# When should I measure the angle of a complex number clockwise or anticlockwise?

In this diagram theta is measured anticlockwise. How would I know from which side to measure the angle?

In this diagram theta is measured clockwise.

• You measure it from whichever side leads the angle to be in the specified desired range, in this case $$-\pi<\theta\leq\pi$$ Commented Aug 15, 2018 at 18:12
• Notice that both are measured anticlockwise. THe second diagram the angle is negative. Commented Aug 15, 2018 at 18:25
• What if I don't know what the desired range is? The range is never specifed in my textbooks. Sometimes the angle measured is +ve and sometime's it's -ve. How would I know which way I should measure it? Commented Aug 17, 2018 at 2:44

Actually, in both cases the angle is measured counterclockwise (as they should). That's why in the second angle the value is $-135^\circ$. If it was being measured clockwise, the value would be just $135^\circ$.

• I don't think that's correct. Sure, the angle that the diagram presents is equal to $135^\circ$, but the reason why it should be $-135^\circ$ is because it's a complex number, and convention is that degrees should land between $-180^\circ<\theta\leq180^\circ$ Commented Aug 15, 2018 at 18:15
• @RushabhMehta I am not aware of such convention. And that doesn't change the fact that if the angle was being measued clockwise, then its value would be $135^\circ$. Commented Aug 15, 2018 at 18:16
• Yes of course, that's obvious. His question is a bit more nuanced. It's asking why they chose to measure it from the clockwise position, rather than counterclockwise. There's a reason the diagram labelled the angle $-135^\circ$ Commented Aug 15, 2018 at 18:18
• I thin Jose's answer is correct and i don't the the question was that nuanced. The angle NEGATIVE 135 is being measured in the COUNTERclockwise direction. why it is being measured in the counterclockwise direction is not stated but is being consistent with counter clockwise. Commented Aug 15, 2018 at 18:29

This is because we want to have $\theta$ such that $$-\pi<\theta\le\pi.$$ This is known as the principal value of an argument, denoted $\text{Arg}\,z$.

• I would say this is plagiarism but it is your own comment lol +1. Commented Aug 15, 2018 at 18:16
• Do you mean that theta must be less than 180° but if it is greater than 180° then the value of the angle would be measured counterclockwise such that it's a -ve angle? For example if anticlockwise, I had theta = 240° then I should measure it clockwise getting theta = -120° ? Am I right or have I misunderstood it? Commented Aug 17, 2018 at 2:52

Actually those are both counter clockwise. Note the second angle is NEGATIVE $135$. A negative angle in the counterclockwise position, will appear to be the same as a positive angle of the same magnitude in the clockwise position.

So both diagrams ARE counterclockwise.

Counter-clockwise means: Increasing values go counter-clockwise. And decreasing values go clockwise. In this diagram we start at $0^{\circ}$ and DEcrease to $- 135^{\circ}$.

Note that in the first diagram, $-1+i$ is equal to $\sqrt {2} \angle 135°$, but in the second diagram the quantity is $-1-i$, which is equal to $\sqrt {2} \angle -135°.$

Negative angles figure heavily into AC circuits, where the phase angle $\tan \theta = \frac {X_L-X_C}{R}$, where $L$ (impedance) and $C$ (capacitance) are imaginary and $R$ is the resistance, the phase angle between the voltage and the current is determined by the sign of the angle. To show that the voltage lags behind the current, we must show the angle as negative, and to do so we take the angle clockwise, rather than anti (counter) clockwise.

The usual representation of arguments (polar angles) of points $(x,y)$, resp. $z=x+iy$, in figures is unfortunate, as is exemplified by the above question. In both linked figures the argument is measured counterclockwise, starting from the positive $x$-axis, but the second figure is showing a strange arrow pointing clockwise, accompagned by a negative number. I therefore suggest to indicate the measurement of arguments in figures as follows:

Argument of a complex number is defined to be the angle between the positive real axis and the line joining the origin and z, represented as a point in the complex plane

NOTICE that no where it is mentioned that it must be measured in anti-clockwise direction or in clockwise direction.

Although it is a convention that we must measure the angle in anticlockwise direction.

But for the principal argument or amplitude of a complex number to be defined we only need the following conditions:

(i) the angle must be measured with the positive real access

(ii)the angle must lie between (-π,π]

So, in your 2nd diagram the angle -135°, if measured from anticlockwise direction with the positive real axis would come out to be 225°. The angle 225 degrees is not lying between(-π,π]. Therefore 225 degree will not be accepted as the principle argument or amplitude of the complex number.

Therefore, measuring the angle with the positive real axis in clockwise direction gives us -135°. And this angle completely satisfies the condition for it to be principal argument

Therefore, the principle argument is -135 degrees

Conclusion : The final conclusion is that we must not consider the direction of the measurement of the angle. Instead we must only measure the angle with the positive real axis.