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.
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).
The absolute value of any number, including complex numbers, is its distance from zero.
You may be confusing the unit circle, which is the general idea for the absolute value of a complex number, with the trigonometry of the unit circle.
But the absolute value uses the Pythagorean theorem: the real number is plotted on the x-axis, and the imaginary number on the y-axis. So to get the distance from the plotted complex number to zero would be the square root of the units on the x-axis squared plus the units on the y-axis squared:
sqrt (9+16) = sqrt 25 = 5.
You can think of the tangent as the slope of the hypotenuse of the right triangle, or sine/cosine. But it's not absolute value...