Difference between revisions of "Arithmetic with complex number"
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| − | The motivating factor for the use of [[complex number]]s is that they form an [[algebraically closed]] [[field]]. In simpler language any algebraic operation that is proformed on a complex number gives a complex number. This is not true for [[real number]]s, for example, <math>x^{2}+1=0</math> is an entirly real expression but <math>x</math> cannot be a real number. The solution is <math>x=\pm i</math>. | + | The motivating factor for the use of [[complex number]]s is that they form an [[algebraically closed]] [[Field_(mathematics)|field]]. In simpler language any algebraic operation that is proformed on a complex number gives a complex number. This is not true for [[real number]]s, for example, <math>x^{2}+1=0</math> is an entirly real expression but <math>x</math> cannot be a real number. The solution is <math>x=\pm i</math>. |
For this we will consider two complex numbers, <math>z_{1}=x_{1}+iy_{1}</math> and <math>z_{2}=x_{2}+iy_{2}</math>, where <math>x_{1},x_{2},y_{1},y_{2}\in\mathbb{R}</math> and <math>i^{2}=-1</math>. | For this we will consider two complex numbers, <math>z_{1}=x_{1}+iy_{1}</math> and <math>z_{2}=x_{2}+iy_{2}</math>, where <math>x_{1},x_{2},y_{1},y_{2}\in\mathbb{R}</math> and <math>i^{2}=-1</math>. | ||
Revision as of 21:32, June 27, 2008
The motivating factor for the use of complex numbers is that they form an algebraically closed field. In simpler language any algebraic operation that is proformed on a complex number gives a complex number. This is not true for real 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 x^{2}+1=0} is an entirly real expression but 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 x} cannot be a real number. The solution 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 x=\pm i} .
For this we will consider 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 z_{1}=x_{1}+iy_{1}} and , where 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 x_{1},x_{2},y_{1},y_{2}\in\mathbb{R}} and .
Addition
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 z_{1}+z_{2}=(x_{1}+x_{2})+i(y_{1}+y_{2})}
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 z_{1}z_{2}=(x_{1}+iy_{1})(x_{2}+iy_{2})=x_{1}x_{2}+ix_{1}y_{2}+iy_{1}x_{2}+i^{2}y_{1}y_{2}=(x_{1}x_{2}-y_{1}y_{2})+i(x_{1}y_{2}+y_{1}x_{2})}
Division
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{z_{1}}{z_{2}}=\frac{x_{1}+iy_{1}}{x_{2}+iy_{2}}=\frac{x_{1}+iy_{1}}{x_{2}+iy_{2}}\cdot\frac{x_{2}-iy_{2}}{x_{2}-iy_{2}}=\frac{(x_{1}x_{2}+y_{1}y_{2})-i(x_{1}y_{2}-y_{1}x_{2})}{(x_{2})^{2}+(y_{2})^2}}
Here we see two important number. The complex conjugate, 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 \bar{z}=\overline{x+iy}=x-iy}
and the length 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 {||z||}^{2}=z.\bar{z}=x^{2}+y^{2}.}