What is the absolute value of 9 + 12i?
The absolute value of a complex number is a measure of its distance from the origin on the complex plane. To find the absolute value of 9 + 12i, we need to calculate the length of the vector formed by this complex number. We can do this by using the Pythagorean theorem.
The Pythagorean theorem states that the square of the hypotenuse (the longest side) of a right triangle is equal to the sum of the squares of the other two sides. In the case of a complex number, we can consider the real part (9) as the length of one side and the imaginary part (12) as the length of the other side.
By applying the formula and simplifying, we can find the absolute value:
Absolute value = √(Real part^2 + Imaginary part^2)
= √(9^2 + 12^2)
= √(81 + 144)
= √(225)
= 15
**Therefore, the absolute value of 9 + 12i is 15.**
FAQs:
1. What is a complex number?
A complex number is a number that comprises a real part and an imaginary part. It is written in the form “a + bi,” where “a” is the real part and “bi” is the imaginary part.
2. How do you identify the real and imaginary parts of a complex number?
In a complex number written as “a + bi,” the real part is represented by “a” and the imaginary part is represented by “bi.”
3. 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. It indicates the magnitude of the complex number.
4. Is the absolute value of a complex number always a real number?
Yes, the absolute value of a complex number is always a real number. It can be a positive value or zero.
5. How is the absolute value of a complex number calculated in general?
To calculate the absolute value of a complex number, you need to find the square root of the sum of the squares of its real and imaginary parts.
6. Can the absolute value of a complex number ever be negative?
No, the absolute value of a complex number is always non-negative. It cannot be negative.
7. What is the significance of the absolute value of a complex number in geometric terms?
The absolute value of a complex number represents the distance between the origin and the point plotted by that complex number on the complex plane.
8. What is the complex plane?
The complex plane, also known as the Argand plane, is a two-dimensional coordinate system where complex numbers are represented. The real axis represents the real part, and the imaginary axis represents the imaginary part.
9. How can the absolute value of a complex number be used to compare two complex numbers?
The absolute value of a complex number can be used to determine which complex number has a greater magnitude. The larger the absolute value, the greater the magnitude.
10. Can the absolute value of a complex number be zero?
Yes, the absolute value of a complex number can be zero. If the real and imaginary parts are both zero, then the absolute value is zero.
11. Can the absolute value of a complex number be complex itself?
No, the absolute value of a complex number is always a real number. It cannot have an imaginary part.
12. How is the absolute value of a complex number useful in mathematical applications?
The absolute value of complex numbers finds applications in various fields, such as engineering, physics, signal processing, and calculus. It is used to analyze and solve problems involving complex numbers, such as impedance calculations in electrical circuits.