A-level Mathematics/MEI/FP2/Complex Numbers

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Modulus-argument form[edit]

Polar form of a complex number[edit]

It is possible to express complex numbers in polar form. The complex number z in the diagram below can be described by the length r and the angle of its position vector in the Argand diagram.

[Argand diagram]

The distance r is the modulus of z, . The angle is measured from the positive real axis and is taken anticlockwise. Adding any whole multiple of however, would give the same vector so a complex number's principal argument, , is where . The following examples demonstrate this in each quadrant.

The following Argand diagram shows the complex number .

[Argand diagram]

This Argand diagram shows the complex number .
This Argand diagram shows the complex number .
This Argand diagram shows the complex number .

When we have a complex number in polar form we can use and to write it in the form: . This is the modulus-argument form for complex numbers.

Multiplication and division[edit]

The polar form of complex numbers can provide a geometrical interpretation of the multiplication and division of complex numbers.


Take two complex numbers in polar form,

and then multiply them together,

The result is a complex number with a modulus of and an argument of . This means that:


Dividing two complex number and in polar form:

Multiply numerator and denominator by .

Then, use distribution to simplify.

Here, factorize by in the numerator and cancel out terms in the denominator. Note that .

Apply the formulas for the cosine of the difference of two angles and for the sine of the difference of two angles:


De Moivre's theorem[edit]

Using the multiplication rules we can see that if


De Moivre's theorem states that this holds true for any integer power. So,

Complex exponents[edit]


If we let we can then differentiate z with respect to .

The general solution to the differential equation is .

This means that

By putting as 0 we get:

So the general definition can be made:

For a complex number , calculating can be done:

Proof of de Moivre's theorem[edit]

We can now give an alternative proof of de Moivre's theorem for any rational value of n:


deMoivre's therom can come in handy for finding simple expressions for infinite series. This usually involves a series multiple angles, cosrθ, as in this next example:

Infinite series are defined by:

In order to find either the sum of C or the sum of S (or both!) you need to add C to jS: Which using deMoivre's theorem can be written as:

It is easier to work with now using the form e^jθ:

and you should be able to see the pattern, that the factor is negative 1/2 to the power of n-1 (the negative being alternately to even then odd powers is what makes it flip between + and -) and the power of e (the number in front of θ in our original equations) is equal to 3(n-1)jθ.

This is a geometric series, with a=

Complex roots[edit]

The roots of unity[edit]

The fundamental theorem of algebra states that a polynomial of degree n should have exactly n (complex) roots. This means that the simple equation has n roots.

Let's take a look at . This has two roots, 1 and -1. These can be plotted on an Argand diagram:

[Argand diagram]

Consider , from the above stated property, we know this equation has three roots. One of these is easily seen to be 1, for the others we rewrite the equation as and use the factor theorem to obtain . From this, we can solve by completing the square on z so that we have . Solving for z you obtain . We have now found the three roots of unity of , they are , and

Solving an equation of the form [edit]

We know has six roots, one of which is 1.

We can rewrite this equation replacing the number 1 with since 1 can be represented in polar form as having a modulus of 1 and an argument which is an integer multiple of .

Now by raising is side to the power of 1/6:

To find all six roots we just change k, starting at 0 and going up to 5:

Applications of complex numbers in geometry[edit]