# 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