Basics of Complex Numbers (I)
1. General
•i≡√−1, so i2=−1, i3=−i,i4= 1 and then it starts over again.
•Any complex number zcan be written as the sum of a real part and an imaginary part:
z= [Re z] + i[Im z],
where the numbers or variables in the []’s are real. So z=x+y i with xand yreal is
in this form but w= 1/(a+b i) is not (see ”Rationalizing” b elow). Thus, Im z=y, but
Re w6= 1/a.
•Complex Conjugate: The complex conjugate of z, which is written as z∗, is found by
changing the sign of every iin z:
z∗= [Re z]−i[Im z] so if z=1
a+b i ,then z∗=1
a−b i .
Note: There may be “hidden” i’s in the variables; if ais a complex number, then
z∗= 1/(a∗−b i).
•Magnitude: The magnitude squared of a complex number zis:
zz∗≡ |z|2= [Re z]2−(i)2[Im z]2= [Re z]2+ [Im z]2≥0,
where the last equality shows that the magnitude is positive (except when z= 0).
Basic rule: if you need to make something real, multiply by its complex conjugate.
2. Rationalizing: We can apply this rule to “rationalize” a complex number such as z=
1/(a+b i). Make the denominator real by multiplying by the complex conjugate on top and
bottom: 1
a+b i ·a−b i
a−b i =a−b i
a2+b2=a
a2+b2+i−b
a2+b2
so Re z=a/(a2+b2) and Im z=−b/(a2+b2).
3. The Complex x–y Plane
•Rectangular form: Any complex number zcan be uniquely represented as a point in
the x–yplane, where the x–coordinate is Re zand the y–coordinate is Im z(see figure).
–You can think of ias a unit vector in the “imaginary” (y) direction.
–The magnitude of zis just the length of the vector from the origin.
•Polar form: We can also write zin polar form as:
z=r eiθ =rcos θ+i r sin θ ,
where rand θare real and equal to the length and angle of the vector.
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