What Is The Absolute Value Of The Complex Number $-4-\sqrt{2}i$?A. $\sqrt{14}$B. $3 \sqrt{2}$C. 14D. 18

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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 the concept of absolute value of a complex number and calculate the absolute value of the complex number −4−2i-4-\sqrt{2}i.

What is the Absolute Value of a Complex Number?

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. It can be 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.

Calculating the Absolute Value of a Complex Number

To calculate the absolute value of the complex number −4−2i-4-\sqrt{2}i, we can use the formula:

|a + bi| = √(a^2 + b^2)

In this case, a = -4 and b = -√2. Plugging these values into the formula, we get:

|-4 - √2i| = √((-4)^2 + (-√2)^2) = √(16 + 2) = √18 = √(9 * 2) = 3√2

Therefore, the absolute value of the complex number −4−2i-4-\sqrt{2}i is 3√2.

Conclusion

In conclusion, the absolute value of a complex number is a measure of its distance from the origin in the complex plane. It can be calculated using the formula |a + bi| = √(a^2 + b^2). In this article, we calculated the absolute value of the complex number −4−2i-4-\sqrt{2}i and found that it is 3√2.

Absolute Value of Complex Numbers: Key Concepts

  • Definition of 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.
  • Formula for Absolute Value: |a + bi| = √(a^2 + b^2)
  • Pythagorean Theorem: The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

Real-World Applications of Complex Numbers

Complex numbers have numerous real-world applications in fields such as:

  • Electrical Engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
  • Signal Processing: Complex numbers are used to represent signals and analyze their frequency content.
  • Navigation: Complex numbers are used to represent positions and velocities in navigation systems.

Common Mistakes to Avoid When Calculating Absolute Value

When calculating the absolute value of a complex number, it's essential to avoid the following common mistakes:

  • Not using the correct formula: Make sure to use the formula |a + bi| = √(a^2 + b^2) to calculate the absolute value.
  • Not simplifying the expression: Simplify the expression under the square root to get the final answer.
  • Not checking the units: Check the units of the answer to ensure that they match the units of the input values.

Conclusion

Frequently Asked Questions

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. It can be calculated using the formula |a + bi| = √(a^2 + b^2).

Q: How do I calculate the absolute value of a complex number? A: To calculate the absolute value of a complex number, simply plug the values of a and b into the formula |a + bi| = √(a^2 + b^2) and simplify the expression under the square root.

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 both refer to the distance of the complex number from the origin in the complex plane.

Q: Can I use the absolute value formula to calculate the absolute value of a complex number with negative coefficients? A: Yes, you can use the absolute value formula to calculate the absolute value of a complex number with negative coefficients. The formula |a + bi| = √(a^2 + b^2) works for all complex numbers, regardless of whether the coefficients are positive or negative.

Q: How do I simplify the expression under the square root when calculating the absolute value of a complex number? A: To simplify the expression under the square root, look for perfect squares and factor them out. For example, if you have √(16 + 4), you can simplify it to √(4 * 4 + 1) = √(4^2 + 1) = 4√2.

Q: Can I use the absolute value formula to calculate the absolute value of a complex number with imaginary coefficients? A: Yes, you can use the absolute value formula to calculate the absolute value of a complex number with imaginary coefficients. The formula |a + bi| = √(a^2 + b^2) works for all complex numbers, regardless of whether the coefficients are real or imaginary.

Q: What is the absolute value of the complex number 3 + 4i? A: To calculate the absolute value of the complex number 3 + 4i, plug the values of a and b into the formula |a + bi| = √(a^2 + b^2). In this case, a = 3 and b = 4, so the absolute value is |3 + 4i| = √(3^2 + 4^2) = √(9 + 16) = √25 = 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, plug the values of a and b into the formula |a + bi| = √(a^2 + b^2). In this case, a = -2 and b = -3, so the absolute value is | -2 - 3i| = √((-2)^2 + (-3)^2) = √(4 + 9) = √13.

Q: Can I use the absolute value formula to calculate the absolute value of a complex number with decimal coefficients? A: Yes, you can use the absolute value formula to calculate the absolute value of a complex number with decimal coefficients. The formula |a + bi| = √(a^2 + b^2) works for all complex numbers, regardless of whether the coefficients are integers, decimals, or fractions.

Q: What is the absolute value of the complex number 0.5 + 0.3i? A: To calculate the absolute value of the complex number 0.5 + 0.3i, plug the values of a and b into the formula |a + bi| = √(a^2 + b^2). In this case, a = 0.5 and b = 0.3, so the absolute value is |0.5 + 0.3i| = √((0.5)^2 + (0.3)^2) = √(0.25 + 0.09) = √0.34.

Conclusion

In conclusion, the absolute value of a complex number is a measure of its distance from the origin in the complex plane. It can be calculated using the formula |a + bi| = √(a^2 + b^2). In this article, we answered frequently asked questions about the absolute value of complex numbers, including how to calculate it, what it represents, and how to simplify the expression under the square root. We also provided examples of calculating the absolute value of complex numbers with different coefficients.