Download Complex Numbers - Art of Problem Solving

Complex Numbers
Let’s start with Cartesian forms, z = a + i*b or real numbers a and b and where i = sqrt(-1).
Example 1: z = 1 + 2i and w = 3 – i. The real part of z is 1 and the imaginary part of z is 2.
z+w=(1+2i)+(3-i) = 4+i
z*w=(1+2i)(3-i) = 3-i+6i-2i*i=5+5i
The conjugate of z, conj(z)=1-2i, where we change the sign on the imaginary part.
The absolute value of z, abs(z) = sqrt(z*conj(z)), abs(z)^2 = (1+2i)*(1-2i)=5.
3  4i  (3  4i) * (3  4i)  5
On an x and iy axis, 3 is the base and 4 the altitude with abs(z) the hypotenuse.
To compute the reciprocal of z, we multiply the numerator and denominator by the conjugate of the
denominator.
z/w = (1+2i)/(3-i)*(3+i)/(3+i)=(1+7i)/10
z 1  2i 3  i 3  2  6i  i 1  7i

*


w 3i 3i
9 1
10
Example:
Solve for the roots and verify: z^2 -2z+5=0.
2  4  20 2  4i

 1  2i
2
2
Verify 1+2i
(1+2i)2-2(1+2i)+5 =1+4i-4-2-4i+5=0
Problems:
1. Verify this formula for the square root of a complex number and find the square root of 3+4i.
a  bi  
2
2

z a i z a

2. In an AC circuit the voltage, E, and the current, I, are related by conductance, G: I=EG.
The conductance G is the reciprocal of impedance, Z. For a parallel connection the conductance
is the sum of the conductances in parallel. Find the impedance of Z1=1+i in parallel with
Z2=3+4i.
May 4, 2011
Dr. G. Boyd Swartz
HeroesGifted.org