Download Rules for the Algebra of Complex Numbers

The Algebra of Complex Numbers
Complex numbers are algebraic expressions containing the factor i ≡ − 1 . A complex
number z consists of a “real” part, Re z ≡ x, and an “imaginary” part, Im z ≡ y, that is,
z = Re z + i Im z = x + iy .
If Im z = 0, then z = x is a “real number”. If Re z = 0, then z = iy is said to be “purely
imaginary.” The real and imaginary parts, x and y, are themselves real numbers.
Addition and subtraction:
Multiplication:
z1 ± z2 = (x1 ± x2 ) + i (y1 ± y 2 )
z1 × z2 = (x1 + iy 1 ) × (x 2 + iy 2 ) = (x1x 2 − y 1y 2 ) + i (x1y 2 + x 2 y 1 )
Remember! i 2 = −1.
Division:
z1
x + iy 1 (x1x 2 + y 1y 2 ) + i (x 2 y 1 − x1y 2 )
= 1
=
z2 x 2 + iy 2
x 22 + y 22
Modulus and Argument:
The modulus of a complex number is defined by z = x 2 + y 2 . (Sometimes called the
“absolute value.”)
⎛y⎞
The argument is arg z = tan −1⎜ ⎟ .
⎝x⎠
Complex Conjugate:
The complex conjugate is defined by z ∗ = x − iy . The following rules apply:
zz∗ = z
2
(z1 + z2 )∗ = z1∗ + z2∗
(z1z2 )∗ = z1∗ z2∗
∗
⎛ z1 ⎞
z∗
⎜⎜ ⎟⎟ = 1∗
z2
⎝ z2 ⎠
PH461/PH561, Fall 2008
page 1
©William W. Warren, Jr., 2008
Some useful relations involving complex conjugates:
(z )
∗ ∗
=z
z + z ∗ = 2 Re z = 2 x
z − z ∗ = 2i Im z = 2iy
.
z ⎛ x 2 − y 2 ⎞ ⎛ 2 xy ⎞
⎟+i ⎜
⎟
=⎜
z ∗ ⎜⎝ x 2 + y 2 ⎟⎠ ⎜⎝ x 2 + y 2 ⎟⎠
Polar representation of complex numbers:
z = re i θ with r and θ real.
r = z =
⎛y⎞
x 2 + y 2 and θ = arg z = tan −1⎜ ⎟ .
⎝x⎠
Euler relations: e ± i θ = cos θ ± i sin θ .
e 2nπi = 1
e iπ / 2 = i
e i θ + e − i θ = 2 cos θ
e iπ = −1
e i 3 π / 2 = −i
e i θ − e − i θ = 2i sin θ
Argand Diagram:
A complex number z = x + iy is represented as a point in the plane Im z – Re z or,
equivalently, as a vector with components x and y. The length of the vector is the
modulus r = |z|, and the direction is determined by the angle θ = arg z.
PH461/PH561, Fall 2008
page 2
©William W. Warren, Jr., 2008