Definitions Definitions Complex numbers and complex plane Algebra of complex numbers Algebra of complex numbers Complex conjugate Polar coordinates form of complex numbers Polar coordinates form of complex numbers Modulus of a complex number
1. Complex numbers
Complex numbers are of the form
Chapter 13: Complex Numbers z = x + iy, x, y ∈ R, i 2 = −1. Sections 13.1 & 13.2 In the above definition, x is the real part of z and y is the imaginary part of z. y
The complex number z=3+2i
z = x + iy may be 1
representedinthe 0 1 x complex plane as the point with cartesian coordinates (x, y).
Chapter 13: Complex Numbers Chapter 13: Complex Numbers
Definitions Complex numbers and complex plane Definitions Complex numbers and complex plane Algebra of complex numbers Complex conjugate Algebra of complex numbers Complex conjugate Polar coordinates form of complex numbers Modulus of a complex number Polar coordinates form of complex numbers Modulus of a complex number
Complex conjugate Modulus of a complex number
The complex conjugate of z = x + iy is defined as z x iy z¯ = x − iy. The absolute value or modulus of = + is √ |z| zz x2 y 2. As a consequence of the above definition, we have = ¯ = + z +¯z z − z¯ It is a positive number. e(z)= , m(z)= , zz¯ = x2 + y 2. (1) i 2 2 Examples: Evaluate the following If z1 and z2 are two complex numbers, then |i| |2 − 3i| z1 + z2 = z1 + z2, z1z2 = z1 z2. (2)
Chapter 13: Complex Numbers Chapter 13: Complex Numbers Definitions Definitions Algebra of complex numbers Algebra of complex numbers Polar coordinates form of complex numbers Polar coordinates form of complex numbers
2. Algebra of complex numbers Algebra of complex numbers (continued)
Youshouldusethe same rules of algebra as for real numbers, Remember that multiplying a complex number by its complex but remember that i 2 = −1. Examples: conjugate gives a real number. # 13.1.1: Find powers of i and 1/i. Examples: Assume z1 =2+3i and z2 = −1 − 7i. z i z − − i z z z Assume 1 =2+3 and 2 = 1 7 .Calculate 1 2 and Find 1 . 2 z (z1 + z2) . 2 z Find 1 . z Get used to writing a complex number in the form 2 m 1 Find z 3 . z =(real part)+i (imaginary part), 1 # 13.2.27: Solve z2 − (8 − 5i)z +40− 20i =0. no matter how complicated this expression might be.
Chapter 13: Complex Numbers Chapter 13: Complex Numbers
Definitions Definitions Euler’s formula Euler’s formula Definitions Definitions Integer powers of a complex number Integer powers of a complex number Algebra of complex numbers Algebra of complex numbers Product and ratio of two complex numbers Product and ratio of two complex numbers Polar coordinates form of complex numbers Polar coordinates form of complex numbers Roots of a complex number Roots of a complex number Triangle inequality Triangle inequality 3. Polar coordinates form of complex numbers Principal value Arg(z)
In polar coordinates, y Because arg(z) is multi-valued, it is convenient to agree on a x = r cos(θ), y = r sin(θ), z=x+iy particular choice of arg(z), in order to have a single-valued y function. where θ The principal value of arg(z), Arg(z), is such that 0 x x r = x2 + y 2 = |z|. y tan (Arg(z)) = , with − π
Chapter 13: Complex Numbers Chapter 13: Complex Numbers Definitions Definitions Euler’s formula Euler’s formula Definitions Definitions Integer powers of a complex number Integer powers of a complex number Algebra of complex numbers Algebra of complex numbers Product and ratio of two complex numbers Product and ratio of two complex numbers Polar coordinates form of complex numbers Polar coordinates form of complex numbers Roots of a complex number Roots of a complex number Triangle inequality Triangle inequality Principal value Arg(z) (continued) Polar and cartesian forms of a complex number
Examples: Find the principal value of the argument of z =1− i. Youneedtobeabletogobackandforthbetweenthepolar Find the principal value of the argument of z = −10. and cartesian representations of a complex number. y z = x + iy = |z| cos(θ)+i|z| sin(θ).
1 In particular, you need to know the values of the sine and cosine of multiples of π/6andπ/4. π π Convert cos + i sin to cartesian coordinates. 0 1 x 6 6 What is the cartesian form of the complex number such that |z| =3andArg(z)=π/4?
Chapter 13: Complex Numbers Chapter 13: Complex Numbers
Definitions Definitions Euler’s formula Euler’s formula Definitions Definitions Integer powers of a complex number Integer powers of a complex number Algebra of complex numbers Algebra of complex numbers Product and ratio of two complex numbers Product and ratio of two complex numbers Polar coordinates form of complex numbers Polar coordinates form of complex numbers Roots of a complex number Roots of a complex number Triangle inequality Triangle inequality Euler’s formula Integer powers of a complex number
To find the n-th power of a complex number z = 0, proceed as Euler’s formula reads follows exp(iθ)=cos(θ)+i sin(θ),θ∈ R. 1 Write z in exponential form, z = |z| exp (iθ) . As a consequence, every complex number z =0canbe written as 2 Then take the n-th power of each side of the above equation n n n z = |z| (cos(θ)+i sin(θ)) = |z| exp(iθ). z = |z| exp (inθ)=|z| (cos(nθ)+i sin(nθ)) .
3 In particular, if z is on the unit circle (|z| = 1), we have This formula is extremely useful for calculating powers and n roots of complex numbers, or for multiplying and dividing (cos(θ)+i sin(θ)) = cos(nθ)+i sin(nθ). complex numbers in polar form. This is De Moivre’s formula.
Chapter 13: Complex Numbers Chapter 13: Complex Numbers Definitions Definitions Euler’s formula Euler’s formula Definitions Definitions Integer powers of a complex number Integer powers of a complex number Algebra of complex numbers Algebra of complex numbers Product and ratio of two complex numbers Product and ratio of two complex numbers Polar coordinates form of complex numbers Polar coordinates form of complex numbers Roots of a complex number Roots of a complex number Triangle inequality Triangle inequality Integer powers of a complex number (continued) Product of two complex numbers
The product of z1 = r1 exp (iθ1)andz2 = r2 exp (iθ2)is Examples of application:
Trigonometric formulas z1 z2 =(r1 exp (iθ1)) (r2 exp (iθ2)) ⎧ = r r exp (i (θ + θ )) . (4) ⎨ cos(2θ)=cos2(θ) − sin2(θ), 1 2 1 2 ⎩ (3) As a consequence, sin(2θ) = 2 sin(θ) cos(θ).
arg(z1 z2)=arg(z1)+arg(z2), |z1z2| = |z1||z2|. Find cos(3θ) and sin(3θ) in terms of cos(θ) and sin(θ). We can use Equation (4) to show that
cos (θ1 + θ2)=cos(θ1) cos (θ2) − sin (θ1) sin (θ2) , (5) sin (θ1 + θ2)=sin(θ1) cos (θ2)+cos(θ1) sin (θ2) .
Chapter 13: Complex Numbers Chapter 13: Complex Numbers
Definitions Definitions Euler’s formula Euler’s formula Definitions Definitions Integer powers of a complex number Integer powers of a complex number Algebra of complex numbers Algebra of complex numbers Product and ratio of two complex numbers Product and ratio of two complex numbers Polar coordinates form of complex numbers Polar coordinates form of complex numbers Roots of a complex number Roots of a complex number Triangle inequality Triangle inequality Ratio of two complex numbers Roots of a complex number n z z To find the -th roots of a complex number = 0, proceed as Similarly, the ratio 1 is given by follows z2 1 Write z in exponential form, z r iθ r 1 1 exp ( 1) 1 z = r exp (i(θ +2pπ)) , = = exp (i (θ1 − θ2)) . z2 r2 exp (iθ2) r2 with r = |z| and p ∈ Z. As a consequence, 2 Then take the n-th root (or the 1/n-th power) √ θ pπ √ θ pπ z z |z | n /n /n +2 n +2 1 1 1 z = z1 = r 1 exp i = r exp i . arg =arg(z1) − arg(z2), = . n n z2 z2 |z2| 3 There are thus n roots of z, given by z Example: z i z − − i 1 √ Assume 1 =2+3 and 2 = 1 7 . Find . n θ +2kπ θ +2kπ z2 zk = r cos + i sin , k =0, ··· , n−1. n n
Chapter 13: Complex Numbers Chapter 13: Complex Numbers Definitions Definitions Euler’s formula Euler’s formula Definitions Definitions Integer powers of a complex number Integer powers of a complex number Algebra of complex numbers Algebra of complex numbers Product and ratio of two complex numbers Product and ratio of two complex numbers Polar coordinates form of complex numbers Polar coordinates form of complex numbers Roots of a complex number Roots of a complex number Triangle inequality Triangle inequality Roots of a complex number (continued) Roots of a complex number (continued) √ The principal value of n z is the n-th root of z obtained by taking θ = Arg(z)andk =0. Examples: The n-th roots of unity are given by Find the three cubic roots of 1. √ n 2kπ 2kπ 1=cos + i sin = ωk , k =0, ··· , n − 1 n n √ Find the four values of 4 i. where ω = cos(2π/n)+i sin(2π/n).
In particular, if w1 is any n-th root of z =0,thenthe n-th roots of z are given by Give a representation in the complex plane of the principal value of the eighth root of z = −3+4i. 2 n−1 w1, w1ω, w1ω , ··· , w1ω .
Chapter 13: Complex Numbers Chapter 13: Complex Numbers
Definitions Euler’s formula Definitions Integer powers of a complex number Algebra of complex numbers Product and ratio of two complex numbers Polar coordinates form of complex numbers Roots of a complex number Triangle inequality Triangle inequality
If z1 and z2 are two complex numbers, then
|z1 + z2|≤|z1| + |z2|.
This is called the triangle inequality. Geometrically, it says that the length of any side of a triangle cannot be larger than the sum of the lengths of the other two sides.
More generally, if z1, z2, ..., zn are n complex numbers, then
|z1 + z2 + ···+ zn|≤|z1| + |z2| + ···+ |zn| .
Chapter 13: Complex Numbers