What is `( sqrt( 3) + i) // ( 1 + sqrt( 3)i)` equal to ?

Options

A. `1+i`
B. `1-i`
C. `sqrt(3) ( 1-i) //2`
D. `( sqrt( 3) - i ) //2`

Answer

To solve the problem \(( \sqrt{3} + i) / ( 1 + \sqrt{3}i)\), we will follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ \frac{\sqrt{3} + i}{1 + \sqrt{3}i} \]

### Step 2: Rationalize the denominator
To simplify the expression, we will multiply the numerator and the denominator by the conjugate of the denominator, which is \(1 - \sqrt{3}i\):
\[ \frac{(\sqrt{3} + i)(1 - \sqrt{3}i)}{(1 + \sqrt{3}i)(1 - \sqrt{3}i)} \]

### Step 3: Calculate the denominator
Using the difference of squares formula:
\[ (1 + \sqrt{3}i)(1 - \sqrt{3}i) = 1^2 - (\sqrt{3}i)^2 = 1 - 3(-1) = 1 + 3 = 4 \]

### Step 4: Calculate the numerator
Now, we expand the numerator:
\[ (\sqrt{3} + i)(1 - \sqrt{3}i) = \sqrt{3} \cdot 1 + \sqrt{3} \cdot (-\sqrt{3}i) + i \cdot 1 + i \cdot (-\sqrt{3}i) \]
This simplifies to:
\[ \sqrt{3} - 3i + i - \sqrt{3}i^2 \]
Since \(i^2 = -1\), we have:
\[ \sqrt{3} - 3i + i + \sqrt{3} = 2\sqrt{3} - 2i \]

### Step 5: Combine the results
Now, we can put the numerator and denominator together:
\[ \frac{2\sqrt{3} - 2i}{4} \]
This simplifies to:
\[ \frac{2\sqrt{3}}{4} - \frac{2i}{4} = \frac{\sqrt{3}}{2} - \frac{i}{2} \]

### Step 6: Final answer
Thus, the final result is:
\[ \frac{\sqrt{3}}{2} - \frac{i}{2} \]

### Summary:
The expression \(\frac{\sqrt{3} + i}{1 + \sqrt{3}i}\) simplifies to \(\frac{\sqrt{3}}{2} - \frac{i}{2}\).

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