What is the absolute value of 5-i?

When dealing with complex numbers, it is common to come across the concept of absolute value. The absolute value of a complex number provides a measure of its magnitude or distance from the origin on the complex plane. In this article, we will explore what the absolute value of 5-i is and how to calculate it.

Understanding Complex Numbers

Before we dive into the absolute value of 5-i, let’s quickly recap what complex numbers are. A complex number consists of a real part and an imaginary part combined together. In this case, 5 is the real part, represented by the vertical axis on the complex plane, and -i is the imaginary part, represented by the horizontal axis.

Calculating the Absolute Value

The absolute value of a complex number z (denoted as |z|) can be calculated by finding the distance between z and the origin on the complex plane. To find the absolute value of 5-i, we need to find how far it is from the origin (0,0).

The distance between two points in a coordinate plane can be calculated using the distance formula, which utilizes the Pythagorean theorem. The formula for finding the distance between two coordinates (a,b) and (c,d) is:

Distance = √((c-a)² + (d-b)²)

In our case, the coordinates of 5-i are (5, -1) as the real part is 5 and the imaginary part is -1. The origin has coordinates (0, 0). Substituting the values into the distance formula, we get:

Distance = √((5-0)² + (-1-0)²)
= √(5² + (-1)²)
= √(25 + 1)
= √26

Therefore, the absolute value of 5-i is √26.

What is the absolute value of a complex number?

The absolute value of a complex number |z| is the distance between that number and the origin on the complex plane.

How do you find the absolute value of a complex number?

To find the absolute value of a complex number, you calculate the distance of that number from the origin on the complex plane using the distance formula.

What does the absolute value represent?

The absolute value represents the magnitude or distance of a complex number from the origin.

Can the absolute value of a complex number be negative?

No, the absolute value of a complex number is always positive or zero.

What if the complex number is a positive real number?

If the complex number is a positive real number, such as 5 + 0i, the absolute value would simply be 5.

What if the complex number is an imaginary number?

If the complex number is an imaginary number, like 0 + 3i, the absolute value would be 3.

Can the absolute value of a complex number be a non-real number?

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

What happens if the complex number is at the origin?

If the complex number is at the origin, such as 0 + 0i, the absolute value would be 0.

How is the absolute value of a complex number represented geometrically?

The absolute value of a complex number represents its distance from the origin on the complex plane, and thus it can be shown as the radius of a circle centered at the origin.

What is the relationship between absolute value and modulus of a complex number?

The modulus of a complex number is another term used to represent its absolute value. Both terms are interchangeable.

Can multiple complex numbers have the same absolute value?

Yes, multiple complex numbers can have the same absolute value if they are equidistant from the origin.

Is the absolute value of a complex number always a rational number?

No, the absolute value of a complex number can be a rational number or an irrational number, depending on the specific complex number.

In conclusion, the absolute value of 5-i is √26. This value represents the distance of 5-i from the origin on the complex plane. Calculating the absolute value is essential for understanding the magnitude of complex numbers and their geometric representation on the complex plane.

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