$$ z = re^{i\theta} $$
I have seen that, when specifying a complex number, most people would rather use radians as the units for $\theta.$ Is it incorrect to use degrees? Why is there a preference for radians?
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1$\begingroup$ It depends. Elementary introduction to Complex Numbers can precede Real Analysis (AKA Calculus). In Analytical Geometry, the domain of the sine and cosine functions are angles, so you don't really gain anything by measuring angles in radians rather than degrees. That is, it does not facilitate solving any Analytical Geometry problems (that I know of). For example, DeMoivre's theorem works fine against angles measured in degrees. ...see next comment $\endgroup$user2661923– user26619232021-11-02 21:06:08 +00:00Commented Nov 2, 2021 at 21:06
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$\begingroup$ However, if you have already begun Real Analysis (or Calculus), then things have changed remarkably. Here, the domain of the sine and cosine functions are dimensionless real numbers, and you grapple with such concepts as the $\displaystyle \lim_{x \to 0} \frac{\sin(x)}{x} = 1.$ The term Radians is ambiguous, and can refer to the unit of measure of an angle (i.e. $\pi$ radians equals $180^\circ$) or it can refer to the dimensionless arc length of a specific arc of the unit circle. Short answer: in Calculus, it is best to stop thinking in terms of angles, measured in degrees. $\endgroup$user2661923– user26619232021-11-02 21:11:33 +00:00Commented Nov 2, 2021 at 21:11
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$\begingroup$ No, it isn't incorrect to use other units (it never is fwiw), it's just that radians are the mathematically more "natural" unit. Maybe you know the formula for the arc length of a circle segment: $l = \frac{2\pi r \alpha}{360^\circ}$ where $\alpha$ is given in degrees. Note that this formula essentially has the conversion to radians built in and becomes way simpler if we express $\alpha$ in radians directly: $l = r \alpha$; so the angle itself tells you how far along the circle you've traveled. $\endgroup$SV-97– SV-972021-11-02 21:13:49 +00:00Commented Nov 2, 2021 at 21:13
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$\begingroup$ This stems from the fact that radians give us a so-called "arc-length parametrization" of the circle via sine and cosine - and we like those because they have a lot of nice properties. And via eulers formula we thus also arrive at radians as the "natural unit" for the angle in the complex exponential function. $\endgroup$SV-97– SV-972021-11-02 21:13:54 +00:00Commented Nov 2, 2021 at 21:13
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1$\begingroup$ @user2661923 I disagree with radians being ambiguous. Even when used as an angle it's a dimensionless number (and the SI standard agrees with this - in fact all angles are technically dimensionless) $\endgroup$SV-97– SV-972021-11-02 21:22:09 +00:00Commented Nov 2, 2021 at 21:22
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3 Answers
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You can use degrees… However, if you do so, then for a flat angle of $180°$ you don’t have the famous relation $$e^{i\pi}=-1$$ which holds if you use radians.
answered Nov 2, 2021 at 21:07
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1$\begingroup$ The rebuttal here is that $e^{i\theta}$ is nothing more than syntactic sugar for $\cos(\theta) + i\sin(\theta)$. You certainly have that $\cos(180^\circ) + i\sin(180^\circ) = -1$. See the comments that I left, following the question. Please flag me with a comment if you wish to disagree. $\endgroup$user2661923– user26619232021-11-02 21:16:00 +00:00Commented Nov 2, 2021 at 21:16
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$\begingroup$ @user2661923. It's not just syntactic sugar. Actually, for all $z\in\mathbb{C}$, one has $$e^z = \sum_{k=0}^{\infty} \frac{z^k}{k!}.$$ If $z=i\theta$ and $\theta$ is in radians you can use this formula directly. If $\theta$ is in degrees you have to replace $z$ with $i\pi\theta/180$ in the sum. $\endgroup$md2perpe– md2perpe2021-11-02 21:22:58 +00:00Commented Nov 2, 2021 at 21:22
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1$\begingroup$ @md2perpe When I say syntactic sugar, I am assuming that that the domain of the sine and cosine functions are real numbers, rather than angles. Then $e^z = e^{x + iy} = e^x \times [\cos(y) + i\sin(y)]$, which can be shown to equal $\displaystyle \sum_{k=0}^\infty \frac{(x + iy)^k}{k!}$. From my perspective, all that the $e^z$ notation achieves is simply a convenience of notation. $\endgroup$user2661923– user26619232021-11-02 21:29:45 +00:00Commented Nov 2, 2021 at 21:29
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$\begingroup$ @user2661923. And that equality is only valid if $y$ is in radians, not if it's in degrees. $\endgroup$md2perpe– md2perpe2021-11-02 21:46:24 +00:00Commented Nov 2, 2021 at 21:46
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1$\begingroup$ @md2perpe No, the equality is also valid if $y$ is in neither radians nor degrees, but is instead a dimensionless real number. This is viable by my assumption that the domain of the sine and cosine functions are dimensionless real numbers. $\endgroup$user2661923– user26619232021-11-02 21:48:13 +00:00Commented Nov 2, 2021 at 21:48
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The formula $z=re^{i\theta}$ is based on the observation that $$\begin{align} e^{i\theta} = \sum_{n=0}^{\infty} \frac{(i\theta)^n}{n!} &= \sum_{n=0\\n\text{ even}}^{\infty} \frac{(i\theta)^n}{n!} + \sum_{n=0\\n\text{ odd}}^{\infty} \frac{(i\theta)^n}{n!} \\ &= \sum_{k=0}^{\infty} \frac{(-1)^k\theta^{2k}}{(2k)!} + i \sum_{k=0}^{\infty} \frac{(-1)^k\theta^{2k+1}}{(2k+1)!} = \cos\theta + i\sin\theta, \end{align}$$ which is only valid if $\theta$ is in radians.
From this we get $$ z=x+iy=r\cos\theta+ir\sin\theta=r(\cos\theta+i\sin\theta)=re^{i\theta}. $$
Radians is the natural unit of angle in mathematics. You should learn and use them almost everywhere in mathematics.
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The angle $73^\circ=1.274\,$rad is a dimensionless quantity: there's no dimension (in the sense of dimensional analysis) involved in measuring $\frac{73}{360}$ of a full turn/circle (a turn, like a cycle, is dimensionless), while the length dimensions get cancelled out when dividing arc length by radius. It's a misconception that unit and dimension are synonyms.
The input of the natural (radian) trigonometric function $\sinθ$ is not actually an angle, but a number corresponding to some angle. E.g., $\theta$ might be $2.51,$ which corresponds to—but doesn't equal—$2.51\,\mathrm{rad}=144^\circ.$
$$z=r(\cos\theta+i\sin\theta)$$ is the polar form of a complex number. By Euler's formula, $$z=re^{iθ},\tag1$$ while by the Taylor series for sine and cosine, $$z=r\left(\left(1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\ldots\right)+i\left(\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\ldots\right)\right)$$$$=r\left(1+(i\theta)+\frac{(i\theta)^2}{2!}-\frac{(i\theta)^3}{3!}+\frac{(i\theta)^4}{4!}+\frac{(i\theta)^5}{5!}+\ldots\right).\tag2$$
Complex Analysis (specifically, applications involving $\exp()$ and $e^{()}$) has been developed based on the natural circular measure radian and the natural (radian) trigonometric functions. For example, the various derivations of formulae $(1)$ and $(2)$ all involve the natural trigonometric functions.
Electing to work in ‘degrees’ is fine if we avoid $\exp()$ and $e^{()}$ and stick to purely geometric analyses/perspectives. In this case, $e^{iθ}$ is not a useful shorthand for $(\cos\theta+i\sin\theta)$ due to potential for confusion; stick to the $\mathrm{cis}\,\theta$ notation instead.