If z = 1 + i, find |z|².

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Question: If z = 1 + i, find |z|².

Options:

Correct Answer: 2

Solution:

|z|² = (1)² + (1)² = 1 + 1 = 2.

If z = 1 + i, find |z|².

If z = 1 + i, find |z|².

Step 1: Identify the complex number z. Here, z = 1 + i.
Step 2: Recall the formula for the magnitude squared of a complex number z = a + bi, which is |z|² = a² + b².
Step 3: In our case, a = 1 and b = 1 (since i is the imaginary unit and corresponds to 1).
Step 4: Substitute the values into the formula: |z|² = (1)² + (1)².
Step 5: Calculate (1)², which is 1.
Step 6: Calculate (1)² again, which is also 1.
Step 7: Add the two results together: 1 + 1 = 2.
Step 8: Conclude that |z|² = 2.

Complex Numbers – Understanding the representation of complex numbers in the form z = a + bi, where a and b are real numbers.
Magnitude of Complex Numbers – Calculating the magnitude (or modulus) of a complex number, which is given by |z| = √(a² + b²).
Square of the Magnitude – Recognizing that |z|² = a² + b² directly without needing to compute the square root.