# Complex Number **An extension of real numbers using the imaginary unit.** > [!NOTE] Rectangular form of a complex number > $z=x+iy$ > - $x,y\in\mathbb{R}$ - *real* and *imaginary part* > - $i=\sqrt{-1}$ - *imaginary unit* The imaginary unit is so called as no real number can satisfy the equation $x^{2}=-1$. It is represented by $j$ in electrical engineering, as $i$ is used to denote electric current. The set of complex numbers is denoted $\mathbb{C}$. The complex numbers also form a *real vector space* of dimension two with $[1,i]$ as its [[Basis#Standard basis|standard basis]]. Thus every complex number can be represented as a [[vector]]. Every complex number also has its own [[complex conjugate]]. ![[ComplexNumber.svg|550]] *A complex number represented as a vector; the argument $\phi$ is notated and is used in polar form* ## Polar form Complex numbers are also commonly expressed in *[[Polar Coordinate System|polar]] form*, either using [[Euler's formula]] or angle notation. Angle notation is the more common form of representing a [[phasor]]. > [!NOTE] Polar form of a complex number > $z=re^{i\phi}=r\angle \phi$ > - $r=|z|=\sqrt{x^2+y^2}$ - *modulus* or *magnitude* - the distance from the origin to the point $z$. > - $\phi=\tan^{-1}\frac{y}{x}$ - *argument* or *[[phase]]* - the angle between the vector $z$ and the positive real axis. The argument, $\phi$ or $\arg (z)$, can take *infinitely many values*, with values separated by integer multiples of $2\pi$. ### Principal argument The *principal argument*, $\text{Arg}(z)$, is the typical choice for a well-defined *single-valued* argument function. It is defined over a restricted range $-\pi<\text{Arg}(z)\le\pi$. $\text{Arg}(z)=\tan^{-1}\left(\frac{y}{x}\right)+ {\begin{cases} 0, & z\text{ in 1st or 4th quadrant} \\ \pi, & z\text{ in 2nd quadrant} \\ -\pi, & z\text{ in 3rd quadrant} \end{cases}}$ ## Operations > [!Addition and subtraction] $z_{1}\pm z_{2}=(x_{1}\pm x_{2})+i(y_{1}\pm y_{2})$ > [!Multiplication] $z_{1}z_{2}=r_{1}r_{2}\angle(\phi_{1}+\phi_{2})$ ^e47b34 > [!Division] $\frac{z_{1}}{z_{2}}=\frac{r_{1}}{r_{2}}\angle(\phi_{1}-\phi_{2})$ ^e3a9f7 > [!Inversion] $\frac{1}{z}=\frac{\bar{z}}{|z|^2}=\frac{1}{r}\angle{-\phi}$ > [!NOTE] Integer [[Exponentiation#Complex base with rational exponents|exponentiation]] > $z^{n}=r^{n}\angle n\phi$ > [[De Moivre's formula]] can be used to simplify the calculation of the integer exponentiation of complex numbers. > [!Fractional exponentiation or root extraction] Fractional exponentiation is [[Exponentiation#Complex base with rational exponents|multi-valued]] but its *principal value* is $z^{\frac{1}{n}}=\sqrt[n]{z}=\sqrt[n]{r}\angle{\frac{\phi}{n}}$ For other values, the root can be found by *extending* [[de Moivre's Formula|de Moivre's formula]].