What Is The Absolute Value Of The Complex Number $-4-\sqrt{2}i$?A. $\sqrt{14}$ B. $3\sqrt{2}$ C. 14 D. 18
Understanding Complex Numbers
A complex number is a number that can be expressed in the form of a + bi, where 'a' is the real part and 'bi' is the imaginary part. The imaginary part is denoted by 'i', which is the square root of -1. Complex numbers are used to represent quantities that have both real and imaginary components.
The Absolute Value of a Complex Number
The absolute value of a complex number is also known as its modulus or magnitude. It is a measure of the distance of the complex number from the origin in the complex plane. The absolute value of a complex number z = a + bi is denoted by |z| and is calculated using the formula:
|z| = √(a^2 + b^2)
Calculating the Absolute Value of a Complex Number
To calculate the absolute value of the complex number -4 - √2i, we need to substitute the values of a and b into the formula.
a = -4 b = -√2
Now, we can plug these values into the formula:
|z| = √((-4)^2 + (-√2)^2) |z| = √(16 + 2) |z| = √18
Simplifying the Absolute Value
We can simplify the absolute value by factoring out the square root:
|z| = √(9 * 2) |z| = √9 * √2 |z| = 3√2
Conclusion
The absolute value of the complex number -4 - √2i is 3√2. This is the distance of the complex number from the origin in the complex plane.
Comparison of Options
Let's compare our answer with the given options:
A. √14 B. 3√2 C. 14 D. 18
Our answer, 3√2, matches option B.
Final Answer
Q: What is a complex number?
A: A complex number is a number that can be expressed in the form of a + bi, where 'a' is the real part and 'bi' is the imaginary part. The imaginary part is denoted by 'i', which is the square root of -1.
Q: What is the absolute value of a complex number?
A: The absolute value of a complex number is also known as its modulus or magnitude. It is a measure of the distance of the complex number from the origin in the complex plane.
Q: How is the absolute value of a complex number calculated?
A: The absolute value of a complex number z = a + bi is calculated using the formula:
|z| = √(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 simplify the absolute value of a complex number?
A: To simplify the absolute value of a complex number, you can factor out the square root:
|z| = √(a^2 + b^2) |z| = √(n * m) |z| = √n * √m
Q: What is the absolute value of the complex number 3 + 4i?
A: To calculate the absolute value of the complex number 3 + 4i, we need to substitute the values of a and b into the formula:
a = 3 b = 4
Now, we can plug these values into the formula:
|z| = √((3)^2 + (4)^2) |z| = √(9 + 16) |z| = √25 |z| = 5
Q: What is the absolute value of the complex number -2 - 3i?
A: To calculate the absolute value of the complex number -2 - 3i, we need to substitute the values of a and b into the formula:
a = -2 b = -3
Now, we can plug these values into the formula:
|z| = √((-2)^2 + (-3)^2) |z| = √(4 + 9) |z| = √13
Q: Can I use a calculator to calculate the absolute value of a complex number?
A: Yes, you can use a calculator to calculate the absolute value of a complex number. Most calculators have a built-in function for calculating the absolute value of a complex number.
Conclusion
We hope this article has helped you understand complex numbers and absolute value. If you have any more questions, feel free to ask!