Angle and Absolute Value of a Complex Number

Every complex number \(z = a + bi\) can be plotted as a single point on the complex plane, where the horizontal axis measures the real part \(a\) and the vertical axis measures the imaginary part \(b\). Once that point is plotted, two natural questions come up: how far is it from the origin, and in what direction does it point?

The answer to the first question is the absolute value, also called the modulus, of the complex number. The answer to the second is the argument, which is simply the angle the segment from the origin to the point makes with the positive real axis. Together, the modulus and argument describe exactly where a complex number sits, which is why they are the foundation for writing complex numbers in polar form.

Since \(a + bi\) corresponds to the point \((a, b)\), the distance from the origin \((0, 0)\) to that point is found the same way you would find the distance between any two points in the plane, using the distance formula. This gives the modulus formula:

\( |z| = |a + bi| = \sqrt{a^2 + b^2} \)

Example 1: Absolute Value of 3 + 4i

For \(z = 3 + 4i\), the real part is \(a = 3\) and the imaginary part is \(b = 4\). Substituting into the formula:

\( |3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)

So the point \(3 + 4i\) is exactly 5 units away from the origin.

Example 2: Absolute Value of 3i

A purely imaginary number like \(3i\) can be written as \(0 + 3i\), so \(a = 0\) and \(b = 3\):

\( |3i| = \sqrt{0^2 + 3^2} = \sqrt{9} = 3 \)

The modulus of a purely imaginary number is just the absolute value of its coefficient, since the point lies straight up or down on the imaginary axis.

A few properties of the modulus are useful to remember when working with complex numbers:

• \( |z| \ge 0 \) for every complex number, and \( |z| = 0 \) only when \(z = 0\).
• \( |z_1 z_2| = |z_1||z_2| \), the modulus of a product equals the product of the moduli.
• \( \left|\dfrac{z_1}{z_2}\right| = \dfrac{|z_1|}{|z_2|} \), the modulus of a quotient equals the quotient of the moduli.
• \( |\bar{z}| = |z| \), a complex number and its conjugate have the same modulus.

These properties become especially handy once you start multiplying and dividing complex numbers in polar form.

The argument of \(z = a + bi\) is the angle \(\theta\) measured from the positive real axis to the segment connecting the origin to the point \((a, b)\). Because \(a\), \(b\), and the modulus form a right triangle, the argument can be found using the tangent ratio:

\( \tan(\theta) = \dfrac{b}{a} \quad \Rightarrow \quad \theta = \tan^{-1}\left(\dfrac{b}{a}\right) \)

This formula alone only gives an angle between \(-90^\circ\) and \(90^\circ\), so you always need to check which quadrant the point \((a, b)\) lies in and adjust the angle accordingly. The principal argument is the value of \(\theta\) chosen so that \(-180^\circ < \theta \le 180^\circ\) (equivalently \(-\pi < \theta \le \pi\)).

Example 3: Argument of -1 + i

Here \(a = -1\) and \(b = 1\), which places the point in the second quadrant. The reference angle satisfies \(\tan(\theta) = \left|\dfrac{1}{-1}\right| = 1\), and this ratio matches the familiar 45-45-90 special right triangle, giving a reference angle of \(45^\circ\). Since the point is in the second quadrant, the principal argument is:

\( \theta = 180^\circ - 45^\circ = 135^\circ \)

The modulus of this number is \( |{-1} + i| = \sqrt{(-1)^2 + 1^2} = \sqrt{2} \), so \(-1 + i\) sits at distance \(\sqrt{2}\) from the origin, at an angle of \(135^\circ\).

Once you can find both the modulus and the argument of a complex number, you have everything needed to describe its exact location on the complex plane without ever plotting it by hand, and to rewrite it in polar form. Practicing with a variety of complex numbers, including purely real, purely imaginary, and numbers in each of the four quadrants, will help you get comfortable choosing the correct quadrant adjustment every time.

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