The [[Complex Numbers|Complex]] [[Plane]] is the shorthand for referring to the two-dimensional [[Plane]] / [[Vector Space]] [[Span|spanned]] by the [[Basis Vectors]] $1$ and $i$. When discussing the complex plane, typically the $i$-axis is represented up and down while the [[Real Numbers|Real]] [[Axis]] is measured left to right as if it were an extension of the [[Real Number Line]]. ### Geometric View The complex plane gives a geometric view of operations on complex numbers. For example, multiplication of two complex numbers on the plane looks the same as making a new number with magnitude $|a||b|$ and argument $\arg(a)+\arg(b)$. Rotating a number about the origin can be done by multiplication with a [[Roots of Unity|root of unity]] $\huge \arg \left((a+bi)*e^{i\theta}\right) = \arg(a+bi)+ \theta $ This can be shown by using polar form $\huge r_{1}e^{i\theta} r_{2}e^{i\phi} = r_{1}r_{2}e^{i(\theta+\phi)} $