/n = ( n ) m This mens tht ANY rdicl cn be written using rtionl exponents. Exmples: 1. Clculte 36 3/2 2. Clculte 48 3/4 6
7 3. Simplify the following nd write without negtive exponents. Assume ll vribles represent positive numbers. ( 8x 5/2 y 6 ) 2/3 54z 3 4. Write the following s single power of x: x 4 x ( 5 x) 2 We rtionlize the denomintor in order to get ny rdicls out of the denomintor. We do this by multiplying top nd bottom by n pproprite fctor. Exmples x 3 7
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