Find the square root of the following complex numbers
i

Answer

To find the square root of the complex number \( i \), we can follow these steps:

### Step 1: Represent the square root
Let \( \sqrt{i} = a + bi \), where \( a \) and \( b \) are real numbers.

### Step 2: Square both sides
Squaring both sides gives us:
\[ i = (a + bi)^2 \]
Using the formula for the square of a binomial, we can expand the right-hand side:
\[ i = a^2 + 2abi + (bi)^2 = a^2 + 2abi - b^2 \]
This simplifies to:
\[ i = (a^2 - b^2) + 2abi \]

### Step 3: Equate real and imaginary parts
Now, we equate the real and imaginary parts from both sides:
- The real part: \( a^2 - b^2 = 0 \)
- The imaginary part: \( 2ab = 1 \)

### Step 4: Solve the equations
From the first equation \( a^2 - b^2 = 0 \), we can deduce that:
\[ a^2 = b^2 \implies a = b \quad \text{or} \quad a = -b \]

Using the second equation \( 2ab = 1 \), we can substitute \( b \) with \( a \) (assuming \( a = b \)):
\[ 2a^2 = 1 \implies a^2 = \frac{1}{2} \implies a = \pm \frac{1}{\sqrt{2}} \]
Thus, if \( a = \frac{1}{\sqrt{2}} \), then \( b = \frac{1}{\sqrt{2}} \) and if \( a = -\frac{1}{\sqrt{2}} \), then \( b = -\frac{1}{\sqrt{2}} \).

### Step 5: Write the final results
Therefore, we have two possible values for \( \sqrt{i} \):
\[ \sqrt{i} = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i \quad \text{or} \quad \sqrt{i} = -\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}i \]
Thus, the square roots of \( i \) are:
\[ \sqrt{i} = \pm \left( \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i \right) \]