The Modulus Of The Complex Number

In mathematics, complex numbers combine real and imaginary components, allowing us to represent quantities that cannot be expressed using real numbers alone. One of the most important concepts when working with complex numbers is the modulus. The modulus of a complex number gives a measure of its size or magnitude, which is crucial in fields ranging from engineering to quantum physics. Understanding how to find and interpret the modulus can make dealing with complex numbers much more intuitive and practical.

Definition of the Modulus

The modulus of a complex number refers to its distance from the origin in the complex plane. If a complex number is written asz = a + bi, whereais the real part andbis the imaginary part, the modulus is given by

|z| = √(a² + b²)

This formula comes directly from the Pythagorean theorem, as the real and imaginary parts can be seen as forming the legs of a right-angled triangle, with the modulus being the hypotenuse.

Geometrical Interpretation

When plotted on the complex plane, each complex number corresponds to a unique point. The horizontal axis represents the real part, and the vertical axis represents the imaginary part. The modulus is simply the straight-line distance from the origin (0, 0) to the point representing the complex number. This distance is always non-negative and is zero only if the complex number itself is zero.

Examples of Finding the Modulus

• Forz = 3 + 4i, the modulus is|z| = √(3² + 4²) = √(9 + 16) = √25 = 5.
• Forz = -5 + 12i, the modulus is|z| = √((-5)² + (12)²) = √(25 + 144) = √169 = 13.
• Forz = 7(purely real), the modulus is simply|z| = √(7² + 0²) = 7.
• Forz = -9i(purely imaginary), the modulus is|z| = √(0² + (-9)²) = 9.

Properties of the Modulus

The modulus of complex numbers follows several useful properties that make calculations and proofs easier

• |z| ≥ 0for all complex numbers.
• |z| = 0if and only ifz = 0.
• |z₁·z₂| = |z₁|·|z₂|(multiplicative property).
• |z₁ / z₂| = |z₁| / |z₂|forz₂ ≠ 0.
• |z₁ + z₂| ≤ |z₁| + |z₂|(triangle inequality).
• |z| = |conjugate(z)|.

The Triangle Inequality

The triangle inequality states that the modulus of the sum of two complex numbers is less than or equal to the sum of their moduli. Geometrically, this reflects the idea that the shortest distance between two points is a straight line. This property is essential in analysis and vector spaces involving complex numbers.

Modulus and Polar Form

One of the most useful applications of the modulus is in representing a complex number in polar form. Instead of writingz = a + bi, we can write

z = r(cos θ + i sin θ)

Here,ris the modulus, andθis the argument (angle) of the complex number. This form is especially helpful in multiplication and division because the moduli multiply or divide directly, and the arguments add or subtract.

Example in Polar Form

Ifz = 1 + i, then

• r = |z| = √(1² + 1²) = √2
• θ = arctan(1 / 1) = π/4 radians

Thus,zcan be written as√2(cos π/4 + i sin π/4).

Applications of the Modulus

The modulus of a complex number has a wide range of applications across mathematics and applied sciences

• Electrical EngineeringUsed to calculate magnitudes of impedance and phasors.
• Quantum MechanicsProbability amplitudes are represented by complex numbers, and their modulus squared gives probabilities.
• Signal ProcessingMagnitudes of frequency components are determined using the modulus of complex Fourier coefficients.
• GeometryUsed in measuring distances in the complex plane.

Modulus in Inequalities

In complex analysis, inequalities involving the modulus often simplify problem-solving. For example, knowing that|z| < 1tells us that the point representingzlies inside the unit circle in the complex plane. Similarly,|z| = Rrepresents a circle of radiusRcentered at the origin.

Example of an Inequality

If|z – 2| ≤ 3, it means all points in the complex plane whose distance from the point (2, 0) is at most 3. This represents a closed disk of radius 3 centered at (2, 0).

Relationship Between Modulus and Conjugate

The conjugate of a complex numberz = a + biisa – bi. The product of a complex number with its conjugate is always equal to the square of its modulus

z à conjugate(z) = a² + b² = |z|²

This property is extremely useful when dividing complex numbers because it allows us to remove imaginary parts from the denominator.

Common Mistakes When Working With Modulus

• Forgetting that the modulus is always non-negative.
• Confusing the modulus with the argument.
• Not squaring both the real and imaginary parts before adding under the square root.
• Misapplying properties of modulus to non-multiplicative operations.

The modulus of a complex number is more than just a formula it is a fundamental concept connecting algebra, geometry, and real-world applications. By understanding its geometric meaning, algebraic properties, and role in polar representation, one can unlock powerful techniques for solving problems across mathematics and science. Whether in theoretical work or applied fields like engineering, the modulus serves as a vital tool for interpreting and manipulating complex numbers with confidence.