If z = 1 + √3i, what is the square of the modulus of z?

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Question: If z = 1 + √3i, what is the square of the modulus of z?

Options:

Correct Answer: 4

Solution:

The modulus of z = 1 + √3i is |z| = √(1² + (√3)²) = √(1 + 3) = √4 = 2. The square of the modulus is 2² = 4.

If z = 1 + √3i, what is the square of the modulus of z?

If z = 1 + √3i, what is the square of the modulus of z?

Step 1: Identify the complex number z, which is given as z = 1 + √3i.
Step 2: Recall that the modulus of a complex number z = a + bi is calculated using the formula |z| = √(a² + b²), where a is the real part and b is the imaginary part.
Step 3: In our case, a = 1 and b = √3.
Step 4: Calculate a², which is 1² = 1.
Step 5: Calculate b², which is (√3)² = 3.
Step 6: Add the results from Step 4 and Step 5: 1 + 3 = 4.
Step 7: Take the square root of the sum from Step 6: |z| = √4 = 2.
Step 8: To find the square of the modulus, calculate 2² = 4.

Modulus of a Complex Number – The modulus of a complex number z = a + bi is calculated using the formula |z| = √(a² + b²), where a is the real part and b is the imaginary part.
Square of a Number – The square of a number x is calculated as x², which is the result of multiplying the number by itself.