Complex Numbers

Explore imaginary unit, complex arithmetic, conjugates and powers.

Lesson 4 of 53 parts Practice Flashcards

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Imaginary Unit

Objective: Explain imaginary unit and use it to solve or interpret a problem.

The imaginary unit i satisfies i²=−1; complex numbers have form a+bi with real a,b. The real and imaginary parts are coordinates on a plane. No, no real square equals −1.

Worked examples

Try itComplex Numbers — practice 1.1: Find the real part of (4+3i)+(5−2i).Show answer

Step 1: Group real parts 4+5. Step 2: Group imaginary parts separately as (3−2)i. Step 3: The real part is 9.

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Imaginary Unit

Objective: Explain imaginary unit and use it to solve or interpret a problem.

The imaginary unit i satisfies i²=−1; complex numbers have form a+bi with real a,b. The real and imaginary parts are coordinates on a plane. No, no real square equals −1.

Worked examples

Try itComplex Numbers — practice 1.1: Find the real part of (4+3i)+(5−2i).Show answer

Step 1: Group real parts 4+5. Step 2: Group imaginary parts separately as (3−2)i. Step 3: The real part is 9.

Complex Arithmetic

Objective: Explain complex arithmetic and use it to solve or interpret a problem.

Add components separately; distribute for products and replace i² by −1. Use a conjugate to make a complex denominator real when dividing. By definition i²=−1.

Worked examples

Try itComplex Numbers — practice 2.1: Find the coefficient of i in (2+4i)−(8+2i).Show answer

Step 1: Distribute subtraction: 2+4i−8−2i. Step 2: Combine imaginary terms (4−2)i. Step 3: The requested coefficient is 2.

Conjugates and Powers

Objective: Explain conjugates and powers and use it to solve or interpret a problem.

The conjugate of a+bi is a−bi, and their product a²+b² is real. Powers of i cycle every four exponents. Because (i²)²=(−1)²=1.

Worked examples

Try itComplex Numbers — practice 3.1: Find the real value of i^44.Show answer

Step 1: Since i²=−1, i⁴=1. Step 2: Group i^44=(i⁴)^11. Step 3: The result is 1^11=1.

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