Geometric analysis of the reproduction of complex numbers?

I've constantly been educated that means to consider complex numbers is as a cartesian room, where the "actual" component is the x part and also the "fictional" component is the y part.

In this feeling, these complex numbers resemble vectors, and also they can be included geometrically like regular vectors can.

Nonetheless, exists a geometric analysis for the reproduction of 2 complex numbers?

I experimented with 2 examination ones, $3+i$ and also $-2+3i$, which increase to $-9+7i$. Yet no geometric value appears to be located.

Exists a geometric value for the reproduction of complex numbers?

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2019-05-18 21:02:13
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Answers: 3

Suppose we increase the complex numbers $z_1$ and also $z_2$. If these numbers are created in the polar kind as $r_1 e^{i \theta_1}$ and also $r_2 e^{i \theta_2}$, the item will certainly be $r_1 r_2 e^{i (\theta_1 + \theta_2)}$. Equivalently, we are extending the first facility number $z_1$ by a variable equivalent to the size of the 2nd facility number $z_2$ and afterwards revolving the extended $z_1$ counter - clockwise by an angle $\theta_2$ to get to the item. There are numerous internet sites that expand upon this instinct with graphics and also even more description. See this website as an example - http://www.suitcaseofdreams.net/Geometric_multiplication.htm

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2019-05-21 02:50:48
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Yes, there is a straightforward geometric definition, yet you require to transform to the polar kind of the complex numbers to see it plainly. $3+i$ has size $\sqrt{10}$ and also angle concerning $18^\circ$ ; $-2+3i$ has size $\sqrt{13}$ and also angle concerning $124^\circ$. Reproduction of the complex numbers increases both sizes, causing $\sqrt{130}$, and also includes both angles, $142^\circ$. To put it simply, you can watch the 2nd number as scaling and also revolving the first (or the first scaling and also revolving the 2nd).

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2019-05-21 02:49:11
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Add the angles and also increase the sizes.

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2019-05-21 02:43:47
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