Modulus–Argument Form with the Principal Argument

Hello! You have already worked with complex numbers in Cartesian form $a+bi$ and plotted them on Argand diagrams. In this first module, you will learn to use those diagrams to choose forms and methods efficiently, especially when an exam question does not tell you what to do next.

Today’s task is to describe a nonzero complex number by its distance from the origin and its direction. The calculation is usually short; the reasoning that matters is choosing the principal argument rather than an angle in the wrong quadrant.

Distance and direction on an Argand diagram

For $z=a+bi$, the point $(a,b)$ gives you two pieces of geometric information. Its modulus, $|z|$, is its distance from the origin. An argument is an angle measured from the positive real axis to the line from the origin to $z$, with anticlockwise angles positive.

Watch the geometric introduction before working with the formulas:

Pythagoras gives the modulus:

There is also an algebraic check using a fact you already know about conjugates:

A direction is less straightforward. Turning through a full revolution leaves you pointing at the same number, so if $\theta$ is an argument, then $\theta +2\pi k$ is also an argument for every integer $k$. To select just one, this lesson uses the principal argument $\mathrm{Arg}(z)$, defined by

Notice the unequal treatment of the endpoints: a point on the negative real axis has principal argument $\pi$, not $-\pi$. Both angles point in the right direction, but only $\pi$ belongs to the chosen interval.

For the precise definitions, read the short opening portion of the following resource:

The restriction $z\ne 0$ is essential. The origin has modulus $0$, but it has no direction from the origin, so $\mathrm{Arg}(0)$ is undefined.

Turning distance and direction into a number

Let $r=|z|>0$ and $\theta =\mathrm{Arg}(z)$. If the line to $z$ has length $r$, its horizontal and vertical components are

Substitute these into $z=a+bi$:

We abbreviate $\cos \theta +i\sin \theta$ as $\mathrm{cis}\theta$. Thus the modulus–argument form using the principal argument is

In this form, $r$ fixes the distance and $\theta$ fixes the direction. The following worked conversion connects that geometric idea to the signs of sine and cosine:

For that example, $r=2$. Since $a=1>0$ and $b=\sqrt{3}>0$, the point is in quadrant I. Its components give $\cos \theta =\frac{1}{2}$ and $\sin \theta =\frac{\sqrt{3}}{2}$, so $\theta =\frac{\pi }{3}$. Therefore

An exam-ready conversion should make three things visible: a positive modulus, an angle consistent with the quadrant, and a final expression in the requested form.

Choosing the principal argument reliably

When neither coordinate is zero, first find the acute reference angle $\alpha$ between the line to $z$ and the nearest part of the real axis:

Then use the quadrant to turn $\alpha$ into the principal argument. This is safer than applying $\arctan (b/a)$ and accepting its output: tangent has the same value in opposite quadrants.

The negative angle in quadrant III is worth pausing over. You could reach the point by turning anticlockwise through $\pi +\alpha$, but that exceeds $\pi$. Turning clockwise through $\pi -\alpha$ reaches the same point and lies in the principal interval; its signed value is $-\pi +\alpha$.

See that choice made in two worked diagrams:

For example, take $z=-\sqrt{3}+i$. Its modulus is $2$, and its reference angle is $\pi /6$. The point is in quadrant II, so

You can verify the result rather than rely solely on the diagram: $2\cos (5\pi /6)=-\sqrt{3}$ and $2\sin (5\pi /6)=1$.

Axis points need no reference-angle calculation. A positive real number has argument $0$; a negative real number has argument $\pi$. A positive or negative pure imaginary number has argument $\pi /2$ or $-\pi /2$, respectively. For instance,

When the conversion is hidden

An unfamiliar-looking question may require you to simplify a complex expression before finding its modulus and argument. Suppose you are asked to express $(1-2i{)}^2$ in modulus–argument form. Using familiar Cartesian arithmetic first,

Now the geometry is visible. The modulus is $5$, and the point lies in quadrant III. Its reference angle is $\alpha =\arctan (4/3)$, so its principal argument is $-\pi +\alpha$. Hence

The angle need not be a familiar fraction of $\pi$: an exact inverse-tangent expression is a valid exact answer. The crucial check is its quadrant. Here, both the cosine and sine of the stated angle are negative, matching the real and imaginary parts of $-3-4i$.

Key takeaways

For a nonzero $z=a+bi$, calculate $r=\sqrt{a^2+b^2}$, locate the point on an Argand diagram, and choose its unique argument in $(-\pi ,\pi ]$. Then write $z=r\mathrm{cis}\theta$. A quick check of the signs of $r\cos \theta$ and $r\sin \theta$ catches many quadrant errors.

In the next lesson, you will use this same distance-and-direction description to interpret what multiplication and division do to points on an Argand diagram.