The value of `sqrti` is

Options

A. 1-i
B. 1+i
C. `+-(1+i)`
D. `(+-1)/(sqrt2)(1+i)`

Answer

To find the value of \(\sqrt{i}\), we can follow these steps:

### Step 1: Express \(i\) in polar form
The complex number \(i\) can be represented in polar form. We know that:
\[ i = 0 + 1i \]
In polar form, this can be expressed as:
\[ i = r \cdot e^{i\theta} \]
where \(r\) is the modulus and \(\theta\) is the argument (angle).

### Step 2: Calculate the modulus \(r\)
The modulus \(r\) of \(i\) is given by:
\[ r = |i| = \sqrt{0^2 + 1^2} = \sqrt{1} = 1 \]

### Step 3: Determine the argument \(\theta\)
The argument \(\theta\) for \(i\) is:
\[ \theta = \frac{\pi}{2} \quad \text{(since it lies on the positive imaginary axis)} \]

### Step 4: Write \(i\) in polar form
Now we can write \(i\) in polar form:
\[ i = 1 \cdot e^{i\frac{\pi}{2}} \]

### Step 5: Find the square root of \(i\)
To find \(\sqrt{i}\), we take the square root of the polar form:
\[ \sqrt{i} = \sqrt{1 \cdot e^{i\frac{\pi}{2}}} = \sqrt{1} \cdot e^{i\frac{\pi}{4}} = 1 \cdot e^{i\frac{\pi}{4}} = e^{i\frac{\pi}{4}} \]

### Step 6: Convert back to rectangular form
Now we convert \(e^{i\frac{\pi}{4}}\) back to rectangular form:
\[ e^{i\frac{\pi}{4}} = \cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}} \]

### Step 7: Include the negative root
Since we are looking for both roots, we also consider the negative:
\[ \sqrt{i} = \pm\left(\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}\right) \]

### Final Answer
Thus, the value of \(\sqrt{i}\) is:
\[ \sqrt{i} = \pm\left(\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}\right) \]

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