Complex numbers

Complex numbers have a real part and an imaginary part. They are typically written in the form $z=a+bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit with $i^2=-1$. Can sometimes be denoted with $j$ instead of $i$ too.

• Complex Conjugate(Liittoluku): The complex conjugate of a complex number $z=a+bi$ is $\overline{z}=a-bi$. It mirrors $z$ across the real axis in the complex plane.
• Polar Form (Osoitinmuoto): A complex number can also be expressed in polar coordinates as $z=r(\cos (\theta )+i\sin (\theta ))$, where $r$ is the modulus (or absolute value) of $z$ and $\theta$ is the argument (or angle) of $z$. This form is particularly useful for multiplying and dividing complex numbers.
• Exponential Form (Eksponenttimuoto): Using Euler’s formula, $e^{i\theta }=\cos (\theta )+i\sin (\theta )$, the exponential form of a complex number is $z=r{e}^{i\theta }$. This form simplifies many operations involving complex numbers, such as exponentiation.
• Division of Complex Numbers: To divide two complex numbers, $z=a+bi$ and $w=c+di$, you multiply the numerator and the denominator by the complex conjugate of the denominator: $\frac{z}{w}=\frac{a+bi}{c+di}\cdot \frac{c-di}{c-di}=\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}$

Examples

1. Complex Conjugate Example: For $z=3+4i$, the complex conjugate $\overline{z}=3-4i$.
2. Polar Form Example: Convert $z=1+i$ to polar form.
   • Modulus: $r=\sqrt{1^2+1^2}=\sqrt{2}$
   • Argument: $\theta ={\tan }^{-1}(\frac{1}{1})=\frac{\pi }{4}$
   • Polar form: $z=\sqrt{2}(\cos (\frac{\pi }{4})+i\sin (\frac{\pi }{4}))$
3. Exponential Form Example: Using the polar coordinates from above, $z=1+i$ can also be written as $z=\sqrt{2}{e}^{i\frac{\pi }{4}}$.
4. Complex Number Division Example: Divide $z=1+i$ by $w=1-i$.
   • Using the division formula: $\frac{1+i}{1-i}\cdot \frac{1+i}{1+i}=\frac{(1+1)+(1-1)i}{1+1}=1$