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I know complex numbers but what is the meaning of "complex" in "Complex Number"? Does "complex" mean "complicated here? Are complex numbers used in easy real world problems? Give me examples please.

Waqar
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    Related. –  Apr 23 '16 at 07:40
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    As far as I know, "complex" here roughly means that they are "compound", consisting of real and imaginary part. – Wojowu Apr 23 '16 at 07:44
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    The term "complex number" was introduced by Gauss in the treatise Theoria residuorum biquadraticorum (Theory of biquadratic residues). The German version is available online; complex numbers are introduced on p. $540$. There's no explanation for the term "complex"; it merely states "Derartige Zahlen werden wir ganze complexe Zahlen nennen" ("We will call such numbers integer complex numbers"). – joriki Apr 23 '16 at 08:04

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As mentioned already, complex numbers are used extensively in electromagnetic theory, which is very practical, for example, to the development of electronics. It's important to note that complex numbers are not essential to this theory in many cases, but they do make it extremely more efficient to solve problems. This is because of the relationship between complex numbers and waves. Euler's formula tells us that

$$e^{i\theta} = \cos{\theta}+i\sin{\theta} $$

This is a very useful relation since exponential functions have nice properties which make them easy to work with. For example, when multiplying two exponentials you simply add the exponents together. The $\sin$ and $\cos$ functions are not as easy to work with, and so it is common to solve problems in E&M using the complex exponential and then to take the real part of the result to obtain a physical solution.

wgrenard
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