SECTION.-.3. Types of Real Numbers The natural numbers, positive integers, or counting numbers, are The negative integers are N = {, 2, 3,...}. {..., 4, 3, 2, } The integers are the positive integers, the negative integers, and zero: Z = {..., 3, 2,, 0,, 2, 3,...} The rational numbers are real numbers that can be written as a fraction where the numerator and denominator (not equals 0) are integers. If a rational number is expressed as a decimal it turns out that it is a repeating decimal. Example: Q = { m n m, n Z, n 0} The Irrational numbers, are the numbers that have non-repeating and nonterminating decimal expansions. They cannot be written as fractions of integers. Example: Diagram:
2 SECTION.-.3 2. Property of Real Numbers Real numbers have two operations, addition (+) and multiplication ( ), that are defined for all real numbers. For any real numbers a, b, and c the following properties hold for addition and multiplication: 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: a(bc)=(ab)c Identity Property of addition: a+0=0+a=a Identity Property of multiplication: a = a = a Inverse Property of addition: a+(-a)=(-a)+a=0 Inverse Property of multiplication: a ( a ) = a a = Distributive Property: a(b + c) = ab + ac (a + b)c = ac + bc Subtraction: a-b=a+(-b) Division: a b = a ( b )
SECTION.-.3 3 3. Number Lines and Absolute Value A number line is a method of picturing the set of real numbers. Each point on the number line corresponds to exactly one real number. Definition. The absolute value of a number x, denoted by x, refers to the distance from that number to the origin. Thus, { x if x 0 x = x if x < 0 Note. An absolute value is ALWAYS POSITIVE! (above x is just using the to change x to positive). Example. Find the absolute value () 30 = (2) 0 = (3) 0 = (4) 2 π = The distance between any two numbers a and b is a b. Example. Find the distance between the numbers -5 and 7
4 SECTION.-.3 4. Exponents and Radicals Definition. If n is a positive integer, then When n = 0, then x 0 =. Examples. x n = x x x x (n times) We define x to a negative power as follows (we assume that x 0) Examples. x = x, x 2 = x 2,, x n = x n Properties of Exponents () a m a n = a m+n (2) a n = a n (3) am = a m n a n (4) a 0 = for a 0 (5) (ab) n = a n b n (6) ( a b )n = an b n (7) (a n ) m = a nm (8) ( a b ) n = bn a n = ( b a )n, a, b 0
Example. Simplify the following expressions: () ( 2x 3 ) 4 = SECTION.-.3 5 (2) 9y6 x 2 = (3) 2y3 y 2 = (4) 3x( 5x6 y 2 )3 = (5) 5x4 3x 3 x 2 = (6) ( 3y5 x ) = (7) 3 2 x 2 y 4 7 0 z 7 (xy) 2 = (8) 255 + 5( 5 ) 00 5 3 25 50 5 0 =
6 SECTION.-.3 5. Radicals and Properties of Radicals Definition. An n-th root of a number a is a number b such that b n = a. We write b = n a. Remarks: () b is called the n-th root of a. (2) n is called the index of the radical. (3) a is called the radicand. (4) Sometimes, a has two n-th roots in the real numbers. In this case, we choose b = n a to have the same sign as a; this b is called the principal n-th root. For example: 4 = 2, not -2. (5) In the real numbers, there are NO even roots of a negative number. Therefore, in this class, n a does not exist if n is even and a is negative. Example. Find the domain of x + 8. Dont forget that radicals and rational exponents mean the same thing. b = a /n = n a means b n = a. b = a m/n = n a m = ( n a) m means b n = a m. Example () 4 ( 2) 4 = (2) ( 4 2) 4 = (3) 36 = (4) 3 27 =
SECTION.-.3 7 Properties of Radicals and Fractional Exponents () a n b = n ab and a n b n = (ab) n n a a a (2) n = n n and = ( a b b b ) n (3) n b n m a = nm a and (a m ) n = (a) mn (4) If n is odd, then n a n = a and (a n ) n = a. (5) If n is even, then n a n = a and (a n ) n = a. Examples: () ( 32) 5 = (2) 25 2/3 = (3) ( 6) 3/2 = (4) 6 3/2 = 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. 3. No radicals are in the denominator. If a problem has a radical sign, use a radical sign, if needed, in the answer. If a problem has fractional exponents, use fractional exponents, if needed, in the answer. Do this unless the problem asks you to convert from one form to the other. Examples Simplify the following: () 63 (2) 4 32 (3) 4 48x 2 y 3 z 8 (4) 4 8 2x 4 5x 2 4x
8 SECTION.-.3 Rationalizing Denominators Single Term: When the denominator is a single term with a radical do:. Simplify the radical in the denominator 2. Multiply both the numerator and the denominator by something that will produce a perfect root in the denominator. (this is just multiplying by ). Rationalizing Denominators - Sum of Terms: When the denominator has a radical and is the sum of two terms, then multiply by the following:. If the denominator is a + b, multiply the fraction by a b. a b. a+b 2. If the denominator is a b, multiply the fraction by a+b Here either a, b, or both contains a radical. The conjugate of the expression a + b is a b, and similary the conjugate of a b is a + b. Combining Radical Expressions: When radicals are being added or subtracted, we can only combine radicals where the index and radicand are identical. n a Examples 7 () 5 (2) (3) 5 3 72 2 6 x6 y 7 z 9x (4) If x > 0 and y > 0, simplify 3 y 4 50x 8 y 5 (5) 3 5 2 3 (6) 2x 2 5 7
SECTION.-.3 9 (7) x2 (8) 3 + 4 5 7 (9) 2 5 + 7 5 7 + 0 5 (0) 7 5 x 6 + 4 8x 5 x 5 32x () 2 8
0 SECTION.-.3 Definition. We define i such that 6. Complex Numbers i = A complex number is a number that can be written in the form a + bi, where a and b are real numbers. a is called the real part. b is called the imaginary part. a + bi is called the standard form of the complex number. Two complex numbers are equal if and only if their real parts and their imaginary parts are equal. Example () Write the number 24 64 8 + 54 in its standard form a + bi. Find the real part and the imaginary part of the number. (2) Find real numbers a and b such that (5a ) 9i = 5 ( b 4 )i
SECTION.-.3 Definition. The absolute value of a complex numer z = a + bi in standard form is defined as z = a + bi = a 2 + b 2 Definition If z and z 2 are two complex numbers, then the distance between these numbers is defined as z z 2. Note that if z is a real number, then the absolute value of z is the same as its real number absolute value. Examples Find the absolute value of the following complex numbers. () 5 3i (2) 5 + 2i (3) (2 i)(3 + 2i) Definition. The conjugate of the complex number z = a + bi is defined as z = a bi. We use z to denote the conjugate. Examples () 5 + 3i =
2 SECTION.-.3 (2) 7i = (3) Calculate the following for z = 5i + 2 and w = 3 2i. (a) z w (6 + i) (b) 2z (z 6) w (4) Put 2 + 3i 4i in standard form