# Questions about the arg of complex numbers

• I
Santiago24
Hi PF community, I'm reading about complex numbers and i have some questions about the argument of a complex number that i can't solve with Google or reading again the same page. Well, my first doubt is about , i can't understand where come this and why there is some random integer, i understand but the previous one is impossible to me. My other doubt is about an example that i saw.
z^3 = 8i
3 arg z = arg(z^3) = arg(8i ) = π/2 + 2kπ,
If someone can answer and clear my doubts i will be very thankful.

Mentor
Hi PF community, I'm reading about complex numbers and i have some questions about the argument of a complex number that i can't solve with Google or reading again the same page. Well, my first doubt is about ,
In the formula above ##\arg z## and ##\text{Arg} z## represent, respectively, the angle that z makes with the horizontal (i.e., real) axis, and the angle in the interval ##[0, 2\pi)##.

In your example below, ##\text{Arg} z= \frac \pi 2## since that's the angle that the imaginary number ##i## makes with the real axis. Keep in mind that ##\frac \pi 2, \frac {5\pi} 2, \frac{9\pi} 2## and so on, are all possible angles for ##i##. Each of these is ##\frac \pi 2## plus some integer multiple of ##2\pi##. All of these angles are possible values for ##\arg z## when ##\text{Arg} z= \frac \pi 2##.
Santiago Perini said:
i can't understand where come this and why there is some random integer, i understand but the previous one is impossible to me. My other doubt is about an example that i saw.
z^3 = 8i
3 arg z = arg(z^3) = arg(8i ) = π/2 + 2kπ,
If someone can answer and clear my doubts i will be very thankful.
Because ##8i## is an imaginary number that makes an angle of ##\frac \pi 2## with the real axis.

• jedishrfu and Santiago24
Santiago24
In the formula above ##\arg z## and ##\text{Arg} z## represent, respectively, the angle that z makes with the horizontal (i.e., real) axis, and the angle in the interval ##[0, 2\pi)##.

In your example below, ##\text{Arg} z= \frac \pi 2## since that's the angle that the imaginary number ##i## makes with the real axis. Keep in mind that ##\frac \pi 2, \frac {5\pi} 2, \frac{9\pi} 2## and so on, are all possible angles for ##i##. Each of these is ##\frac \pi 2## plus some integer multiple of ##2\pi##. All of these angles are possible values for ##\arg z## when ##\text{Arg} z= \frac \pi 2##.
Because ##8i## is an imaginary number that makes an angle of ##\frac \pi 2## with the real axis.

• 