Difference between revisions of "Complex Numbers"

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Multiplication: <math> (a + bi)(c + di) = ac + bci +  adi + bdi^2 = ac + bci + adi - bd = (ac - bd) + (bc + ad)i</math>
 
Multiplication: <math> (a + bi)(c + di) = ac + bci +  adi + bdi^2 = ac + bci + adi - bd = (ac - bd) + (bc + ad)i</math>
  
Division: Division works a bit differently with complex numbers. The reciprocal of a complex number <math> \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}</math>. So, <math> \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2}</math>.
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Division: Division works a bit differently with complex numbers. The reciprocal of a complex number <math> \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}</math>.
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So, <math> \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}</math> = .
  
 
==Resources==
 
==Resources==
 
* [https://tutorial.math.lamar.edu/classes/alg/ComplexNumbers.aspx Complex Numbers], Paul's Online Notes
 
* [https://tutorial.math.lamar.edu/classes/alg/ComplexNumbers.aspx Complex Numbers], Paul's Online Notes
 
* [https://courses.lumenlearning.com/collegealgebra2017/chapter/introduction-complex-numbers/ Intro to Complex Numbers], Lumen Learning
 
* [https://courses.lumenlearning.com/collegealgebra2017/chapter/introduction-complex-numbers/ Intro to Complex Numbers], Lumen Learning

Revision as of 13:19, 20 September 2021

A complex number is a number of the form where is the real part of the complex number, and is the imaginary part of the complex number. If , then is a real number. If and b is not equal to 0, the complex number is called an imaginary number. The imaginary unit , and can be used to express other imaginary numbers (for example, ). Note that , , , , and so on.

Operations with Complex Numbers

Addition: Given two complex numbers and , . For example, .

Subtraction: .

Multiplication:

Division: Division works a bit differently with complex numbers. The reciprocal of a complex number .

So, = .

Resources