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 $i^{2}=-1<math>,<math>i^{3}=-i$, $i^{4}=1$, $i^{5}=i$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^6 = -1 } , and so on.

Operations with Complex Numbers

Addition: Given two complex numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a + bi } and $c+di$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a + bi) + (c + di) = (a + c) + (b + d)i} . For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (4 + 3i) + (-3 - i) = 1 + 2i } .

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

Multiplication: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{a + bi} = \frac{a - bi}{(a + bi)(a - bi)} = \frac{a - bi}{a^2 + abi - abi - b^2i^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