SHORT NOTES: Complex Numbers

$N\subset W\subset Z\subset Q\subset R$

Natural no.(N) ,Whole no.(W) , Integers(Z or I) , Rational(Q) & Irrational no.(Q'), Real no(R).

Imaginary Number is square root of a negative real number e.g.

$x^2+1=0\Rightarrow x=\pm \sqrt{-1}$

which is imaginary.so the quantity $\sqrt{-1}$ is denoted by $i$ .Thus $i=\sqrt{-1}$

$i^n=(-1)^\frac{n}{2}$ if n is even

$i^n=(-1)^\frac{n-1}{2}$ if n i odd

Complex Number: $z=x+iy$ = Re z + i Im z

Algebra of complex numbers:

$a+ib=c+id\Rightarrow a=c,b=d$

$(a+ib)+(c+id)=(a+c)+i(b+d)$ Addition

$(a+ib)(c+id)=(ac-bd)+i(ad+bc)$ Multiplication

$\overline{a+ib}=a-ib$ Conjugation

Polar representation:

$a=rcos\theta , b=rsin\theta\\z=a+ib=r(cos\theta+isin\theta)$

Modulus of a complex number:

$\left | z \right |=\sqrt{(x^2+y^2)}$

Properties of modulus:

Hint: The above property can be remembered by thinking $z_1,z_2$ as two sides of a triangle so that third side which is resultant of $z_1,z_2$ is always less than equal to sum of individual side and always greater than equal to difference of two sides which is basic theorem of triangle.

Principal argument:

$Arg(z)=\theta=tan^{-1}\frac{b}{a}$

The value of $\theta$ where $-\pi< \theta< \pi$ is called principle argument.

Euler's notation:

$z=re^{i\theta}=r(cos\theta+isin\theta)$

cube roots of unity:

$x=\sqrt[3]{1}\Rightarrow x=1,\frac{-1+i\sqrt{3}}{2},\frac{-1-i\sqrt{3}}{2}\Rightarrow x=1,\omega,\omega^2$

properties of $\omega$ :

$\omega^{3r}=1,\omega^{3r+1}=\omega,\omega^{3r+2}=\omega^2\\$

$1+\omega^r+\omega^{2r}=o$ if 'r' is not multiple of '3'

$1+\omega^r+\omega^{2r}=3$ if 'r' is multiple of '3'

Geometrical interpretation:

$\left | z-z_0 \right |=a$ circumference of a circle with center $z_0$ and radius a

$\left | \frac{z-z_1}{z-z_2} \right |=k$ Circle if $k\neq 1$

perpendicular bisector if $k=1$

Logarithm of complex no.

$\log z=\log (re^{i(\theta+2n\pi ))})=\log r+i(\theta+2n\pi)=\log\left | z \right |+iArgz+2n\pi$

if we put n=0 we get principle value of log z.

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Complex Numbers

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