Complex Numbers

A complex number is a number of the form $a+bi$ where $a$ is the real part of the complex number, and $bi$ is the imaginary part of the complex number. If $b=0$, then $a+bi$ is a real number. If $a=0$ and b is not equal to 0, the complex number is called an imaginary number. The imaginary unit $i={\sqrt {-1}}$, and can be used to express other imaginary numbers (for example, ${\sqrt {-25}}=5{\sqrt {-1}}=5i$). Note that $i^{2}=-1$, $i^{3}=-i$, $i^{4}=1$, $i^{5}=i$, $i^{6}=-1$, and so on. The "complex conjugate" of $a+bi$ is $a-bi$, and $(a+bi)(a-bi)=a^{2}+b^{2}$, which is a real number. The complex conjugate is useful for simplifying expressions involving complex numbers (for example, see complex division below).

Operations with Complex Numbers

Addition: Given two complex numbers $a+bi$ and $c+di$, $(a+bi)+(c+di)=(a+c)+(b+d)i$. For example, $(4+3i)+(-3-i)=1+2i$.

Subtraction: $(a+bi)-(c+di)=(a-c)+(b-d)i$.

Multiplication: $(a+bi)(c+di)=ac+bci+adi+bdi^{2}=ac+bci+adi-bd=(ac-bd)+(bc+ad)i$

Division: Division works a bit differently with complex numbers. The reciprocal of a complex number ${\frac {1}{a+bi}}={\frac {a-bi}{(a+bi)(a-bi)}}={\frac {a-bi}{a^{2}+abi-abi-b^{2}i^{2}}}={\frac {a-bi}{a^{2}+b^{2}}}$.

So, ${\frac {a+bi}{c+di}}={\frac {(a+bi)(c-di)}{c^{2}+d^{2}}}={\frac {(ac+bd)+(bc-ad)i}{c^{2}+d^{2}}}={\frac {ac+bd}{c^{2}+d^{2}}}+{\frac {bc-ad}{c^{2}+d^{2}}}i$. Note that c and d cannot both be equal to 0.

Resources

• Complex Numbers, Paul's Online Notes
• Intro to Complex Numbers, Lumen Learning
• Multiplying and Dividing Complex Numbers, Lumen Learning