A complex number expands the concept of a real number by including an imaginary component, typically expressed as \(a + bi\). Here, \(a\) represents the real part and \(bi\) represents the imaginary part. Complex numbers require two dimensions, unlike real numbers which exist on a simple number line. The magnitude measures the “size” or “length” of a complex number, providing a single, quantifiable value for a number existing in this two-dimensional space. This measurement is necessary because the simple rules for comparing real numbers do not apply.
Defining the Absolute Value
The term magnitude is mathematically synonymous with the absolute value when discussing complex numbers. Just as the absolute value of a real number represents its distance from zero on the number line, the magnitude of a complex number signifies its distance from the origin in the complex number system. The notation used is represented by vertical bars around the complex number \(z\), written as \(|z|\).
This definition establishes the magnitude as a measure of scale rather than direction. The resulting value is a single, real-valued distance that does not retain the imaginary component. For example, \(2 + 3i\) and \(-2 – 3i\) share the same magnitude because they are the same distance from the origin. This distance measurement is fundamental in many areas of physics and engineering, particularly those dealing with oscillations and wave mechanics.
Calculating the Magnitude
Finding the magnitude of a complex number requires a specific algebraic formula incorporating both the real and imaginary components. For a complex number \(z\) expressed as \(a + bi\), the magnitude \(|z|\) is calculated using the square root of the sum of the squares of \(a\) and \(b\). This relationship is summarized by the equation \(|z| = \sqrt{a^2 + b^2}\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary part. The imaginary unit \(i\) itself is not included in the calculation, only its coefficient \(b\).
To illustrate this calculation, consider the complex number \(z = 3 + 4i\). The real part \(a\) is 3, and the imaginary coefficient \(b\) is 4. Applying the formula involves first squaring these two parts (\(3^2=9\) and \(4^2=16\)), and then summing the squared values (\(9 + 16 = 25\)).
The final step in determining the magnitude is to calculate the square root of this sum. Therefore, the magnitude \(|3 + 4i|\) is \(\sqrt{25}\), which simplifies to 5. This process provides a single real number, 5, representing the length of the vector extending from the origin to the point \((3, 4)\) in the complex plane.
Visualizing the Complex Plane (Using Geometry)
The algebraic formula used to calculate the magnitude is directly derived from the Pythagorean theorem. Complex numbers are typically plotted on a two-dimensional coordinate system called the complex plane, often referred to as an Argand diagram. In this plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis.
Plotting a complex number \(z = a + bi\) involves locating the point \((a, b)\) by moving \(a\) units along the real axis and \(b\) units along the imaginary axis. The magnitude \(|z|\) is visualized as the straight-line distance from the origin \((0, 0)\) to this point \((a, b)\). This line segment forms the hypotenuse of a right-angled triangle.
The two legs of this triangle are defined by the coordinates \(a\) and \(b\). According to the Pythagorean theorem, the square of the hypotenuse (the magnitude) is equal to the sum of the squares of the other two sides: \(|z|^2 = a^2 + b^2\). Taking the square root of both sides yields the formula \(|z| = \sqrt{a^2 + b^2}\). This geometric interpretation provides the fundamental justification for the algebraic method.