What is `(sqrt3+i)//(1+sqrt3i)` equal to ?

Options

A. `1+i`
B. `1-i`
C. `sqrt3(1-i)//2`
D. `(sqrt3-i)//2`

Explanation

To solve the problem \(\frac{\sqrt{3} + i}{1 + \sqrt{3}i}\), we will rationalize the denominator. Here are the steps:

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

### Step 2: Multiply by the conjugate of the denominator
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(1 - \sqrt{3}i\):
\[ z = \frac{(\sqrt{3} + i)(1 - \sqrt{3}i)}{(1 + \sqrt{3}i)(1 - \sqrt{3}i)} \]

### Step 3: Expand 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:
\[ \sqrt{3} - 3i + i - \sqrt{3} = \sqrt{3} - 3i + i - \sqrt{3} = -2i \]

### Step 4: Expand the denominator
Now we expand the denominator:
\[ (1 + \sqrt{3}i)(1 - \sqrt{3}i) = 1^2 - (\sqrt{3}i)^2 = 1 - 3(-1) = 1 + 3 = 4 \]

### Step 5: Combine the results
Now we can combine the results:
\[ z = \frac{-2i}{4} = -\frac{1}{2}i \]

### Step 6: Write in standard form
In standard form, this is:
\[ z = 0 - \frac{1}{2}i \]

### Final Answer
Thus, the final answer is:
\[ z = 0 - \frac{1}{2}i \]

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