Which Complex Number Has An Absolute Value Of 5?A. { -3 + 4i$}$ B. ${ 2 + 3i\$} C. ${ 7 - 2i\$} D. ${ 9 + 4i\$}
Introduction
In mathematics, complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation i^2 = -1. The absolute value of a complex number is a measure of its distance from the origin in the complex plane. In this article, we will explore which complex number has an absolute value of 5.
Understanding Absolute Value
The absolute value of a complex number a + bi is defined as the square root of the sum of the squares of a and b. Mathematically, it can be represented as:
|a + bi| = √(a^2 + b^2)
Calculating Absolute Value
To determine which complex number has an absolute value of 5, we need to calculate the absolute value of each option and compare it to 5.
Option A: -3 + 4i
The absolute value of -3 + 4i is:
| -3 + 4i | = √((-3)^2 + 4^2) | -3 + 4i | = √(9 + 16) | -3 + 4i | = √25 | -3 + 4i | = 5
Option B: 2 + 3i
The absolute value of 2 + 3i is:
| 2 + 3i | = √(2^2 + 3^2) | 2 + 3i | = √(4 + 9) | 2 + 3i | = √13 | 2 + 3i | ≈ 3.61
Option C: 7 - 2i
The absolute value of 7 - 2i is:
| 7 - 2i | = √(7^2 + (-2)^2) | 7 - 2i | = √(49 + 4) | 7 - 2i | = √53 | 7 - 2i | ≈ 7.28
Option D: 9 + 4i
The absolute value of 9 + 4i is:
| 9 + 4i | = √(9^2 + 4^2) | 9 + 4i | = √(81 + 16) | 9 + 4i | = √97 | 9 + 4i | ≈ 9.85
Conclusion
Based on the calculations above, only option A, -3 + 4i, has an absolute value of 5.
Final Answer
Introduction
Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including engineering, physics, and computer science. In this article, we will provide a comprehensive Q&A guide to complex numbers, covering their definition, properties, and operations.
Q: What is a complex number?
A: 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, which satisfies the equation i^2 = -1.
Q: What is the imaginary unit i?
A: The imaginary unit i is a mathematical concept that is used to extend the real number system to the complex number system. It is defined as the square root of -1, denoted by i = √(-1).
Q: How do you add complex numbers?
A: To add complex numbers, you simply add the real parts and the imaginary parts separately. For example, if you have two complex numbers a + bi and c + di, their sum is (a + c) + (b + d)i.
Q: How do you subtract complex numbers?
A: To subtract complex numbers, you simply subtract the real parts and the imaginary parts separately. For example, if you have two complex numbers a + bi and c + di, their difference is (a - c) + (b - d)i.
Q: How do you multiply complex numbers?
A: To multiply complex numbers, you use the distributive property and the fact that i^2 = -1. For example, if you have two complex numbers a + bi and c + di, their product is (ac - bd) + (ad + bc)i.
Q: How do you divide complex numbers?
A: To divide complex numbers, you multiply the numerator and the denominator by the conjugate of the denominator. For example, if you have two complex numbers a + bi and c + di, their quotient is ((ac + bd) + (bc - ad)i) / (c^2 + d^2).
Q: What is the absolute value of a complex number?
A: The absolute value of a complex number a + bi is defined as the square root of the sum of the squares of a and b, denoted by |a + bi| = √(a^2 + b^2).
Q: How do you find the absolute value of a complex number?
A: To find the absolute value of a complex number, you simply calculate the square root of the sum of the squares of the real and imaginary parts. For example, if you have a complex number a + bi, its absolute value is |a + bi| = √(a^2 + b^2).
Q: What is the conjugate of a complex number?
A: The conjugate of a complex number a + bi is defined as a - bi.
Q: How do you find the conjugate of a complex number?
A: To find the conjugate of a complex number, you simply change the sign of the imaginary part. For example, if you have a complex number a + bi, its conjugate is a - bi.
Q: What is the modulus of a complex number?
A: The modulus of a complex number a + bi is defined as the absolute value of the complex number, denoted by |a + bi| = √(a^2 + b^2).
Q: How do you find the modulus of a complex number?
A: To find the modulus of a complex number, you simply calculate the square root of the sum of the squares of the real and imaginary parts. For example, if you have a complex number a + bi, its modulus is |a + bi| = √(a^2 + b^2).
Conclusion
In this article, we have provided a comprehensive Q&A guide to complex numbers, covering their definition, properties, and operations. We hope that this guide has been helpful in understanding complex numbers and their applications.