Complex Number

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• Imaginary Number

 

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If a one-dimensional vector rotates +90 degrees and rotates +90 degrees once more, the result can be -1.

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• Complex Number

 

 

$$\begin{array}{}& x+jy\end{array}$$

 

If a number is not expressed as a single number such as 1, 2, or 3 but as a field composed of several units, it is called a complex number, x is called the real part, and y is called the imaginary part. 

 

For example, the equation 

has no real solution, because the square of a real number cannot be negative.

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The ways of Complex Number expression.

 

1. Cartesian coordinate

 

A coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines

 

$$\begin{array}{}& (2,3):2+j3\end{array}$$ $$\begin{array}{}& (0,0):0+j0\end{array}$$ $$\begin{array}{}& (-<!--\; -\; -->3,1):-<!--\; -\; -->3+j1\end{array}$$ $$\begin{array}{}& (-<!--\; -\; -->2,-<!--\; -\; -->2):-<!--\; -\; -->2-<!--\; -\; -->j2\end{array}$$

 

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2. Polar Coordinate

 

Express using the distance from the origin and the angle from the horizontal axis.

 

$$\begin{array}{}& \{\begin{array}{l}r=\sqrt{x^2+y^2}\\ \mathrm{\theta <!--\; \theta \; -->}={\tan }^{-<!--\; -\; -->1}<!--\; \; -->(y/x)\end{array}\end{array}$$ The pair of $(r,\mathrm{\theta <!--\; \theta \; -->})$ can be expressed $$\begin{array}{}& r\mathrm{\angle <!--\; \angle \; -->}\mathrm{\theta <!--\; \theta \; -->}\end{array}$$ If the angle is not within the range of $-<!--\; -\; -->180°<<!--\; <\; -->\mathrm{\theta <!--\; \theta \; -->}<<!--\; <\; -->180°$, add or subtract a multiple of 360° and make it within 180° such as $3\mathrm{\angle <!--\; \angle \; -->}280°$ to $3\mathrm{\angle <!--\; \angle \; -->}-<!--\; -\; -->80°$

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For example, let's calculate the distance and the degree of this.

 

$\mathrm{z(Complex\; Number)}=1+j0.5$ $$\begin{array}{}& r=\sqrt{1^2+{0.5}^2}=1.118\end{array}$$ $$\begin{array}{}& \mathrm{\theta <!--\; \theta \; -->}={\tan }^{-<!--\; -\; -->1}<!--\; \; -->\left(\frac{0.5}{1}\right)=0.4636\text{ rad}\end{array}$$

 

https://angeloyeo.github.io/2019/06/15/imaginary_number.html