Complex Number With Absolute Value Of 5
When delving into the realm of complex numbers, a fundamental concept to grasp is the absolute value, also known as the modulus. The absolute value of a complex number represents its distance from the origin (0, 0) in the complex plane. This article will guide you through the process of determining the absolute value of complex numbers and, more importantly, will help you solve the question: Which complex number has an absolute value of 5? We will meticulously explore the options provided and utilize the formula for calculating the absolute value to arrive at the correct answer. This exploration will not only solve the specific problem but also deepen your understanding of complex numbers and their properties.
Complex Numbers: A Quick Recap
Before we dive into the problem, let's briefly recap what complex numbers are. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1 (i² = -1). The a part is called the real part, and the b part is called the imaginary part. Complex numbers extend the real number system by incorporating this imaginary unit, allowing us to solve equations that have no solutions within the real number system alone.
Visualizing Complex Numbers
Complex numbers can be visualized on a complex plane, which is similar to the Cartesian coordinate plane. The horizontal axis represents the real part, and the vertical axis represents the imaginary part. Thus, a complex number a + bi can be plotted as a point (a, b) on this plane. This visual representation is crucial for understanding the geometric interpretation of operations on complex numbers, including the absolute value.
Absolute Value: The Distance from the Origin
The absolute value (or modulus) of a complex number a + bi is the distance from the point (a, b) to the origin (0, 0) in the complex plane. We calculate it using the Pythagorean theorem. The formula for the absolute value, denoted as |a + bi|, is:
|a + bi| = √(a² + b²)
This formula is derived directly from the distance formula, where a and b represent the lengths of the legs of a right triangle, and the absolute value is the length of the hypotenuse. Understanding this geometric interpretation is key to intuitively grasping the concept of the absolute value of a complex number.
Solving the Problem: Which Complex Number Has an Absolute Value of 5?
Now that we have refreshed our understanding of complex numbers and absolute value, let's tackle the problem. We are given four complex numbers and need to determine which one has an absolute value of 5. We will apply the formula |a + bi| = √(a² + b²) to each option and see which one yields 5.
The given options are:
A. -3 + 4i B. 2 + 3i C. 7 - 2i D. 9 + 4i
Let's evaluate each option step-by-step.
Option A: -3 + 4i
For the complex number -3 + 4i, a = -3 and b = 4. Plugging these values into the formula, we get:
|-3 + 4i| = √((-3)² + (4)²)
|-3 + 4i| = √(9 + 16)
|-3 + 4i| = √25
|-3 + 4i| = 5
So, the absolute value of -3 + 4i is 5. This option satisfies our condition.
Option B: 2 + 3i
For the complex number 2 + 3i, a = 2 and b = 3. Applying the formula:
|2 + 3i| = √((2)² + (3)²)
|2 + 3i| = √(4 + 9)
|2 + 3i| = √13
Since √13 is not equal to 5, this option is incorrect.
Option C: 7 - 2i
For the complex number 7 - 2i, a = 7 and b = -2. Calculating the absolute value:
|7 - 2i| = √((7)² + (-2)²)
|7 - 2i| = √(49 + 4)
|7 - 2i| = √53
As √53 is not equal to 5, this option is also incorrect.
Option D: 9 + 4i
Finally, for the complex number 9 + 4i, a = 9 and b = 4. Let's find its absolute value:
|9 + 4i| = √((9)² + (4)²)
|9 + 4i| = √(81 + 16)
|9 + 4i| = √97
Since √97 is not equal to 5, this option is incorrect as well.
Conclusion: The Complex Number with an Absolute Value of 5
After meticulously calculating the absolute value of each given complex number, we found that the complex number -3 + 4i has an absolute value of 5. Therefore, the correct answer is A. -3 + 4i. This detailed step-by-step solution not only provides the answer but also reinforces the understanding of how to calculate the absolute value of complex numbers.
Understanding complex numbers and their absolute values is crucial in various fields, including mathematics, physics, and engineering. The ability to manipulate and interpret complex numbers opens doors to solving a wide range of problems. This exercise has demonstrated a practical application of the absolute value concept, which is a building block for more advanced topics in complex analysis.
Further Exploration of Complex Numbers
To further enhance your understanding of complex numbers, consider exploring the following topics:
- Operations on Complex Numbers: Learn how to add, subtract, multiply, and divide complex numbers.
- Complex Conjugates: Understand the concept of complex conjugates and their properties.
- Polar Form of Complex Numbers: Explore the polar representation of complex numbers and its advantages.
- Euler's Formula: Discover the connection between complex exponentials and trigonometric functions.
By delving deeper into these areas, you will gain a more comprehensive grasp of complex numbers and their applications. The journey into the world of complex numbers is fascinating and rewarding, offering new perspectives and tools for problem-solving.
The Significance of Absolute Value in Complex Numbers
The absolute value of a complex number is more than just a mathematical calculation; it carries significant geometric and algebraic implications. Understanding its significance can deepen your appreciation for complex numbers and their applications.
Geometric Interpretation
As we discussed earlier, the absolute value represents the distance from the origin in the complex plane. This geometric interpretation allows us to visualize complex numbers as vectors, with the absolute value being the magnitude of the vector. This visualization is particularly useful in understanding operations such as addition and subtraction of complex numbers, which can be seen as vector addition and subtraction.
Applications in Physics and Engineering
Complex numbers and their absolute values are widely used in physics and engineering, particularly in areas such as electrical engineering and quantum mechanics. For example, in electrical engineering, complex numbers are used to represent alternating current (AC) circuits, where the absolute value of the complex impedance represents the magnitude of the impedance. In quantum mechanics, complex numbers are fundamental to the mathematical formulation of quantum states, and the absolute value squared of the wave function gives the probability density of finding a particle in a particular state.
Algebraic Properties
The absolute value has several important algebraic properties that make it a powerful tool in mathematical analysis. Some key properties include:
- |z| ≥ 0 for all complex numbers z, and |z| = 0 if and only if z = 0.
- |z₁ * z*₂| = |z₁| * |z₂| for any complex numbers z₁ and z₂.
- |z₁ / z₂| = |z₁| / |z₂| for any complex numbers z₁ and z₂ (where z₂ ≠ 0).
- |z₁ + z₂| ≤ |z₁| + |z₂| (the triangle inequality).
These properties are crucial for proving theorems and solving problems involving complex numbers. The triangle inequality, in particular, has wide-ranging applications in various areas of mathematics.
Conclusion: Mastering the Absolute Value of Complex Numbers
In conclusion, understanding the absolute value of complex numbers is essential for anyone studying mathematics, physics, or engineering. It provides a geometric interpretation of complex numbers, has practical applications in various fields, and possesses important algebraic properties. By mastering this concept, you will be well-equipped to tackle more advanced topics in complex analysis and related areas. Remember, the absolute value is not just a formula; it's a fundamental concept that unlocks a deeper understanding of the world of complex numbers.
