The Complex Plane (College Board AP® Precalculus): Revision Note

Complex numbers & rectangular coordinates

What is a complex number?

• A complex number is a number of the form $a+bi$, where

  • $a$ and $b$ are real numbers

  • $i$ is the imaginary unit, defined by $i^2=-1$

    • or equivalently, $i=\sqrt{-1}$

• The real number $a$ is called the real part of the complex number

• The real number $b$ is called the imaginary part of the complex number

  • E.g. the complex number $3-5i$ has real part $3$ and imaginary part $-5$

• Every real number is also a complex number

  • A real number $a$ can be written as $a+0i$

    • i.e. with real part $a$ and imaginary part $0$

How are complex numbers represented using rectangular coordinates?

• A complex number can be represented as a point in a plane, called the complex plane

  • The horizontal axis is the real axis (Re)

  • The vertical axis is the imaginary axis (Im)

• The complex number $a+bi$ is represented by the point with rectangular coordinates $(a, b)$

  • The real part $a$ gives the horizontal coordinate

  • The imaginary part $b$ gives the vertical coordinate

• E.g. the complex number $4+2i$ corresponds to the point $(4, 2)$ in the complex plane

  • or going the other way, the point with rectangular coordinates $(-3, -3)$ in the complex plane corresponds to the complex number $-3-3i$

• Note that a real number like $2=2+0i$ is located on the real axis

  • and a purely imaginary number like $i=0+i$ is located on the imaginary axis

Complex numbers & polar coordinates

How are complex numbers represented using polar coordinates?

• Since a complex number corresponds to a point in the complex plane, it can also be described using polar coordinates $(r, \theta )$

  • $r$ is the signed radius value

  • $\theta$ is the angle in standard position whose terminal ray includes the point

• If a complex number has polar coordinates $(r, \theta )$

  • then its rectangular coordinates are $(r\cos \theta , r\sin \theta )$

    • This follows directly from the standard conversion formulas

      • $x=r\cos \theta$ and $y=r\sin \theta$

  • Substituting into the $a+bi$ form gives the polar form of the complex number:

    • $\left(r\cos \theta \right)+i\left(r\sin \theta \right)$

• E.g. the complex number with polar coordinates $\left(4, \frac{\pi }{3}\right)$ can be written as $\left(4\cos \frac{\pi }{3}\right)+i\left(4\sin \frac{\pi }{3}\right)$

  • which simplifies to $2+2\sqrt{3} i$ in the $a+bi$ form

    • i.e. because $\cos \frac{\pi }{3}=\frac{1}{2}$ and $\sin \frac{\pi }{3}=\frac{\sqrt{3}}{2}$

• Going the other way, to convert a complex number $a+bi$ to polar form

  • use the same rectangular-to-polar conversion formulas from the polar coordinate system

    • $r=\sqrt{a^2+b^2}$

    • $\theta =\arctan \left(\frac{b}{a}\right)$ for $a>0$

    • $\theta =\arctan \left(\frac{b}{a}\right)+\pi$ for $a<0$

• E.g. to convert $-\sqrt{3}+ i$ to polar form

  • $r=\sqrt{(-\sqrt{3}{)}^2+1^2}=\sqrt{4}=2$

  • Since $a=-\sqrt{3}<0$

    • $\theta =\arctan \left(\frac{1}{-\sqrt{3}}\right)+\pi =-{\displaystyle \frac{\pi }{6}}+\pi ={\displaystyle \frac{5\pi }{6}}$

  • So the polar form is $\left(2\cos \frac{5\pi }{6}\right)+i\left(2\sin \frac{5\pi }{6}\right)$

    • with polar coordinates $\left(2,\frac{5\pi }{6}\right)$