Difference between revisions of "Complex Numbers"
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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). | + | 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). |
===Operations with Complex Numbers=== | ===Operations with Complex Numbers=== | ||
Revision as of 13:30, 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 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 = 0 } and b is not equal to 0, the complex number is called an imaginary number. The imaginary unit 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 = \sqrt{-1}} , and can be used to express other imaginary numbers (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 \sqrt{-25} = 5\sqrt{-1} = 5i } ). Note that 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^2 = -1 } , 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^3 = -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^4 = 1 } , 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^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. The "complex conjugate" of 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 } is 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 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)(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 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 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 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: 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} .
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, 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{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
