What is the absolute value of 7-i?

The absolute value of a complex number represents its distance from the origin on the complex plane. To determine the absolute value of a complex number, we use the Pythagorean theorem. In this case, let’s find the absolute value of 7 – i.

To calculate the absolute value of 7 – i, we first need to understand that the real part of the complex number is 7, and the imaginary part is -1. Now, let’s apply the Pythagorean theorem: the absolute value is the square root of the sum of the squares of the real and imaginary parts.

So, the absolute value of 7 – i is given by √(7^2 + (-1)^2):

Absolute value of 7 – i = √(49 + 1) = √50.

**Therefore, the absolute value of 7 – i is √50.**

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 is the absolute value of a complex number calculated?

To calculate the absolute value of a complex number, use the Pythagorean theorem: take the square root of the sum of the squares of its real and imaginary parts.

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

Yes, the absolute value of a complex number is always a positive value or zero if the complex number is at the origin.

4. Can the absolute value of a complex number be zero?

Yes, the absolute value of a complex number can be zero. This occurs when the complex number is at the origin (0, 0) on the complex plane.

5. How do I represent the absolute value of a complex number in mathematical notation?

The absolute value of a complex number is usually represented using vertical bars or absolute value symbols. For example, |z| denotes the absolute value of the complex number z.

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

The absolute value of a purely real complex number is equal to the absolute value of its real part. For example, the absolute value of 9 + 0i is simply 9.

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

The absolute value of a purely imaginary complex number is equal to the absolute value of its imaginary part. For example, the absolute value of 0 + 5i is 5.

8. How does the absolute value of a complex number affect its argument?

The absolute value of a complex number does not affect its argument or phase angle. The argument represents the angle that the complex number forms with the positive real axis.

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

No, the absolute value of a complex number is always non-negative. It represents a distance, which is a positive value or zero.

10. What are the applications of complex numbers in real life?

Complex numbers have various applications in fields like electrical engineering, quantum mechanics, control systems, signal processing, and computer graphics.

11. Can complex numbers be graphically represented?

Yes, complex numbers can be graphically represented on a complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.

12. How is the concept of the absolute value of a complex number related to vectors?

The absolute value of a complex number can be thought of as the magnitude of a vector in the complex plane. Complex numbers can be viewed as vectors in two dimensions.

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