Complex Numbers – HSC Math

Posted In: Mathematics

A complex number is a number that can be expressed in the form $a + ib$, where a and b are real numbers, and $i$ is a solution of the equation $x^2 = −1$. Because no real number satisfies this equation, $i$ is called an imaginary number.

  1. Cartesian form
    $z=x+iy; x=r\ cos\ θ, t=r\ sin\ θ,$
    $r=√{x^2+y^2}, θ=tan^{-1}|y/x|$
  2. Polar form
    $Z=re^{iθ}; e^{iθ}=cos\ θ+i\ sin\ θ$
  3. Euler’s formula
    $cos\ θ+i\ sin\ θ=e^{iθ}$ and
    $cos\ θ-i\ sin\ θ=e^{-iθ}$
  4. Solution to different types of complex number
    $z=x+iy; θ=tan^{-1}|y/x|$
    $z=-x+iy; θ=π-tan^{-1}|y/x|$
    $z=-x-iy; θ=-π+tan^{-1}|y/x|$
    $z=x-iy; θ=-tan^{-1}|y/x|$
  5. if $z=x+iy$ is a complex number, $z’=x-iy$ is the conjugate complex number of that complex number.
  6. if $z_1,z_2$ are two complex numbers, then $|z_1z_2|=|z_1|+|z_2|$
  7. Power of complex number
    $i^{4n+0}=1$
    $i^{4n+1}=i$
    $i^{4n+2}=-1$
    $i^{4n+3}=-i$
  8. Cubic toots of unity
    $1, ω, ω^2$
    $ω={-1+i√3}/2$
    $ω^2={-1-i√3}/2$
    $ω^3=1$
    $1+ω+ω^2=0$

Post Tags: HSC | Number Sets


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