What Is The Modulus Of 6 + 7i

What is the Modulus of 6 + 7i? A Deep Dive into Complex Numbers

The seemingly simple question, "What is the modulus of 6 + 7i?", opens a door to a fascinating world of complex numbers. This article will not only answer that question directly but will also explore the concept of modulus in detail, its applications, and related concepts within the broader field of complex analysis. We'll delve into the geometrical interpretation, explore practical examples, and provide a solid foundation for understanding this crucial aspect of complex numbers.

Understanding Complex Numbers

Before diving into the modulus, let's establish a firm understanding of complex numbers themselves. A complex number is a number that can be expressed in the form a + bi, where:

• a is the real part (a real number)
• b is the imaginary part (a real number)
• i is the imaginary unit, defined as the square root of -1 (√-1)

The set of all complex numbers is denoted by ℂ. Examples include 3 + 2i, -1 - i, 5 (which is 5 + 0i), and 4i (which is 0 + 4i). These numbers extend the real number system, allowing us to solve equations that have no solutions within the real numbers, such as x² + 1 = 0.

The Modulus: Magnitude in the Complex Plane

The modulus of a complex number, often denoted as |z|, represents the magnitude or absolute value of that number. Geometrically, it corresponds to the distance between the origin (0, 0) and the point representing the complex number in the complex plane. The complex plane is a two-dimensional plane where the x-axis represents the real part and the y-axis represents the imaginary part of a complex number.

For a complex number z = a + bi, the modulus is calculated using the Pythagorean theorem:

|z| = √(a² + b²)

This formula stems directly from the distance formula in the Cartesian coordinate system. The real part (a) and the imaginary part (b) act as the coordinates (a, b) in the complex plane.

Calculating the Modulus of 6 + 7i

Now, let's apply this knowledge to answer our initial question: What is the modulus of 6 + 7i?

In this case, a = 6 and b = 7. Substituting these values into the modulus formula:

|6 + 7i| = √(6² + 7²) = √(36 + 49) = √85

Therefore, the modulus of 6 + 7i is √85. This is an irrational number, approximately equal to 9.22.

Geometrical Interpretation and Visualization

Visualizing complex numbers and their moduli in the complex plane enhances understanding. Consider the complex number 6 + 7i. In the complex plane, this corresponds to the point (6, 7). The modulus, √85, represents the distance from the origin (0, 0) to the point (6, 7). You can draw a right-angled triangle with vertices at (0,0), (6,0), and (6,7). The hypotenuse of this triangle represents the modulus.

This geometrical representation provides a powerful intuitive grasp of the modulus's meaning: it measures the distance of the complex number from the origin.

Properties and Applications of the Modulus

The modulus possesses several important properties that make it a valuable tool in complex analysis:

• Non-negativity: The modulus of a complex number is always non-negative (|z| ≥ 0). It's zero only if the complex number itself is zero.
• Multiplicative Property: The modulus of a product of two complex numbers is the product of their moduli: |z₁z₂| = |z₁||z₂|
• Division Property: The modulus of the quotient of two complex numbers is the quotient of their moduli: |z₁/z₂| = |z₁|/|z₂| (provided z₂ ≠ 0)
• Triangle Inequality: For any two complex numbers z₁ and z₂, the following inequality holds: |z₁ + z₂| ≤ |z₁| + |z₂|. This is a fundamental property used in many proofs and applications.

These properties are crucial in various applications, including:

• Signal Processing: In signal processing, the modulus represents the amplitude of a complex signal.
• Quantum Mechanics: The modulus squared of a complex wave function represents the probability density of finding a particle in a particular state.
• Fractals: The Mandelbrot set, a famous fractal, is defined using the modulus of complex numbers iteratively.
• Electrical Engineering: The modulus is utilized extensively in analyzing AC circuits, representing the magnitude of impedance or voltage.

Polar Form and the Modulus

The modulus is intrinsically linked to the polar form of a complex number. The polar form expresses a complex number using its modulus (r) and its argument (θ), which is the angle it makes with the positive real axis in the complex plane.

The polar form is given by: z = r(cos θ + i sin θ), where r = |z|

This form simplifies many calculations, particularly those involving multiplication and division of complex numbers.

Advanced Concepts Related to Modulus

For a deeper dive, consider these advanced concepts related to the modulus:

• Complex Conjugate: The complex conjugate of a complex number z = a + bi is denoted as z̄ and is given by z̄ = a - bi. The product of a complex number and its conjugate is always a real number equal to the square of the modulus: z * z̄ = |z|².
• Distance between two Complex Numbers: The distance between two complex numbers z₁ and z₂ is given by |z₁ - z₂|. This is a direct application of the modulus, extending its geometric interpretation.
• Regions in the Complex Plane: The modulus is used to define regions in the complex plane, such as circles (|z - z₀| = r) and disks (|z - z₀| ≤ r), centered at z₀ with radius r.

Conclusion

The modulus of a complex number is a fundamental concept with profound implications across various fields of mathematics, science, and engineering. Understanding its calculation, geometrical interpretation, and properties is crucial for anyone working with complex numbers. By grasping these concepts, you gain a more comprehensive understanding of this vital aspect of complex analysis and its widespread applications. The answer to "What is the modulus of 6 + 7i?" – √85 – is only the starting point for a much deeper and more rewarding exploration of the world of complex numbers.