The modulus of the complex number z =a +ib is

Options

A. `z.barz`
B. a+b
C. `a^(2)+b^(2)`
D. `sqrt(z.barz)`

Answer

To find the modulus of the complex number \( z = a + ib \), we follow these steps:

### Step 1: Understand the definition of modulus
The modulus of a complex number \( z = a + ib \) is defined as:
\[ |z| = \sqrt{a^2 + b^2} \]

### Step 2: Identify the components of the complex number
In the complex number \( z = a + ib \):
- \( a \) is the real part.
- \( b \) is the imaginary part.

### Step 3: Apply the modulus formula
Using the definition from Step 1, we can substitute \( a \) and \( b \) into the formula:
\[ |z| = \sqrt{a^2 + b^2} \]

### Step 4: Conjugate of the complex number
The conjugate of \( z \), denoted as \( \overline{z} \), is given by:
\[ \overline{z} = a - ib \]

### Step 5: Multiply \( z \) by its conjugate
Now, we multiply \( z \) by its conjugate:
\[ z \cdot \overline{z} = (a + ib)(a - ib) = a^2 - (ib)^2 = a^2 + b^2 \]

### Step 6: Relate the product to the modulus
From the multiplication in Step 5, we know:
\[ |z|^2 = z \cdot \overline{z} = a^2 + b^2 \]
Thus, taking the square root gives us:
\[ |z| = \sqrt{a^2 + b^2} \]

### Conclusion
The modulus of the complex number \( z = a + ib \) is:
\[ |z| = \sqrt{a^2 + b^2} \]