What is the Argument of a Complex Number?

In mathematics, specifically in complex analysis, the argument of a complex number z (denoted as arg(z)) is the angle between the positive real axis and the line representing the number in the complex plane (Argand diagram). If you have a complex number in the form z = x + iy, where x is the real part and y is the imaginary part, the argument helps define the number's orientation in polar coordinates.

How to Calculate the Argument Step-by-Step

Calculating the argument is more than just finding the inverse tangent of y/x. Because the inverse tangent function only returns values between -π/2 and π/2, we must consider the signs of both x and y to determine which quadrant the number lies in:

• Quadrant I (x > 0, y > 0): θ = arctan(y/x)
• Quadrant II (x < 0, y ≥ 0): θ = arctan(y/x) + π (or 180°)
• Quadrant III (x < 0, y < 0): θ = arctan(y/x) - π (or -180°)
• Quadrant IV (x > 0, y < 0): θ = arctan(y/x)

Frequently Asked Questions

What is the Principal Argument?

The principal argument, usually denoted as Arg(z) (with a capital A), is the unique value of the argument that falls within the interval (-π, π] or (-180°, 180°]. Our calculator focuses on providing this principal value.

Why is the argument undefined for 0?

For the complex number 0 + 0i, the point lies exactly at the origin. Since there is no distance from the origin to create a vector, no specific angle can be defined, making the argument indeterminate or undefined.

How do radians relate to degrees?

The argument is commonly expressed in radians for calculus and advanced physics, where π radians equals 180 degrees. To convert from radians to degrees, multiply the result by 180/π.