Find the modulus and amplitude of the following complex numbers and hence express them into polar form
`-2i`

Answer

To find the modulus and amplitude of the complex number \(-2i\) and express it in polar form, we can follow these steps:

### Step 1: Identify the complex number
The given complex number is:
\[ z = -2i \]
In the standard form \(z = x + iy\), we can identify:
\[ x = 0 \quad \text{and} \quad y = -2 \]

### Step 2: Calculate the modulus
The modulus \(r\) of a complex number \(z = x + iy\) is given by the formula:
\[ r = \sqrt{x^2 + y^2} \]
Substituting the values of \(x\) and \(y\):
\[ r = \sqrt{0^2 + (-2)^2} = \sqrt{0 + 4} = \sqrt{4} = 2 \]

### Step 3: Calculate the amplitude (argument)
The amplitude (or argument) \(\theta\) can be found using the relationships:
\[ \cos \theta = \frac{x}{r} \quad \text{and} \quad \sin \theta = \frac{y}{r} \]
Substituting the values we have:
\[ \cos \theta = \frac{0}{2} = 0 \quad \text{and} \quad \sin \theta = \frac{-2}{2} = -1 \]
From \(\cos \theta = 0\), we know that \(\theta\) can be \(\frac{\pi}{2}\) or \(\frac{3\pi}{2}\). Since \(\sin \theta = -1\), we find:
\[ \theta = \frac{3\pi}{2} \]

### Step 4: Express in polar form
The polar form of a complex number is expressed as:
\[ z = r(\cos \theta + i \sin \theta) \]
Substituting the values of \(r\) and \(\theta\):
\[ z = 2\left(\cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2}\right) \]

### Final Result
Thus, the modulus of \(-2i\) is \(2\), the amplitude is \(\frac{3\pi}{2}\), and the polar form is:
\[ z = 2\left(\cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2}\right) \]

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