Solving |2z| = 18 A Comprehensive Guide To Complex Modulus Equations

Solve for z in the equation |2z| = 18.

In mathematics, particularly in complex number theory, understanding the modulus of a complex number is crucial. The modulus, often denoted as |z| for a complex number z, represents the distance of the complex number from the origin in the complex plane. This article delves into solving the equation |2z| = 18, providing a comprehensive explanation and step-by-step solution. This exploration will enhance your understanding of complex numbers and their geometric interpretation. We will break down the problem, discuss the properties of the modulus, and arrive at a general solution, ensuring you grasp the underlying concepts thoroughly.

Understanding Complex Numbers and the Modulus

To effectively solve the given equation, a solid understanding of complex numbers and their modulus is essential. A complex number, typically represented as z = a + bi, consists of two parts: a real part (a) and an imaginary part (b), where 'i' is the imaginary unit defined as the square root of -1. The complex plane is a two-dimensional plane where the horizontal axis represents the real part and the vertical axis represents the imaginary part. A complex number can be visualized as a point or a vector in this plane.

The modulus of a complex number z = a + bi, denoted as |z|, is calculated using the Pythagorean theorem. It is the distance from the origin (0, 0) to the point (a, b) in the complex plane. Mathematically, the modulus is given by |z| = √(a² + b²). This value is always a non-negative real number. The modulus represents the magnitude or absolute value of the complex number. Understanding this geometric interpretation is key to solving equations involving complex numbers.

Properties of the Modulus

Before diving into the solution, it's important to understand some key properties of the modulus. These properties will help simplify the equation and provide insights into the solution. One crucial property is the modulus of a product. For any two complex numbers z₁ and z₂, the modulus of their product is the product of their moduli, i.e., |z₁z₂| = |z₁||z₂|. This property is particularly useful when dealing with equations involving multiplication of complex numbers. Another relevant property is that for any complex number z and a scalar (real number) k, |kz| = |k||z|. This property directly applies to our problem where we have |2z|.

Property Application in our Problem

Applying the property |kz| = |k||z| to our equation |2z| = 18, we can rewrite it as |2||z| = 18. Since 2 is a real number, its modulus is simply its absolute value, which is 2. Thus, the equation becomes 2|z| = 18. This simplified form makes it easier to isolate |z| and find its value. Understanding and applying these properties is fundamental in complex number manipulations and problem-solving.

Solving the Equation |2z| = 18

Now, let's proceed to solve the equation |2z| = 18. As discussed earlier, we can use the property |kz| = |k||z| to simplify the equation. Applying this property, we get |2||z| = 18. Since the modulus of 2 is simply 2, the equation becomes 2|z| = 18. To find |z|, we divide both sides of the equation by 2, resulting in |z| = 9. This tells us that the distance of the complex number z from the origin in the complex plane is 9.

Expressing z in Complex Form

To find the specific values of z that satisfy this condition, we express z in its general complex form, z = a + bi, where a and b are real numbers. Then, the modulus of z is given by |z| = √(a² + b²). Since we know |z| = 9, we can write √(a² + b²) = 9. Squaring both sides of the equation, we get a² + b² = 81. This equation represents a circle in the complex plane centered at the origin with a radius of 9. Any complex number whose real and imaginary parts satisfy this equation is a solution to the original equation.

General Solution and Geometric Interpretation

The equation a² + b² = 81 represents an infinite set of solutions. Geometrically, these solutions correspond to all the points on the circle centered at the origin with a radius of 9 in the complex plane. We can express the general solution using polar coordinates. Let a = r cos(θ) and b = r sin(θ), where r is the radius and θ is the angle with the positive real axis. In our case, r = |z| = 9. Thus, a = 9 cos(θ) and b = 9 sin(θ). The complex number z can then be written as z = 9 cos(θ) + 9i sin(θ), or z = 9(cos(θ) + i sin(θ)).

Euler's Formula and Representation

Using Euler's formula, which states that e^(iθ) = cos(θ) + i sin(θ), we can further simplify the general solution to z = 9e^(iθ), where θ can be any real number. This representation provides a concise and elegant way to express all the solutions to the equation |2z| = 18. The angle θ determines the specific location of the complex number on the circle. As θ varies from 0 to 2π, the complex number z traces the entire circle. This geometric interpretation of the solutions is crucial for a complete understanding of the problem.

Examples of Solutions

To further illustrate the solutions, let's consider a few specific values of θ. When θ = 0, z = 9e^(i*0) = 9(cos(0) + i sin(0)) = 9. This corresponds to the point (9, 0) on the complex plane. When θ = π/2, z = 9e^(iπ/2) = 9(cos(π/2) + i sin(π/2)) = 9i. This corresponds to the point (0, 9) on the complex plane. When θ = π, z = 9e^(iπ) = 9(cos(π) + i sin(π)) = -9. This corresponds to the point (-9, 0) on the complex plane. When θ = 3π/2, z = 9e^(i3π/2) = 9(cos(3π/2) + i sin(3π/2)) = -9i. This corresponds to the point (0, -9) on the complex plane.

Infinite Solutions

These examples highlight that there are infinitely many solutions to the equation |2z| = 18, all lying on the circle with radius 9 centered at the origin. Each value of θ gives a unique complex number z that satisfies the equation. This understanding is vital for solving similar problems and appreciating the nature of solutions in complex number theory. The use of Euler's formula provides a powerful tool for expressing these solutions in a compact and meaningful way.

Conclusion

In conclusion, solving the equation |2z| = 18 involves understanding the properties of the modulus of complex numbers and their geometric representation in the complex plane. By applying the property |kz| = |k||z|, we simplified the equation to 2|z| = 18, leading to |z| = 9. This result indicates that the solutions lie on a circle with a radius of 9 centered at the origin. Expressing the complex number z in the form z = a + bi and using the definition of the modulus, we derived the equation a² + b² = 81, confirming our geometric interpretation.

General Solution and Implications

The general solution can be expressed as z = 9e^(iθ) using Euler's formula, where θ is any real number. This representation encapsulates all the possible solutions, demonstrating that there are infinitely many complex numbers that satisfy the given equation. Each value of θ corresponds to a specific point on the circle. The detailed step-by-step solution and the geometric interpretation provide a comprehensive understanding of the problem. This knowledge is essential for tackling more complex problems in complex number theory and related fields. By grasping these fundamental concepts, one can confidently approach and solve a wide range of equations involving the modulus of complex numbers, enhancing their problem-solving skills and deepening their mathematical insight.