De Moivre’s Theorem Proof

De Moivre’s theorem states that the strength of a complex quantity in polar shape is equal to elevating the modulus to the same strength and multiplying the argument with the aid of equal power.

We first attain some intuition for de Moivre’s theorem through thinking about what takes place when we multiply a complex range with the aid of itself. This shows that by squaring a complex number, the argument is improved by way of two and the absolute value is squared. This theorem helps us find the energy and roots of complicated numbers easily. Abraham De Moivre used to be one of the mathematicians to use complex numbers in trigonometry.

History of the complex number

The set of complex range or imaginary diversification that we work with these days has the fingerprints of many mathematical giants.

Abraham De Moivre in addition extended the learning about such numbers when he published Miscellanea Analytica in 1730, using trigonometry to represent the power of complex numbers. De Moivre tackles the important catch 22 situation of the time, the factorization of the polynomial into quadratics.

Complex Numbers – Polar Form

We can characterize a complex quantity the usage of trigonometry an awful lot like we characterize vectors in trigonometric form. We additionally called this representation the ‘’polar form’’ of complicated numbers. Rather than the use of a coordinate for the proper phase and the imaginary part, we use the absolute value of the complex range and the directed point of view from the fantastic X-axis or polar axis to the line section connecting the complex factor to the pole, measured in a counter-clockwise course

Powers of Complex Number

In order to compute the powers of difficult numbers, we ought to think about the manner of repeated multiplication.

Given z = r(cos 𝜃+ isin 𝜃)

z2 = [r (cos 𝜃+isin 𝜃)][r(cos 𝜃+isin 𝜃)]=r2 (cos2 𝜃+isin2 𝜃)

z3 = [r2 (cos2 𝜃+ isin2 𝜃)][r(cos 𝜃+isin 𝜃)]=r3 (cos3 𝜃+isin3 𝜃)

z4 = [r3 (cos3 𝜃+ isin 3 𝜃)][r(cos 𝜃+isin 𝜃)]=r4 (cos4 𝜃+isin4 𝜃)

z5 = [r4 (cos4 𝜃+isin4 𝜃)][r(cos 𝜃+isin 𝜃)]=r5 (cos5 𝜃+isin5 𝜃)

After seeing the strength of z, we can see a pattern evolvinging. This sample is the core of the theorem named after the French mathematician Abraham De Moivre.

De Moivre’s Theorem: If z = r(cos 𝜃+isin 𝜃) is a complex number and n is a positive integer,

Then,

zn = [r(cos 𝜃+isin 𝜃]n = rn (cos n 𝜃+ isin n 𝜃).

Using this theorem we can easily compute the power of a complex number such as z=(2+2i)

De Moivre’s Theorem: If z = r(cosq+isinq) is a complex number and n is a positive integer,

then,

z n = [r(cosq+isinq)] n = r n (cosnq + i sinnq).

Using this theorem we can effortlessly compute the energy of a complicated wide variety such as z = (2 +2i). First, we have to convert the complex variety to its polar form:

z =(2+2i) = 2√2(cos45°+i sin45°)

With

r = √22 +22 = √8= 2√2, and tan-1 [2/2]=45°

Where z falls in the 1st quadrant.

Then

z6 = (2+2i)6=[2√2(cos45°+isin45°)]6=(2√2)6(cos 270°+sin 270°)= -512i

The proof of De Moivre’s Theorem

To prove De Moivre’s theorem, we use a simple proof by induction. Given a complex number,

z = (cos 𝜃+isin 𝜃)

we can easily show using repeated multiplication that for n =0, 1, 2, 3, 4,

z0 = [r0 (cos0 𝜃+isin0 𝜃)]=1 (cos0+isin0)=1+i0=1

z1 = [r1 (cos 𝜃+ isin 𝜃)]1=r (cos 𝜃+isin 𝜃)

z2 = [r2 (cos 𝜃+isin 𝜃)]2=[r(cos 𝜃+isin 𝜃)][r(cos 𝜃+isin 𝜃)]=r2 (cos2 𝜃+isin2 𝜃)

z3 = [r3 (cos 𝜃+ isin𝜃)]3=[ r2 (cos2 𝜃+isin2 𝜃)][ r(cos 𝜃+isin 𝜃)]= r3 (cos3 𝜃+isin3𝜃)

z4 =[r4 (cos 𝜃+ isin𝜃)]4=[ r3 (cos3 𝜃+isin3 𝜃)][ r(cos 𝜃+isin 𝜃)]= r4 (cos4 𝜃+isin4𝜃)

Now, let us assume that zn = [r(cos 𝜃+isin 𝜃)]n = rn (cos n𝜃+isin𝜃)

Is true for some n ∈ Z+.

Then we must indicate that this implies it is true for all n + 1, that is,

[r(cos 𝜃+isin 𝜃)] n+1 = r n+1 (cos(n +1) 𝜃+isin(n +1) 𝜃).

Then given

[r(cos 𝜃+isin 𝜃)] n = rn (cos n 𝜃+isin n 𝜃)

We multiply both sides of the equation by [r(cos 𝜃+isin 𝜃)]

Then,

[r(cos 𝜃+isin 𝜃)][r(cos 𝜃+isin 𝜃)] n = rn [ cosn 𝜃+isinn 𝜃][r(cos 𝜃+isin 𝜃)]

Therefore

[r(cos 𝜃+isin 𝜃)]n+1 = rn r[cosn 𝜃cos 𝜃+cosn 𝜃sin 𝜃i+isinn 𝜃cos 𝜃 -sin n 𝜃sin 𝜃].

We then employ the use of the common trigonometric formulas for the sum of the angle for sine and cosine,

sin(x + y)= sin x cos y +cos x sin y and cos(x + y)= cos x cos y -sin x sin y.

We let x = n 𝜃, and y = 𝜃 and we have

  r n+1 [cos(n 𝜃+ 𝜃)+isin(n 𝜃+ 𝜃)]= r n+1 [cos(n +1) 𝜃+isin(n +1) 𝜃],

as desired for all positive integers.

We must also consider Z—n for

  z -n = [r(cosq+isinq)]-n = r-n [cos(-n 𝜃)+isin(-n 𝜃)].

Since cosine and sine are even and odd functions respectively, we have

 cos(-n 𝜃)+isin(-n 𝜃) = cos(n 𝜃)-isin(n 𝜃)

  Therefore,

r -n [cos(-n 𝜃)+ isin(n 𝜃)]= 1/ rn [cosn 𝜃- isinn 𝜃]

Conclusion

We use this theorem to simplify complex numbers raised to a power. It resolves positive kinds of equations or discovers the nth roots of a complex number. This will show this deductively for fantastic quintessential values of n and we will prove it inductively for all n. This theorem is useful in calculating fractional powers and roots of a complex number as well.