For the complex number 3 + 4i, what is the absolute value and the argument? Is it not tan (4/3)?
A complex number z = x + yi, can be expressed as a line with a length equal to |z| or sqrt (x^2+ y^2). The line makes an angle A with the positive x-axis that is known as its argument and equal to A= arc tan (y/x). The argument is kept a positive angle by the use of either adding or subtracting pi radians from arc tan (y/x).
For the complex number 3 + 4i, the absolute value is sqrt (3^2 + 4^2) = sqrt (9 + 16) = sqrt 25 = 5.
The argument is arc tan (4/3), which is equal to 0.9273 radians, and the absolute value is 5.
check Approved by eNotes Editorial
We write a complex number in the rectangular form x+y*i, where x and y are real numbers and i = (-1)^(1/2).
A complex number z = x+yi has the absolute value |z| = (x^2+y^2)^(1/2). And the argument of x+iy is the arg(x+yi) = arc tan (y/x)
So the for given complex number has the absolute value = (3^2+4^2)^1/2) = 5 and argument = arc tan (4/3).
So absolute value of (3+4i) is 5 and arg(3+4i) is arc tan (4/3).