Find The Absolute Value Of { -3 - 5i$}$.A. 4 B. { \sqrt{34}$}$ C. { \sqrt{41}$}$ D. ${ 2 \sqrt{5}\$}
Introduction
In mathematics, the absolute value of a complex number is a measure of its distance from the origin in the complex plane. It is also known as the modulus or magnitude of the complex number. In this article, we will learn how to find the absolute value of a complex number and apply this concept to solve a specific problem.
What is a Complex Number?
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. The real part of the complex number is a, and the imaginary part is b.
The Absolute Value of a Complex Number
The absolute value of a complex number a + bi is denoted by |a + bi| and is calculated using the formula:
|a + bi| = √(a^2 + b^2)
This formula is derived from the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.
Finding the Absolute Value of -3 - 5i
Now, let's apply the formula to find the absolute value of the complex number -3 - 5i.
| -3 - 5i | = √((-3)^2 + (-5)^2) | -3 - 5i | = √(9 + 25) | -3 - 5i | = √34
Therefore, the absolute value of the complex number -3 - 5i is √34.
Conclusion
In this article, we learned how to find the absolute value of a complex number using the formula |a + bi| = √(a^2 + b^2). We applied this concept to find the absolute value of the complex number -3 - 5i and obtained the result √34.
Answer
The correct answer is B. √34.
Comparison of Options
Let's compare the options given in the problem:
A. 4: This is not the correct answer because the absolute value of -3 - 5i is not 4.
B. √34: This is the correct answer because the absolute value of -3 - 5i is indeed √34.
C. √41: This is not the correct answer because the absolute value of -3 - 5i is not √41.
D. 2√5: This is not the correct answer because the absolute value of -3 - 5i is not 2√5.
Final Thoughts
Introduction
In our previous article, we learned how to find the absolute value of a complex number using the formula |a + bi| = √(a^2 + b^2). In this article, we will answer some frequently asked questions about the absolute value of complex numbers.
Q: What is the absolute value of a complex number?
A: The absolute value of a complex number a + bi is a measure of its distance from the origin in the complex plane. It is also known as the modulus or magnitude of the complex number.
Q: How do I find the absolute value of a complex number?
A: To find the absolute value of a complex number a + bi, you can use the formula |a + bi| = √(a^2 + b^2).
Q: What is the difference between the absolute value and the magnitude of a complex number?
A: The absolute value and the magnitude of a complex number are the same thing. They are both measures of the distance of the complex number from the origin in the complex plane.
Q: Can the absolute value of a complex number be negative?
A: No, the absolute value of a complex number cannot be negative. The absolute value is always a non-negative real number.
Q: How do I find the absolute value of a complex number with a negative real part?
A: To find the absolute value of a complex number with a negative real part, you can use the formula |a + bi| = √(a^2 + b^2), just like you would for a complex number with a positive real part.
Q: Can I use the absolute value of a complex number to determine its sign?
A: No, the absolute value of a complex number does not determine its sign. The sign of a complex number is determined by the sign of its real part.
Q: How do I find the absolute value of a complex number with a zero imaginary part?
A: To find the absolute value of a complex number with a zero imaginary part, you can simply take the absolute value of the real part. For example, if the complex number is 3 + 0i, then its absolute value is |3 + 0i| = |3| = 3.
Q: Can I use the absolute value of a complex number to determine its imaginary part?
A: No, the absolute value of a complex number does not determine its imaginary part. The imaginary part of a complex number is determined by the difference between the complex number and its real part.
Q: How do I find the absolute value of a complex number with a zero real part?
A: To find the absolute value of a complex number with a zero real part, you can simply take the absolute value of the imaginary part. For example, if the complex number is 0 + 4i, then its absolute value is |0 + 4i| = |4i| = 4.
Q: Can I use the absolute value of a complex number to determine its real part?
A: No, the absolute value of a complex number does not determine its real part. The real part of a complex number is determined by the sum of the complex number and its imaginary part.
Conclusion
In this article, we answered some frequently asked questions about the absolute value of complex numbers. We hope that this article has provided you with a better understanding of this concept and has helped you to develop your problem-solving skills.
Final Thoughts
The absolute value of a complex number is an important concept in mathematics, and it has many applications in fields such as engineering, physics, and computer science. In this article, we learned how to find the absolute value of a complex number using the formula |a + bi| = √(a^2 + b^2) and answered some frequently asked questions about this concept. We hope that this article has provided you with a better understanding of this concept and has helped you to develop your problem-solving skills.