In mathematics, the absolute value of a complex number is a measure of its distance from the origin of the complex plane. It determines the magnitude or modulus of the number, regardless of its direction. So, let’s calculate the absolute value of 6-3i.
The absolute value of a complex number a+bi can be found using the Pythagorean theorem, which states that the modulus can be determined using the formula: √(a^2 + b^2).
For the given complex number 6-3i, we have a = 6 and b = -3. Plugging these values into the formula, we get:
What is the absolute value of 6-3i?
The absolute value of 6-3i is √(6^2 + (-3)^2) = √(36 + 9) = √45.
Therefore, the absolute value of 6-3i is √45.
Now, let’s address some related frequently asked questions about absolute values of complex numbers:
FAQs:
1. What is the absolute value of a complex number?
The absolute value of a complex number is a measure of its distance from the origin of the complex plane, also known as its modulus or magnitude.
2. How do you calculate the absolute value of a complex number?
To calculate the absolute value of a complex number a+bi, use the formula √(a^2 + b^2), where a and b are the real and imaginary parts of the number, respectively.
3. What does the absolute value of a complex number represent?
The absolute value of a complex number represents its distance from the origin of the complex plane, disregarding its direction.
4. What is the geometric interpretation of the absolute value of a complex number?
The absolute value of a complex number represents the length of the line segment between the origin and the point representing the complex number on the complex plane.
5. Can the absolute value of a complex number be negative?
No, the absolute value of a complex number is always non-negative. It represents a distance and distances cannot be negative.
6. What is the absolute value of a purely real complex number?
For a purely real complex number of the form a+0i, where a is a real number, the absolute value is equal to the absolute value of the real component, |a|.
7. What is the absolute value of a purely imaginary complex number?
For a purely imaginary complex number of the form 0+bi, where b is a real number, the absolute value is equal to the absolute value of the imaginary component, |b|.
8. Can the absolute value of a complex number be zero?
Yes, the absolute value of a complex number can be zero. It occurs when the complex number is at the origin of the complex plane, having no magnitude.
9. How does the absolute value of a complex number relate to its conjugate?
The absolute value of a complex number is equal to the absolute value of its conjugate. If z is a complex number, then |z| = |z*|, where z* represents the complex conjugate of z.
10. What is the absolute value of the sum of two complex numbers?
The absolute value of the sum of two complex numbers is less than or equal to the sum of their absolute values. It follows the triangle inequality: |a + b| ≤ |a| + |b|.
11. What is the absolute value of the product of two complex numbers?
The absolute value of the product of two complex numbers is equal to the product of their absolute values. In mathematical notation, |a * b| = |a| * |b|.
12. How is the absolute value of a complex number represented in the complex plane?
In the complex plane, the absolute value of a complex number corresponds to the distance from the origin to the point representing the number. The magnitude increases as the distance from the origin increases.