Argument in complex numbers

[Mentallic]

When given a complex number $z=x+iy$ and transforming this into its mod-arg form giving $rcis\theta$ where $r=\sqrt{(x^2+y^2)}$ and $\theta=arctan(y/x)$, we are assuming that $-\pi/2<\theta<\pi/2$.

What if however a student is asked to convert the complex number -1-i into mod-arg form? If they just start to plug-and-chug they'll quickly end up with the result $\theta=\pi/4$ and all of a sudden they've changed the complex number into it's negative, 1+i.

How does one avoid this dilemma?

 

[Tinyboss]

With the complex log function.

 

[Mentallic]

Thanks

 

[mathman]

When given a complex number $z=x+iy$ and transforming this into its mod-arg form giving $rcis\theta$ where $r=\sqrt{(x^2+y^2)}$ and $\theta=arctan(y/x)$, we are assuming that $-\pi/2<\theta<\pi/2$.

What if however a student is asked to convert the complex number -1-i into mod-arg form? If they just start to plug-and-chug they'll quickly end up with the result $\theta=\pi/4$ and all of a sudden they've changed the complex number into it's negative, 1+i.

How does one avoid this dilemma?

Be careful with the angle domain. You have a semicircle, while it should be a full circle. The signs of x and y have to be taken into account so that you end up in the correct quadrant.

 

[CellCoree]

I would respond by saying that the mod-arg form is not the only way to represent a complex number. While it is a useful and commonly used form, there are other forms that can also be used to represent complex numbers, such as the rectangular form (x+iy) or the polar form (re^{i\theta}).

In the case of converting -1-i into mod-arg form, it is important to consider the quadrant in which the complex number lies. In this case, the complex number lies in the third quadrant, where both x and y are negative. Therefore, the angle \theta should be in the range of -\pi<\theta<-\pi/2. This would result in the correct mod-arg form of r\cis\theta, where r=\sqrt{2} and \theta=-\pi/4.

To avoid the dilemma of changing the complex number into its negative, it is important to carefully consider the signs of both x and y when calculating the angle \theta. This can be done by drawing a diagram or using the quadrant system to determine the correct angle.

In conclusion, while the mod-arg form is a useful representation of complex numbers, it is important to consider the signs and quadrant of the complex number when calculating the angle \theta to avoid converting it into its negative form.

 

[pmacias]

As a scientist, it is important to approach complex numbers with a clear understanding of their properties and how they behave in mathematical operations. The mod-arg form, also known as polar form, is a useful representation of complex numbers that allows us to visualize their magnitude and direction.

In the given scenario, the student is asked to convert the complex number -1-i into mod-arg form. However, blindly plugging in the values of x and y into the formulas for r and \theta may lead to an incorrect result. This is because the arctan function has a range of -\pi/2 to \pi/2, as stated in the argument. But for the complex number -1-i, the value of \theta is actually -3\pi/4, which falls outside of this range.

To avoid this dilemma, it is important to understand the concept of principal values in trigonometry. The principal value of an angle is the value between -\pi and \pi, and it is the value that is used in most mathematical calculations. In the case of the complex number -1-i, the principal value of \theta is \pi/4, which is the same as the result obtained by blindly plugging in the values.

Therefore, to accurately convert a complex number into mod-arg form, it is important to first determine the principal value of \theta. This can be done by adding or subtracting multiples of 2\pi until the resulting angle falls within the range of -\pi/2 to \pi/2. In this case, adding 2\pi to -3\pi/4 gives us \pi/4, which is the correct principal value of \theta.

In conclusion, understanding the concept of principal values and applying it to the conversion of complex numbers into mod-arg form can help avoid the dilemma of changing the sign of the complex number. It is important to approach complex numbers with a thorough understanding of their properties and to be mindful of their behavior in mathematical operations.