Complex Numbers
Original Notes adopted from November 13, 2001 (Week 10)
© P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong
Polynomial Equations with Integer Coefficients
Eg. 3x + 2 = 0.
No Solution in Z, solution in Q
Every linear equation with coeffients in Z, has a root in Q.
Eg. x2 – 2 = 0. No root in Q. Has root in R.
Eg. x2 + 1 = 0, no root in R.
To define set of complex numbers, C
C = { a + bi, a,b ∈ R }
i2 = -1
Define ( a + bi ) + (c + di) = (a + c) + ( b + d) i
Define ( a + bi ) ( c + di ) = ( ac – bd ) + ( ad + bc) i
Given a + bi, we say " a is the real part" of a + bi, & write R ( a + bi ) , & we say "b is the imaginary part"
of a + bi, & write Im( a + bi) = b.
a + 0i, write as a,
R is embedded in C:
0 + bi, write as bi
to a ∈ R , corresponds a + 0i.
Numbers of form bi for b ≠ 0 are "pure imaginary" numbers
0 = 0 + 0i
a + bi + 0 = a + bi ∀ a + bi.
Given a + bi, there is an additive inverse.
(a + bi ) + (-a + (-b)i) = a + (-a) + ( b + (-b)) i = 0.
1 = 1 + 0i
1( a + bi) = ( a + bi) 1 = a + bi.
Eg. ( 2 + 3 i ) ( a + bi ) = 1.
Want 1/2+3i to be a + bi, some a, b ∈ R.
Note: a + bi = c + di is equivalent to a = c & b = d.
Given a + bi, define its modulus | a + bi | to bi √a2 + b2
|a + bi| = distance from a + bi to 0.
Argand diagram
a + bi ⇒ (a, b) in R2
The complex conjugate of a + bi is a = bi.
Conjugate of 3 - √2i is 3 + √2i
Notation: a + bi
= a – bi
(a+bi)(a+bi) = (a + bi) ( a – bi) = a2 + b2 = |a + bi|2
Eg. Want
1
= 1
2 – 3 i = 2 – 3i
2 + 3i 2+ 3i
2 -3i
13
Generally,
if a + bi ≠ 0
1
= 1
a + bi a+ bi
= 2/13 – 3/13 i
a-bi = a
a-bi
a2 + b2
=
∴ (a + bi) (
1
a2 + b2
-b
a2 + b2
i
(a – bi)
1 ( a – bi) )
a2 + b2
1 (a2 + b2 ) = 1.
2
a + b2
∴ Every complex number ≠ 0 has a multiplicitive inverse.
The complex numbers with the operations are a "field" (as Q,R). The argument of a + bi is the angle that
line from (0, 0) to (a,b) makes with positive x –axis.
If a + bi has modulus r & argument θ,
r = √ a2 + b2
θ = arctan b/a
a = r cos θ, d = r sinθ
a + bi = r cosθ + irsinθ = r ( cos θ + isin θ)
|cos θ + i sinθ| = √cos 2 θ+ sin2 θ =1
[ r (cosθ + isinθ) ] [ s (cos φ + isinφ)]
= r s [( cosθcosφ - sinθsinφ ) + i (sinθcosφ + cosθsinφ)]
= r s ( cosθ + φ ) + i sin( θ + φ)
To multiply complex numbers, multiply modulus & add arguments.
In Particular,
(r (cosθ + isinθ))2 = r2 ( cos2θ + isin2θ)
De Moivre's Theorem (For n ∈ N)
[r (cosθ + isinθ))n = rn ( cosnθ + isin nθ)
Proof: By Induction, n = 1 – Obvious.
Assume n = k.
To show (r (cosθ + isinθ))k+1 = rk+1(cos(k+1) θsin (k+1) θ)
[r (cosθ + isinθ)]k+1 = [r(cos θ + isin θ)] k[r(cos θ + sin θ)]
= r k (cosk θ + isin kθ)[r(cos θ + isin θ)]
= r k r(cos(k θ +θ) + isin(kθ + θ) )
= rk+1(cos(k+1) θsin (k+1) θ)
8
Eg. (1 + i)
(1+ i)
π/4
1 + i = √2( cos π/4 + isinπ/4)
(1 + i) 8= [√2 cos π/4 + isinπ/4)] 8
= 24 (cos 2π + isin2π)
= 16( 1 + i0)
= 16
(1 + i) 100= [√2 cos π/4 + isinπ/4)] 100
= 250 (cos 25π + isin25π)
= 250 (cos π + isinπ)
= 250(-1 + 0) = -250
2
Eg. z = -1
=i
=-i
Eg. z 2= 1
= 1, -1.
-1
1
Eg. z 3= 1
z =1
|z 3| = |z |3= 1
∴ |z| = 1.
if |z| = 1, z = cosθ + sinθ , some θ.
z 3= cos3θ + isin3θ = 1 = cos ( 0 + k2π) + isin( 0 + k2π)
Can have 3θ = 2πk for any k ∈ Z.
In particular, 3θ = 0 z = cosθ + isinθ = 1
θ=0
k = 1: 3θ = 2π
θ = 2π/3
z = cos 2π/3 + isin 2π/3
= -1/2 + i √3/2
k = 2: 3θ = 4π
θ = 4π/3
z = cos 4π/3 + isin 4π/3
= -1/2 - i √3/2
k = 3: 3θ = 6π
θ = 2π
z = cos 2π + isin 2π
=1
Get only 3 solutions to z3 =1.
Eg. The fourth roots of unity are the solutions of z4 =1.
z = r (cosθ + isinθ). Know r =1.
z4 = cos4θ + isin4θ = 1 = cos(0 + 2πk) + isin(0 + 2πk)
4θ = 0 ⇒ θ = 0.
4θ = 2π ⇒ θ = π/2
4θ = 4π ⇒ θ = π
4θ = 6π ⇒ θ = 3/2 π
Eg. The fifth roots of unity are the solutions of z5 =1.
z = r (cosθ + isinθ). Know r =1.
z5 = cos5θ + isin5θ = 1 = cos(0 + 2πk) + isin(0 + 2πk)
5θ = 0 ⇒ θ = 0.
5θ = 2π ⇒ θ = 2π/5
z = cos2π/5 + isin2π/5.
Eg. Solve z3 =|1 + i |
|1 + i | = √2
if |z| = r, r3 = 21/2, so r = 21/6
z = r (cosθ + isinθ). 1 + i = √2(cosπ/4 + isinπ/4)
z3 = r3 (cos3θ + isin3θ)
3θ = π/4 or 3θ = π/4 + 2π OR 3θ = π/4 + 4π.
3θ = 9π/4
θ1 = π/12
θ2 = 3π/4
θ3 = 17π/12
z1 = 21/6 (cosπ/12 + isinπ/12)
z2 = 21/6 (cos 7π/12 + isin 3π/4)
z3 = 21/6 (cos 17π/12 + isin 17π/12)