What is the absolute value of 8-6i?

The absolute value of a complex number represents its distance from the origin on the complex plane. To calculate the absolute value of the complex number 8-6i, we need to find the magnitude of the vector formed by its real and imaginary parts.

To find the absolute value of 8-6i, we use the formula:

|z| = √((Re(z))^2 + (Im(z))^2)

Where Re(z) represents the real part of z and Im(z) represents the imaginary part of z.

In this case, 8 is the real part and -6 is the imaginary part. Plugging these values into the formula, we get:

|8-6i| = √((8)^2 + (-6)^2)

Simplifying and calculating:

|8-6i| = √(64 + 36)

|8-6i| = √100

Answer: The absolute value of 8-6i is 10.

Now, let’s address some frequently asked questions related to the absolute value of complex numbers:

1. What does the absolute value of a complex number represent?

The absolute value of a complex number represents its distance from the origin on the complex plane.

2. How do you calculate the absolute value of a complex number?

To calculate the absolute value of a complex number, use the formula: |z| = √((Re(z))^2 + (Im(z))^2), where Re(z) and Im(z) are the real and imaginary parts of the complex number, respectively.

3. Can the absolute value of a complex number be negative?

No, the absolute value of a complex number is always a non-negative real number.

4. What is the absolute value of a purely real complex number?

The absolute value of a purely real complex number is simply the absolute value of the real part.

5. What is the absolute value of a purely imaginary complex number?

The absolute value of a purely imaginary complex number is the absolute value of the imaginary part.

6. What is the absolute value of a complex number when the real and imaginary parts are both zero?

The absolute value of a complex number with both real and imaginary parts equal to zero is zero.

7. How does the absolute value relate to the modulus of a complex number?

The absolute value of a complex number is often referred to as its modulus. They represent the same concept and can be used interchangeably.

8. Can the absolute value of a complex number be a complex number?

No, the absolute value of a complex number is always a non-negative real number, or equivalently, it can be zero.

9. What is the absolute value of the conjugate of a complex number?

The absolute value of the conjugate of a complex number is the same as the absolute value of the original complex number.

10. How do you find the absolute value of a complex number in polar form?

In polar form, a complex number is represented as z = r(cosθ + isinθ), where r is the magnitude and θ is the angle. The absolute value of a complex number in polar form is simply the magnitude (r).

11. Can the absolute value of a complex number ever be larger than its real part?

No, the absolute value of a complex number is always less than or equal to the sum of the absolute values of its real and imaginary parts.

12. Is the absolute value of a complex number always positive?

Yes, the absolute value of a complex number is always a non-negative real number, including zero.

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