Complex Numbers

1

Step 1: We know i² = -1. Step 2: i⁴ = (i²)² = (-1)² = 1. Step 3: This is a key result: i⁴ = 1, meaning the powers of i repeat with period 4. Final Answer: i⁴ = 1. Remember the cycle: i, -1, -i, 1 keeps repeating for every 4 powers. This cycle is very important for solving higher power problems.

-4

Step 1: Any complex number is written in the form z = a + bi. Step 2: Here z = -4 + 9i, so a = -4 and b = 9. Step 3: Re(z) = a = -4 and Im(z) = b = 9. Final Answer: The real part is -4. A common mistake is taking the magnitude 4 instead of -4. The sign must be preserved.

10

Step 1: |z| = √(a² + b²) where z = a + bi. Step 2: Here a = 6, b = 8. Step 3: |z| = √(6² + 8²) = √(36 + 64) = √100. Final Answer: |z| = 10. This uses the 6-8-10 Pythagorean triple. A common error is adding 6+8=14 without squaring. Always square each part before adding.

0 + 5i

Step 1: A purely imaginary number has real part = 0 and imaginary part ≠ 0. Step 2: Check each option: 3+0i has real part 3 (not zero), so it's a real number. Step 3: 0+5i has real part 0 and imaginary part 5 ≠ 0, so it is purely imaginary. Step 4: 2+3i has both parts non-zero. 0+0i = 0 is neither purely real nor purely imaginary. Final Answer: 0 + 5i is purely imaginary.