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De Moivre's Theorem -Properties
De Moivre's theorem -Properties
1. Sum of \(n^{th}\) roots of unity i zero
2.Product of \(n^{th}\) roots of unity is \((-1)^{n-1}\)
3.\(n^{th}\) roots of unity form a G.P with common ratio \(e^{\frac {i\:2\pi} {n} }\)
4.\(n^{th}\) roots of unity lie on unit circle \(|z|=1\)
In particular if \(n=3\) then
i) \(1+\omega +\omega^2=0\)
ii) \(\omega^2=1\)
De Moivre's Theorem
De Moivre's theorem
\(z=r(cis\: \theta)\)
\(z^n =r (cis \:n \theta)\)
\(n^{th}\) roots of Unity
\(z^n=1\)
\(z=cis\Big( \frac {2k\pi} {n}\Big) ,k=0,1,2,......,n-1\)
If \(n=3,z=1,\omega,\omega^2\) are cube roots of unity
Representation Of Complex Numbers
Complex numbers
A number \(z=x+iy\) where \(x,y\in R\); \(x=\text{Real part or Re(z)}\) ;\(y=\text{Imaginary part or Im(z)}\)
| Magnitude | Argument | complex conjugate |
| \(|z|=\sqrt{x^2+y^2}\) | \(amp(z)=arg(z)=\theta=\tan^{-1} \frac y x\) | If \(z=x+iy\) |
| \(|z|=|\bar{z}|\) | General argument:\(2n\pi+\theta ,n\in N\) | then the conjugate of z is |
|
Principal Argument : \(-\pi <\theta\le\pi\) |
\(\bar{z}=x-iy\) | |
| Least postive Argument:\(0<\theta\le 2\pi\) |
Representation
| Polar representation | Exponential form | vector representation |
| \(x=r\cos \theta,y=r \sin \theta\) | \(z=re^{i\theta}\)( where \(e^{i\theta}=(\cos \theta+i \sin \theta)\) | \(z=x+iy\) may be considered as postion vector of point P |
Conjugate & Modulus
| Properties of complex conjugate | Properties of modulus |
| If \(z=a+ib\Rightarrow\bar{z}=a-ib\) | \(z \bar{z}=|z|^2\) |
| \(\bar{\bar{z}}=z\) | \(z^{-1}=\frac {\bar{z}} {|z|^2}\) |
| \(z+\bar{z}=2a=2 Re(z)=\text{purely real}\) | \(|z_1+z_2|^2=|z_1|^2+|z_2|^2\pm2Re(z_1\bar{z_2})\) |
| \(z-\bar{z}=2ib=2i \,Im(z)=\text{purely imaginary}\) | \(|z_1+z_2|^2+|z_1-z_2|^2=2[|z_1|^2+|z_2|^2|]\) |
| \(z\bar{z}=a^2+b^2=|z|^2=\{Re(z)\}^2+\{Im(z)\}^2\) | |
| \(z+\bar{z}=0 \) or \(z=-\bar{z} \Rightarrow z=0\) or \(z\) is purely Imaginary | |
| \(z=\bar{z}\Rightarrow z\) is purely real |
Properties Of Arugument
Properties of Arugument of a complex number
If \(z,z_1\) and \(z_2\) are complex numbers,then
1. Arg( any real positive numbers )=0
2.\(Arz(z-\bar{z})=\pm \frac {\pi} {2}\)
3.\(Arg(z_1.\bar{z_2})=arg(z_1)-arg(z_2)\)
4.\(|z_1+z_2|^2+|z_1-z_2|^2=2[|z_1|^2+|z_2|]\)
Square Roots Of Complex Numbers
Square roots of a complex Numbers
The Square root of \(z=a+ib\) is
\(\boxed {\sqrt{a+ib}=\pm \big[\sqrt{\frac {|z|+a} {2}} +i \sqrt{\frac {|z|-a} {2}}}\) for \(b>0\)
and
\(\boxed {\sqrt{a-ib}=\pm \big[\sqrt{\frac {|z|+a} {2}} -i \sqrt{\frac {|z|-a} {2}}}\) \(b<0\)
Inequalities & Iota
Inequalities
Triangle Inequalities
1.\(|z_1\pm z_2|\le |z_1|+|z_2|\)
2.\(|z_1\pm z_2|\ge |z_1|-|z_2|\)
Parallelogram Identity
1.\(|z_1+z_2|^2+|z_1-z_2|^2=2[|z_1|^2+|z_2|^2]\)
Integral powers of iota
\(i=\sqrt{-1}\) so \(i^2=-1\) ;
\(i^3=-i\) and \(i^4=1\)
\(i^{4n+3}=-i\)
\(i^{4n}\) or \(i^{4n+4}=1\)
\(i^{4n+1}=i\)
\(i^{4n+2}=-1\)
Geometrical Properties Of Complex Numbers
Geometrical Properties of complex numbers
1.If ABC is an equilateral triangle having vertices \(z_1,z_2,z_3\) then
\(z_1^2+z_2^2+z_3^3=z_1z_2+z_2z_3+z_3z_1\) or
\(\frac {1} {z_1-z_2} +\frac {1} {z_2-z_3}+\frac {1} {z_3-z_1}=0\)
2.If \(z_1,z_2,z_3,z_4\) are vertices of parallelogram then \(z_1+z_3=z_2+z_4\)
3.If \(z_1,z_2,z_3\) are fixes of the points A,B and C in the Argand plane,then
(a) \(\angle BAC=arg\Big( \frac {z_3-z_1} {z_2-z_1}\Big)\)
(b) \(\frac {z_3-z_1} {z_2-z_1}= \frac {|z_3-z_1|}{ |z_2-z_1|} (\cos \alpha+i \sin \alpha)\) ,where \(\alpha=\angle BAC\)
4.The equation of a circle whose centre is at point having affix \(z_0\) and radius
R is \(|z-z_0|=R\)
5.If a,b are positive real numbers then \(\sqrt{-a}\times \sqrt{-b}=-\sqrt{ab}\)
Statement Of Roots Of Unity
Complex Numbers -Roots of Unity
Statement :-
1. If \(n\in Z\) (the set of integers),then \((\cos \theta+i \sin \theta)^n=\cos (n \theta) +i \sin (n \theta)\)
2.If \(n\in Q\)(the set of rational number),then \(\cos (n \theta) +i \sin (n \theta)\) one of the values of \((\cos \theta+i \sin \theta)^n\)