# 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 $$-\pi<\theta\leq\pi$$ – Don Thousand Aug 15 '18 at 18:12
• Notice that both are measured anticlockwise. THe second diagram the angle is negative. – fleablood Aug 15 '18 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? – Aida Ghani Aug 17 '18 at 2:44

## 5 Answers

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 $-180^\circ<\theta\leq180^\circ$ – Don Thousand Aug 15 '18 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 $135^\circ$. – José Carlos Santos Aug 15 '18 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 $-135^\circ$ – Don Thousand Aug 15 '18 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. – fleablood Aug 15 '18 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. – Don Thousand Aug 15 '18 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? – Aida Ghani Aug 17 '18 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: 