Argument of Complex Number: Begin at (-3,2) - 135 Degrees

[Solidmozza]

Hi everybody!
Could somebody please assist me with an explanation as to why the following: arg (z+3-2i) = 135degrees : has its centre at -3,2 and that is the place where you begin the argument (ie go 135 degrees)
Please note, just beginning complex numbers. Sorry if can't understand question.
Thanks.

 

[HallsofIvy]

It would be nice to tell us exactly what you are trying to find! I.e. what do you go 135 degrees to?

The statement that "arg(z+ 3- 2i)= 135 degrees" means that the line from the number z+3- 2i to the origin (0) makes a 135 degree angle with the real axis. I imagine the problem asks you to determine possible values for z.
Write z+ 3- 2i as z- (-3+ 2i). The complex number -3+ 2i is represented on the Argand diagram by the point (-3, 2). We can think of z- (-3+2i) as a vector with its tail at (-3, 2) and its point at z. Now, starting from a horizontal line (parallel to the real axis), sweep 135 degrees counter clockwise (that gets you to the "upper left" quadrant, 45 degrees from either axis). z could be any point on that line. If this line were through (0,0) on an xy- plane, it would be the line y= -x: slope -1. Since it must go through (-3, 2), it has equation y= -(x+3)+ 2= -x- 1: z = x+(-x-1)i for x any real number.

 

[Architeuthis Dux]

Sure, I'd be happy to help with your question about complex numbers. The argument of a complex number refers to the angle that the complex number makes with the positive real axis on the complex plane. In this case, the complex number is z+3-2i, which can be written as (x+3)+i(y-2), where x and y are the real and imaginary parts, respectively.

Starting at the point (-3,2) on the complex plane, we can see that the real part of z+3-2i is -3, and the imaginary part is 2. This means that the complex number is located in the third quadrant, as both the real and imaginary parts are negative.

Now, the argument is the angle that the complex number makes with the positive real axis, which is the horizontal axis on the complex plane. Since the complex number is in the third quadrant, the angle will be measured in the clockwise direction from the positive real axis.

In order to determine the argument, we can use the inverse tangent function, also known as arctan, to find the angle. In this case, we have arctan(2/-3) = -33.69 degrees. However, since the angle is measured clockwise from the positive real axis, we need to subtract this value from 180 degrees to get the argument of 135 degrees.

Therefore, the argument of z+3-2i, starting at (-3,2), is 135 degrees. This means that the complex number makes an angle of 135 degrees with the positive real axis, and its centre is located at (-3,2) on the complex plane. I hope this helps clarify your understanding of complex numbers and their arguments. Keep up the good work in learning about this fascinating topic!