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Complex Numbers

The complex plane represents $a+ib$ as the point with Cartesian coordinates $(a,b)$.
The complex plane represents a+iba+ib as the point with Cartesian coordinates (a,b)(a,b).
Complex numbers extend the real line by adjoining a number ii whose square is −1-1. Every complex number has the form
z=a+ib,\begin{equation*}z=a+ib,\end{equation*}
where aa and bb are real numbers. The number aa is the real part, and bb is the imaginary part.
COMPLEX NUMBERS
The set of complex numbers is
C={a+ib:a,b∈R, i2=−1}.\begin{equation*}\mathbb{C}=\{a+ib:a,b\in\mathbb{R},\ i^2=-1\}.\end{equation*}
For z=a+ibz=a+ib, we write
Re⁡z=a,Im⁡z=b.\begin{equation*}\operatorname{Re}z=a,\qquad \operatorname{Im}z=b.\end{equation*}
The same point can be described by Cartesian coordinates (a,b)(a,b) or by polar coordinates (r,θ)(r,\theta).
Two basic geometric quantities attached to z=a+ibz=a+ib are its modulus and conjugate:
∣z∣=a2+b2,\begin{equation*}|z|=\sqrt{a^2+b^2},\end{equation*}
The modulus is the distance from the origin to the point zz.
The polar form of a complex number is based on the Euler formula.
Euler's formula places eiθe^{i\theta} on the unit circle and reiθre^{i\theta} on the same ray.
EULER FORMULA
For every real number θ\theta,
eiθ=cos⁡θ+isin⁡θ.\begin{equation*}e^{i\theta}=\cos\theta+i\sin\theta.\end{equation*}
Thus a complex number with modulus rr and argument θ\theta can be written as
z=reiθ.\begin{equation*}z=re^{i\theta}.\end{equation*}

References

  • [Needham2023]Needham, Tristan. Visual Complex Analysis. 25th Anniversary Edition. Oxford University Press, 2023. First edition published in 1997.