De Moivre's Theorem

The results of z, from the 1st power to the 12th power are:

z^1=1+i

z^2=2i

z^3=-2+2i

z^4=-4

z^5=-4-4i

z^6=-8i

z^7=8-8i

z^8=16

z^9=16+16i

z^10=32i

z^11=-32+32i

z^12=-64

These values are then plotted on the complex plane, in order to clearly see the relationship between them.

These twelve powers are displayed as points on an exponential spiral. z, from the 1st power to the 12th is then expressed in polar form so the pattern will be more obvious. The polar form expresses a complex number in terms of a distance from the origin and an angle from the positive Re(z) axis. Suppose z=x+yi is represented by the point P (r,θ) on the complex plane.

The distance of the point P, r, can be found using Pythagoras.

〖r^2=x〗^2+y^2

r=√(x^2+y^2 )

Using the trigonometric properties of a right-angled triangle, point P can also be expressed in polar coordinates as follows. We have:

cos⁡〖θ=A/H〗

cos⁡〖θ=x/r〗

x=r cos⁡θ

sin⁡〖θ=O/H〗

sin⁡〖θ=y/r〗

y=r sin⁡θ

Substitute x and y and into z=x+yi :

z=r cos⁡θ+r sin⁡θ i

z=r 〖(cos〗⁡θ+i sin⁡〖θ)〗

A complex number z=x+yi can also be represented in polar form as written below:

z=r 〖(cos〗⁡θ+i sin⁡〖θ)〗

Where r is the absolute value of the complex number and θ is the argument of the complex number.

Now we shall express z, from the 1st power to the 12th in polar form.

For z^1 (1+i):

Determine the radius:

r=√(x^2+y^2 )

r=√(1^2+1^2 )

r=√2

Determine the argument:

tan⁡θ=O/A

tan⁡θ=1/1

θ=45o

∴z^1=√2(cos⁡〖45+isin⁡〖45)〗 〗

For z^2 (2i):

Determine the radius:

r=√(x^2+y^2 )

r=√(0^2+2^2 )

r=√4

r=2

Determine the argument:

θ=90o because 2i is perpendicular to the positive Re(z) axis.

∴z^2=2(cos⁡〖90+isin⁡〖90)〗 〗

For z^3 (-2+2i):

Determine the radius:

r=√(x^2+y^2 )

r=√(〖(-2)〗^2+2^2 )

r=√(4+4)

r=√8

r=2√2

Determine the argument:

tan⁡θ=O/A

tan⁡θ=2/(-2)

θ=-45o

-2+2i is in the quadrant 2, so:

θ=180o-45o

θ=135o

∴z^3=2√2(cos⁡〖135+isin⁡〖135)〗 〗

For z^4 (-4):

Determine the radius:

r=√(x^2+y^2 )

r=√(〖(-4)〗^2+0^2 )

r=√16

r=4

Determine the argument:

θ=180o because it lies on the negative Re(z) axis.

∴z^4=4(cos⁡〖180+isin⁡〖180)〗 〗

For z^5 (-4-4i):

Determine the radius:

r=√(x^2+y^2 )

r=√(〖(-4)〗^2+〖(-4)〗^2 )

r=√(16+16)

r=√32

r=4√2

Determine the argument:

tan⁡θ=O/A

tan⁡θ=(-4)/(-4)

θ=45o

-2+2i is in the quadrant 3, so:

θ=180o+45o

θ=225o

∴z^5=4√2(cos⁡〖225+isin⁡〖225)〗 〗

For z^6 (-8i):

Determine the radius:

r=√(x^2+y^2 )

r=√(0^2+〖(-8)〗^2 )

r=√64

r=8

Determine the argument:

θ=270o because -8i lies on the negative Im(z) axis.

∴z^6=8(cos⁡〖270+isin⁡〖270)〗 〗

For z^7 (8-8i):