What is the absolute value of 9+12i?
The absolute value of a complex number is a measure of its distance from the origin in the complex plane. To find the absolute value of 9+12i, we need to calculate the square root of the sum of the squares of its real and imaginary parts.
Let’s break it down. The real part of the complex number 9+12i is 9, and the imaginary part is 12. To find the absolute value, we follow these steps:
Step 1: Take the square of the real part (9^2 = 81).
Step 2: Take the square of the imaginary part (12^2 = 144).
Step 3: Add the results of step 1 and step 2 (81 + 144 = 225).
Step 4: Take the square root of the sum from step 3 (√225 = 15).
Therefore, the absolute value of 9+12i is **15**.
FAQs about the absolute value of complex numbers:
1. What is a complex number?
A complex number is a number that consists of a real part and an imaginary part. It is expressed in the form a+bi, where a is the real part and bi represents the imaginary part.
2. What is the real part of a complex number?
The real part of a complex number represents the horizontal component on the complex plane. In the number a+bi, the real part is ‘a’.
3. What is the imaginary part of a complex number?
The imaginary part of a complex number represents the vertical component on the complex plane. In the number a+bi, the imaginary part is ‘bi’.
4. How do you calculate the absolute value of a complex number?
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.
5. What does the absolute value of a complex number represent?
The absolute value of a complex number represents its distance from the origin in the complex plane.
6. 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.
7. What is the absolute value of a purely real number?
The absolute value of a purely real number (a+0i) is equivalent to the absolute value of the corresponding real number. For example, the absolute value of 9 is simply 9.
8. What is the absolute value of a purely imaginary number?
The absolute value of a purely imaginary number (0+bi) is equivalent to the absolute value of the corresponding imaginary number. For example, the absolute value of 12i is 12.
9. How does the absolute value of a complex number relate to its magnitude?
The absolute value of a complex number is equal to its magnitude. It represents the length or size of the vector generated by the complex number on the complex plane.
10. What is the significance of the absolute value of a complex number in applications?
The absolute value of a complex number is useful for calculating distances, determining signal strengths in communication systems, and representing magnitudes in physical phenomena involving complex quantities.
11. Can the absolute value of a complex number be zero?
Yes, the absolute value of a complex number can be zero if and only if the complex number itself is zero. In other words, if the real and imaginary parts of a complex number are both zero, its absolute value will be zero as well.
12. How does the absolute value of a complex number change with scaling?
The absolute value of a complex number scales proportionally when the complex number is multiplied or divided by a real scalar. In other words, multiplying or dividing both the real and imaginary parts by a constant does not alter the absolute value.