Complex Numbers. Basic algebra. Definitions. part of the complex number a+ib. ffl Addition: Notation: We i write for 1; that is we

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1 Complex Numbers Definitions Notation We i write for 1; that is we define p to be p 1 so i 2 = 1. i Basic algebra Equality a + ib = c + id when a = c b = and d. Addition A complex number is any expression of the form (a + ib) + (c + id) = (a + c) + i(b + d) Subtraction a + ib where a and b are real numbers. We call a the real part and b the imaginary part of the complex number a+ib. (a + ib) (c + id) = (a c) + i(b d) Multiplication (a+ib)(c+id) = (acbd)+i(ad+bc) T-47 T-48

2 The complex conjugate of a + ib is defined to be a ib If is a complex number we write for its complex conjugate. Solving quadratic equations To take the square root of a negative number d < 0 recall that p 1 = i so p p p 1 d = d p d i = To solve the general quadratic equation Division To divide two complex numbers multiply and divide by the complex conjugate of the denominator. 2 + b + c = 0 a use the formula = b ± p b 2 4ac 2a The algebraic operations for complex numbers satisfy the same properties as the corresponding operations for real numbers. T-49 Note The quantity = b 2 4ac determines the nature of the solutions If > 0 there are 2 real solutions. If = 0 there is a single real solution. If < 0 there are 2 complex solutions. These solutions satisfy = 2 1. T-50

3 The complex plane Modulus of a complex number Imaginary axis The modulus of the complex number a + ib is defined to be r i p 2 + b 2 a and is written ja + as ibj. 3 2 r 2 + i Real - axis The modulus of a complex number gives us an idea of how big this number is r r 3 2i Note In the complex plane jj is the distance of the complex number from the origin. 3 3i 4 T-51 T-52

4 } Ω ΩΩ - Re - Re Argument of a complex number Suppose = a + ib is a complex number which determines an angle with the real axis. A point in the complex plane determines an angle with the positive real axis. Im b 6 r r = jj s ρ ρρ ρ ρρρ Im 6 Ω a Ω ΩΩ We have cos( ) = a r and sin( ) = b r so = sin( ) tan( ) = b a cos( ) We can choose so ß <» that ß. In particular Above the real axis we 0»» ß have Below the real axis we ß < < 0 have Warning For a a b given and there are many values tan( ) = of with! b a These values differ by multiples of ß. tan Using on your calculator gives the value 1 between and ß=2 ß=2. T-53 T-54

5 ρ ρ Suppose is a complex number. The principal argument of is the angle that makes with the positive real axis, chosen so that ß <» ß. Polar form of a complex number Let be a complex number with modulus r and principal argument. The principal argument of is written as Arg(). Im 6 s r To find Arg(a + ib) 1. Draw a picture showing the angle. 2. Find tan 1 b a. Then we have ρ ρρ ρ ρρ - Re 3. Possibly adjust by ±ß to get the angle in the correct range. = r cos( ) + i sin( ) This is called the polar form of. T-55 T-56

6 cos( ) + i sin( ) To write a complex number in polar form Multiplication in polar form 1. Calculate the modulus, r = jj. Suppose and w are written in polar form 2. Calculate the principal argument, = Arg(). = r ; = s cos(ffi) + i sin(ffi) w 3. Write as Then = r cos( ) + i sin( ) w = (rs) cos( + ffi) + i sin( + ffi) T-57 T-58

7 The exponential function The exponential of a complex number is defined by e = ! 3! + Exponential form of a complex number Suppose a complex number is written in polar form as = r cos( ) + i sin( ) Using Euler s formula we can also write = re i This is known as the exponential form of. Euler s formula e i Basic property To write a complex number in exponential form 1. Calculate r, the modulus of. = cos( ) + i sin( ) 2. Calculate, the principal argument of. e +w = e e w 3. Write = re i. T-59 T-60

8 p p + i sin cos = q q Solving equations Powers of a complex number Suppose a complex number is written in exponential form as = re i. Then To solve the equation a + ib = apply the following steps k 1. Write in polar form as n = r n e in De Moivre s Theorem. If p and q are natural numbers then cos( ) + i sin( ) p =q 2. Use De Moivre s Theorem to express in polar form. k 3. Compare the moduli of and a + ib to get r. k = r cos( ) + i sin( ) 4. Compare the real and imaginary parts of and a + ib. k 5. Solve for take care with multiples of and the range of! 2ß T-61 T-62

9 Summary The following are the learning outcomes for the material covered in Chapter 5 The final word Fundamental Theorem of Algebra. Any polynomial equation has all of its solutions in the complex numbers. Define complex numbers. Apply the algebra of complex numbers to manipulate expressions involving complex numbers. Calculate the modulus and argument of a complex number. Understand the polar form of a complex number. Define the exponential function and know its properties. T-63 T-64

10 Understand the exponential form of a complex number. Interpret geometrically certain equations and inequalities in the complex plane. Know De Moivre s Theorem. Use De Moivre s Theorem to compute powers of complex numbers. Use De Moivre s Theorem to find the nth roots of complex numbers. Solve equations involving complex numbers. T-65