Find the modulus and amplitude of the complex number -5i.

To find the modulus and amplitude of the complex number -5i, we can use the following definitions:

Modulus (Magnitude): The modulus of a complex number is the distance from the origin (0, 0) to the point representing the complex number in the complex plane. It is calculated as the square root of the sum of the squares of the real and imaginary parts of the complex number.

Amplitude (Argument): The amplitude of a complex number is the angle between the positive real axis and the vector representing the complex number in the complex plane. It is calculated using the inverse tangent function.

Let's calculate the modulus and amplitude of the complex number -5i:

Modulus:
The complex number -5i has a real part of 0 and an imaginary part of -5. Therefore, the modulus is calculated as:
|z| = sqrt(0^2 + (-5)^2) = sqrt(0 + 25) = sqrt(25) = 5

Hence, the modulus (magnitude) of the complex number -5i is 5.

Amplitude:
To calculate the amplitude, we need to determine the angle between the positive real axis and the vector representing the complex number in the complex plane.
The complex number -5i lies on the negative imaginary axis. The angle between the positive real axis and the negative imaginary axis is -90 degrees or -π/2 radians.

Therefore, the amplitude (argument) of the complex number -5i is -90 degrees or -π/2 radians.

In summary:
Modulus (magnitude): 5
Amplitude (argument): -90 degrees or -π/2 radians