Argument of a Complex Number¶

Definition¶

The argument of a nonzero complex number $z$, denoted $\arg z$, is the angle between the positive real axis and the line joining the origin and $z$. It is a multivalued function due to the periodicity of the angle in the polar coordinate system. The principal argument, denoted as $\operatorname{Arg} z$, is the unique value of the argument that lies in the interval $(-\pi, \pi].$

Interpretation¶

The argument records the direction of a nonzero complex number $z$, as the modulus records its length; together they are the polar coordinates of $z$. Because a full turn brings a point back to itself, if $\varphi$ is one argument of $z$, then $\varphi+2\pi k$ is another for every integer $k$.

Both points lie on the same circle around the origin, so only their directions differ. For $z_1$ the angle $\varphi_1$ is measured anticlockwise from the positive real axis. For $z_2$ in the lower half-plane the principal argument is measured clockwise and is negative, $\varphi_2=-130^\circ$$-130^{\circ }$. The anticlockwise angle $\varphi_2+2\pi$$2\pi$, that is $230^\circ$, is also an argument of $z_2$, but not the principal one, since it lies outside $(-\pi,\pi]$.

The principal argument picks one value, so that $\operatorname{Arg}$ is a genuine function on the nonzero complex numbers. The interval $(-\pi,\pi]$ is half-open so that every direction gets exactly one value: the negative real axis gets $\pi$, not $-\pi$. The price is a jump there: points just above the negative real axis have $\operatorname{Arg}$ close to $\pi$, points just below it close to $-\pi$.

A common slip is to compute $\operatorname{Arg}(a+\mathrm{i}b)$ as $\arctan(b/a)$. This is correct only for $a>0$$0$: for $-1-\mathrm{i}$ it gives $\pi/4$, the direction of $1+\mathrm{i}$$\mathrm{i}$, while $\operatorname{Arg}(-1-\mathrm{i})=-3\pi/4$$-3\pi /4$. The 2-argument arctangent handles all quadrants. The many values of $\arg z$ are also why the complex logarithm has many values.

Key Results¶

• Each nonzero complex number has a unique principal argument. The argument of $0$ is undefined.
• For $(a,b)\ne(0,0)$$(0,0)$, the principal argument of $a+\mathrm{i} b$ is $\operatorname{atan2}(b,a)$, where $\operatorname{atan2}$ is the 2-argument arctangent.

Examples¶

• $\operatorname{Arg} \mathrm{i}=\pi /2$$\pi /2$.
• $\operatorname{Arg}(1-\mathrm{i})=-\pi/4$$-\pi /4$.

Prerequisites¶

• Polar Representation of Complex Numbers
• Complex Number

Related Concepts¶

• 2-Argument Arctangent

Sources¶

• Math II: Linear Algebra · Week 1

Symbols: phi, pi

Last updated: 20 October 2026 03:02