Complex Numbers

1  Introduction

A complex number, Complex_files\Complex_MathML_0.jpg, is a combination of a real number and an imaginary number, and is written in the form Complex_files\Complex_MathML_1.jpg or Complex_files\Complex_MathML_2.jpg. Complex_files\Complex_MathML_3.jpg is called the real part of Complex_files\Complex_MathML_4.jpgand Complex_files\Complex_MathML_5.jpg is called the imaginary part of Complex_files\Complex_MathML_6.jpg If Complex_files\Complex_MathML_7.jpg, Complex_files\Complex_MathML_8.jpg is called a pure imaginary number and if Complex_files\Complex_MathML_9.jpg, Complex_files\Complex_MathML_10.jpg is called a pure real number. Two complex numbers are equal if and only if both their real and imaginary parts are equal i.e. if Complex_files\Complex_MathML_11.jpg, Complex_files\Complex_MathML_12.jpgand Complex_files\Complex_MathML_13.jpgthen Complex_files\Complex_MathML_14.jpg and Complex_files\Complex_MathML_15.jpg.

2  The Argand Diagram


The easiest way to geometrically represent a complex number is by using an Argand Diagram (see above). The point Complex_files\Complex_MathML_16.jpgrepresents the complex number Complex_files\Complex_MathML_17.jpg. All pure real numbers exist on the real axis and all pure imaginary numbers exist on the imaginary axis. The set of all points representing all complex numbers is called the complex plane.

3  Modulus and Argument

As an alternative to using Cartesian co-ordinates, it is possible to specify the point Complex_files\Complex_MathML_18.jpg by the length of the line connecting the origin with the point and the angle this line makes with the positive real axis (see diagram below).


The length, Complex_files\Complex_MathML_19.jpg, is called the modulus of Complex_files\Complex_MathML_20.jpg and is denoted either by Complex_files\Complex_MathML_21.jpg or Complex_files\Complex_MathML_22.jpgThe angle, Complex_files\Complex_MathML_23.jpg, is called the argument of Complex_files\Complex_MathML_24.jpg, denoted by Complex_files\Complex_MathML_25.jpg. Since Complex_files\Complex_MathML_26.jpgall represent the same point, it is conventional to restrict the range of the argument to Complex_files\Complex_MathML_27.jpg Complex_files\Complex_MathML_28.jpg. This is called the principal value of the argument. It will be noted that, by pythagoras’ theorem, Complex_files\Complex_MathML_29.jpg, so that Complex_files\Complex_MathML_30.jpg.

Also, Complex_files\Complex_MathML_31.jpg, so that Complex_files\Complex_MathML_32.jpg. Care must be taken when calculating the argument. For example, if the point is in the third quadrant, so that both Complex_files\Complex_MathML_33.jpg and Complex_files\Complex_MathML_34.jpg are negative, Complex_files\Complex_MathML_35.jpg gives the value of the angle in the first quadrant, whereas the true value is Complex_files\Complex_MathML_36.jpg.

3.1  Examples

Determine the modulus and argument of Complex_files\Complex_MathML_37.jpg where Complex_files\Complex_MathML_38.jpg

i)Complex_files\Complex_MathML_39.jpg Complex_files\Complex_MathML_40.jpg

ii)Complex_files\Complex_MathML_41.jpg Complex_files\Complex_MathML_42.jpg

iii) Complex_files\Complex_MathML_43.jpg


i) It is generally a good idea to sketch an Argand diagram for each case.


Complex_files\Complex_MathML_44.jpg .





Complex_files\Complex_MathML_47.jpg .




Look at the Argand diagram - the point is in the 3rd quadrant.

Complex_files\Complex_MathML_49.jpg - the argument of Complex_files\Complex_MathML_50.jpgis therefore given by


4  Polar form of a Complex number

Using elementary trigonometry it can be seen that Complex_files\Complex_MathML_52.jpgand Complex_files\Complex_MathML_53.jpg.

Any complex number can therefore be written in the alternative form


This is called the polar form of the complex number (as distinct from the cartesian form shown above), and is frequently abbreviated by either Complex_files\Complex_MathML_55.jpg or Complex_files\Complex_MathML_56.jpg.

4.1  Examples

Express the following complex numbers in polar form:

i)Complex_files\Complex_MathML_57.jpg Complex_files\Complex_MathML_58.jpg

ii)Complex_files\Complex_MathML_59.jpg Complex_files\Complex_MathML_60.jpg

iii) Complex_files\Complex_MathML_61.jpg


The modulus and argument for these numbers have already been calculated in 4.2.1 above.

i) Complex_files\Complex_MathML_62.jpg.

Complex_files\Complex_MathML_63.jpg is therefore written in polar form as Complex_files\Complex_MathML_64.jpg.

ii) Complex_files\Complex_MathML_65.jpg.

Complex_files\Complex_MathML_66.jpgis therefore written in polar form as Complex_files\Complex_MathML_67.jpg.

iii) Complex_files\Complex_MathML_68.jpg.

Complex_files\Complex_MathML_69.jpgis therefore written in polar form as Complex_files\Complex_MathML_70.jpg.

Note that although Complex_files\Complex_MathML_71.jpgis negative, we have converted it to a positive value in the polar representation.

4.2  Conversion from polar form to cartesian form

Whilst conversion from cartesian to polar form is straightforward, conversion from polar to cartesian form can be a little more tricky. As always, it is a good idea to sketch an Argand diagram.

4.3  Examples

Express the following complex numbers in cartesian form:

i) Complex_files\Complex_MathML_72.jpg


iii) Complex_files\Complex_MathML_74.jpg


i) Both the cosine and the sine terms are positive, and Complex_files\Complex_MathML_75.jpg, so the point is in the first quadrant.


The value of Complex_files\Complex_MathML_76.jpgis given by Complex_files\Complex_MathML_77.jpgand the value of Complex_files\Complex_MathML_78.jpgis given by Complex_files\Complex_MathML_79.jpg

Therefore, in cartesian form, Complex_files\Complex_MathML_80.jpg.

ii) Both the sine and cosine terms are positive, and Complex_files\Complex_MathML_81.jpg, so the point is in the second quadrant.


The value of Complex_files\Complex_MathML_82.jpg is given by Complex_files\Complex_MathML_83.jpg. The value of Complex_files\Complex_MathML_84.jpg is given by Complex_files\Complex_MathML_85.jpg. The complex number in cartesian form is therefore Complex_files\Complex_MathML_86.jpg.

iii) Both the cosine and sine terms are positive, and Complex_files\Complex_MathML_87.jpg, so the point is in the fourth quadrant.


The value of Complex_files\Complex_MathML_88.jpgis given by Complex_files\Complex_MathML_89.jpgand the value of Complex_files\Complex_MathML_90.jpgis given by Complex_files\Complex_MathML_91.jpg

Therefore, in cartesian form, Complex_files\Complex_MathML_92.jpg.