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Complex Numbers - Nomenclature

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    Okay, I had a question where I was given a simple complex number in the form a+bi, and asked to put it in polar form. So I changed it to \sqrt{a^2+b^2}[cos(\theta )+isin(\theta )], \theta =\frac{b}{a}. However the answer was in the form of \sqrt{a^2+b^2}e^i{\theta }, \theta =\frac{b}{a}. I was under the impression that this was the "Euler form".

    Can someone just clear up which form is which so I am not confused from now on !
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    Polar form just means in terms of it's distance from the origin (its 'radius') and its angle from 1. Both of the forms you have given would qualify as some kind of polar form but writing it as an exponent of e is the most common way since it has the same amount of information and is easier to write.
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    I would always write a complex number in the form  r\ cis(\theta), where r is the modulus of the complex number and  cis(\theta) is another way of saying  cos(\theta)+i\ sin(\theta) . That's polar form to me.

    I almost never use Euler's formula unless I'm proving its validity by the Maclaurin series.
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    At degree level I'd say re^{i\theta} is very much the canonical form for a polar representation.
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    I would agree that re^{i\theta} is much more standard than r(\cos \theta + i \sin \theta) or the quite ugly r \operatorname{cis}\theta.

    However, as usual, if you are answering a question from your course, then what your lecturer/teacher means by 'polar form' will be contained within the content of your course i.e. will have been mentioned in lessons/lectures/the book/lecture notes etc. So the first call would be to reexamine those. I am nearly sure, however, that if you have covered Euler's formula then your teacher/lecturer will either:

    a) decree re^{i\theta} as standard polar form or
    b) accept either way of writing it.

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Updated: April 19, 2012
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