Jens Siebel (University of Applied Sciences Kaiserslautern) An Interactive Introduction to Complex Numbers
1. Introduction We know that some polynomial equations do not have any solutions on R/ .
Example 1.1: Solve x2 +1 = 0 for x : x2+=10 ⇔ x 2 =− 1 ⇔ xx =−−∨=− 11
Problem: The equation has no solution on R/ , as −1 ∉R / .
An exit strategy is to introduce an “imaginary unit“ i with i = − 1 or i2 = − 1, alternatively. Then, x2 +1 = 0 has two solutions: x= − i and x= i .
Rule 1.1 The imaginary unit i is defined as i = − 1 or i2 = − 1.
Example 1.2: Solve x2 −2 ⋅ x + 5 = 0 for x :
2 2 2 −−22 −− 22 xx−⋅+=⇔=−250 x − −∨=− 5 x + −⇔ 5 22 22 xx=−−∨=+−14 14 ⇔ x =− 141 ⋅−∨=+() x 141 ⋅−() ⇔ x=−⋅−∨=+⋅−121 x 121 ⇔ xixi =−⋅∨=+⋅ 12 12
The solutions x=1 − 2 ⋅ i and x=1 + 2 ⋅ i are “complex“ numbers. In cartesian form they are composed from a real number and an imaginary number. Normally, complex numbers are denoted z . According to the fundamental theorem of algebra, any polynomial equation of grade n∈ N / has n complex solutions.
1 Jens Siebel: An Interactive Introduction to Complex Numbers 1. Introduction Cartesian form Rule 1.2 A complex number z in cartesian form is written z= a + bi ⋅ , a, b∈ R / with • a= Re( z ) : real part of z • b= Im( z ) : imaginary part of z
• i : imaginary unit.
Example 1.3:
z1 =1 − 2 ⋅ i and z2 =1 + 2 ⋅ i are the solutions of example 1.2. We have
a1=Re( z 1 ) = 1 , b1=Im( z 1 ) = − 2
a2=Re( z 2 ) = 1 , b2=Im( z 2 ) = 2
Rule 1.3 C/ is the set of complex numbers z . It comprises all other sets of numbers.
N/ Z/ Q/ R/ C/
Figure 1.1: Sets of numbers
Any real number x is a complex number z= x +0 ⋅ i . Real numbers do not have imaginary units.
Rule 1.4 Complex numbers are presented as points or position vectors in a cartesian diagram (“complex plane”). Any real number is located on the real axis.
2 Jens Siebel: An Interactive Introduction to Complex Numbers 1. Introduction
Im(z)
z= a + bi ⋅ b⋅ i
i
Re(z) a Figure 1.2: Complex plane
Example 1.4:
Open the Applet Basic Calculations and enter z1 =1 + 2 ⋅ i and z2 =−0.5 − 1.5 ⋅ i in cartesian form.
Polar form In studying the Applet Basic Calculations in example 1.4 you may have noticed that the complex number z1 =1 + 2 ⋅ i has also been displayed as
63.4349 °i z1 =2.2361 ⋅ exp( 63.4349 °⋅= i) 2.2361 ⋅ e . This is the so-called “exponential form” - a special case of the “polar form”. Vector algebra allows us to turn any complex number z from cartesian form into polar form.
3 Jens Siebel: An Interactive Introduction to Complex Numbers 1. Introduction
Im(z)
b⋅ i z= a + bi ⋅
r
ϕ
a Re(z) Figure 1.3: Geometric presentation of z in polar form
In order to rewrite z= a + bi ⋅ in polar form we have to determine the value of ϕ and the length r of the position vector z= a + bi ⋅ .
Rule 1.5 z ≥ 0 is the absolute value of a complex number. It corresponds the length r∈ R / ≥0 of vector z= a + bi ⋅ . Applying Pythagoras’ equation we get zr= = a2 + b 2 .
Rule 1.6 ϕ is the angle between the positive real axis and the position vector z= a + bi ⋅ . ϕ is also called the „argument of z “: ϕ = arg z . It can be given both in degree or radian.
Due to vector algebra we have a= r ⋅ cos ϕ and b= r ⋅ sin ϕ . Then, we can rewrite z= a + bi ⋅ as zr=⋅cosϕ +⋅⋅ ri sin ϕ =⋅ r( cos ϕϕ +⋅ i sin ) .
Rule 1.7 zr=⋅cosϕ +⋅⋅ ri sin ϕ =⋅ r( cos ϕϕ +⋅ i sin ) is the “trigonometric form“ of complex number z= a + bi ⋅ with a= r ⋅ cos ϕ and b= r ⋅ sin ϕ .
b Regarding figure 1.3 we see that tan ϕ = and hence a
4 Jens Siebel: An Interactive Introduction to Complex Numbers 1. Introduction b arctanifa> 0, b ≥ 0ϕ ∈°°[ 0 , 90 ) a 90°ifa = 0, b > 0 b ϕ =arg z = 180°+ arctanif a < 0ϕ ∈°°() 90 , 270 (degree) a 270°ifa = 0, b < 0 b 360°+ arctanifa >≤ 0, b 0ϕ ∈°°( 270 , 360 ] a or
b π arctanifa> 0, b ≥ 0ϕ ∈ 0, a 2 π ifa=0, b > 0 2 b π 3 ϕπ==+argz arctan ifa <∈⋅ 0 ϕπ , (radian). a 2 2 3 ⋅π ifa =0, b < 0 2 b 3 2⋅+π arctanifa >≤ 0, b 0 ϕππ ∈⋅⋅ , 2 a 2
Example 1.5:
The trigonometric form of z1 =1 + 2 ⋅ i in example 1.4 is 2 z=2.2361 ⋅( cos63.4349 °+⋅ i sin 63.4349 ° ) , as ϕ =arctan = 63.4349 ° and 1 1 1
2 2 r1 =1 + 2 = 2.2361 .
The Euler equation enables us to convert z=⋅ r(cosϕ +⋅ i sin ϕ ) into z= r ⋅ e ϕ⋅i .
Rule 1.8 z= r ⋅ e ϕ⋅i is the exponential form of complex number z .
In General The polar form of z is depicted by r and ϕ . It can be displayed in either trigonometric form or exponential form. Note that it requires the trigonometric form to turn the exponential form into the cartesian form.
5 Jens Siebel: An Interactive Introduction to Complex Numbers 1. Introduction Throughout the following and in the applets we will prefer the exponential form to the trigonometric form, as the first one has a shorter notation and a less demanding algebra. Note that both applets display the exponential form as z= r ⋅exp (ϕ ⋅ i ) .
Example 1.6:
45 °i 2.5 ⋅i Open the Applet Basic Calculations and enter z1 =1 ⋅ e and z2 =2 ⋅ e in exponential form. Note that ϕ1=argz 1 = 45 ° is in degree, whereas ϕ2=argz 2 = 2.5 is in radian. When entering ϕ in degree you have to add “°”.
Example 1.7:
45 °i 2.5 ⋅i In order to rewrite z1 =1 ⋅ e and z2 =2 ⋅ e from example 1.6 in cartesian form, one has to calculate as follows:
45 °i ze1 =⋅=⋅1 1( cos 45 °+⋅ i sin 45 °=⋅) 1( 0.7071 +⋅ i 0.7071) = 0.7071 + 0.7071 ⋅ i
2.5 ⋅i ze2 =⋅=⋅2 2( cos 2.5 +⋅ i sin 2.5) =− 1.6023 + 1.1969 ⋅ i
Exercise 1.1: Rewrite z=+⋅13, iz =−+⋅ 13, iz =−−⋅ 13 i and z=1 − 3 ⋅ i in exponential form. Use the Applet Basic Calculations to check your answers.
Exercise 1.2: Rewrite z=⋅2 e75°i , zez = 90 ° i ,2 =⋅ e 285 ° i and z=2 ⋅ e 0.25 ⋅πi in cartesian form. Use the
Applet Basic Calculations to check your answers. You can enter 2 as “sqrt(2)“ and 0.25 ⋅π as “0.25*pi“ or “pi/4”.
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