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| − | A complex number is a number of the form <math> a + bi </math> where <math> a </math> is the real part of the complex number, and <math> bi </math> is the imaginary part of the complex number. If <math> b = 0 </math>, then <math> a + bi </math> is a real number. If <math> a = 0 </math> and b is not equal to 0, the complex number is called an imaginary number. The imaginary unit <math> i = \sqrt{-1}</math>, and can be used to express other imaginary numbers (for example, <math> \sqrt{-25} = 5\sqrt{-1} = 5i </math>). Note that <math> i^2 = -1 </math>, <math> i^3 = -i </math>, <math> i^4 = 1 </math>, <math> i^5 = i </math>, <math> i^6 = -1 </math>, and so on. The "complex conjugate" of <math> a + bi </math> is <math> a - bi </math>, and <math> (a + bi)(a - bi) = a^2 + b^2 </math>, which is a real number. The complex conjugate is useful for simplifying expressions involving complex numbers (for example, see complex division below).
| + | ==Licensing== |
| − | | + | Content obtained and/or adapted from: |
| − | ===Operations with Complex Numbers===
| + | * [https://en.wikipedia.org/wiki/Complex_number Complex number, Wikipedia] under a CC BY-SA license |
| − | Addition: Given two complex numbers <math> a + bi </math> and <math> c + di </math>, <math> (a + bi) + (c + di) = (a + c) + (b + d)i</math>. For example, <math> (4 + 3i) + (-3 - i) = 1 + 2i </math>.
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| − | Subtraction: <math> (a + bi) - (c + di) = (a - c) + (b - d)i</math>.
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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>
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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} = \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.
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| − | ==Resources==
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| − | * [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
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| − | * [https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/multiply-and-divide-complex-numbers/ Multiplying and Dividing Complex Numbers], Lumen Learning
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