which of the following is the modulus of 2 + I ?

Options

A. `sqrt(2)`
B. 2
C. `sqrt(3)`
D. `sqrt(5)`

Explanation

To find the modulus of the complex number \(2 + i\), we follow these steps:

### Step 1: Identify the components of the complex number
The complex number is given as \(2 + i\), where:
- \(a = 2\) (the real part)
- \(b = 1\) (the imaginary part)

### Step 2: Use the formula for the modulus of a complex number
The modulus of a complex number \(a + bi\) is calculated using the formula:
\[ |z| = \sqrt{a^2 + b^2} \]

### Step 3: Substitute the values of \(a\) and \(b\) into the formula
Substituting \(a = 2\) and \(b = 1\) into the formula gives:
\[ |2 + i| = \sqrt{2^2 + 1^2} \]

### Step 4: Calculate \(a^2\) and \(b^2\)
Now we calculate \(2^2\) and \(1^2\):
\[ 2^2 = 4 \quad \text{and} \quad 1^2 = 1 \]

### Step 5: Add the squares
Now, we add the results:
\[ 4 + 1 = 5 \]

### Step 6: Take the square root
Finally, we take the square root of the sum:
\[ |2 + i| = \sqrt{5} \]

### Conclusion
Thus, the modulus of the complex number \(2 + i\) is \(\sqrt{5}\).

### Answer
The correct option is **d) \(\sqrt{5}\)**.

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