Complex Numbers

-1

Step 1: Evaluate each power of i separately using i²=-1, i³=-i, i⁴=1. Step 2: i² = -1. Step 3: i⁴ = (i²)² = (-1)² = 1. Step 4: i⁶ = (i²)³ = (-1)³ = -1. Step 5: Sum = i² + i⁴ + i⁶ = (-1) + 1 + (-1) = -1. Final Answer: -1. Common mistake: Students sometimes say i⁴ = -1 (confusing it with i²). Remember i⁴ = 1 because (i²)² = (-1)² = 1.

2

Step 1: We have z = 1 + i. First find the conjugate z̄ = 1 - i. Step 2: Compute z × z̄ = (1 + i)(1 - i). Step 3: Use the identity (a+b)(a-b) = a² - b². Here a=1, b=i. Step 4: (1+i)(1-i) = 1² - i² = 1 - (-1) = 1 + 1 = 2. Final Answer: z × z̄ = 2 = |z|². This is consistent with |z| = √(1²+1²) = √2, so |z|² = 2. Key property: z × z̄ = |z|² is always a non-negative real number.

π

Step 1: We have z = -1 = -1 + 0i. This is a purely real negative number. In the Argand plane, it lies on the negative real axis. Step 2: The principal argument is the angle θ such that -π < θ ≤ π. Step 3: The negative real axis makes an angle of π radians (180°) with the positive real axis. Step 4: Since π satisfies -π < π ≤ π, the principal argument is π. Final Answer: Arg(-1) = π. Common mistake: Students confuse -π and π. Note that π is included in the range but -π is not, so we use π for negative real numbers.

5 + i

Step 1: We need to multiply (2 + 3i)(1 - i). Expand using FOIL or distributive law. Step 2: (2 + 3i)(1 - i) = 2(1) + 2(-i) + 3i(1) + 3i(-i). Step 3: = 2 - 2i + 3i - 3i². Step 4: Since i² = -1, substitute: = 2 - 2i + 3i - 3(-1) = 2 - 2i + 3i + 3. Step 5: Combine real parts: 2 + 3 = 5. Combine imaginary parts: -2i + 3i = i. Final Answer: 5 + i. Common mistake: Forgetting to substitute i² = -1, which would give 2 - 3 = -1 instead of 2 + 3 = 5 for the real part.