Difference between revisions of "Complex Numbers"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
Line 10: Line 10:
 
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>.
 
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} = \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</math>.
+
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} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i</math>. Note that c and d cannot both be equal to 0.
  
 
==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 13:23, 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, . Note that c and d cannot both be equal to 0.

Resources