Operations on Complex Numbers

A complex number has the standard form a + bi, where a and b are real numbers. You can add, subtract, and multiply complex numbers using the same algebraic rules as those for real numbers and then simplify the final answer so it’s in the standard form.

For the most part, the i works just like any other variable. In general, you combine all real numbers, change all powers of i to 1, –1, i, or –i, and then combine all terms with i’s in them.

Sample question

1. Given the two complex numbers 3 + 4i and –5 + 2i, perform three separate operations: Add them, subtract the second from the first, and multiply them together.

   Add: –2 + 6i; subtract: 8 + 2i; multiply: –23 – 14i. Adding, you have (3 + 4i) + (–5 + 2i). Rearrange the terms so that the like terms are together. You get 3 – 5 + 4i + 2i = –2 + 6i.

   Subtracting, you have to negate each term in the second complex number: (3 + 4i) – (–5 + 2i) = 3 + 4i + 5 – 2i = 3 + 5 + 4i – 2i = 8 + 2i.

   To multiply the complex numbers, first use the FOIL method (First, Outer, Inner, Last): (3 + 4i)(–5 + 2i) = –15 + 6i – 20i + 8i² = –15 – 14i + 8i². However, this answer isn’t in standard form; you have to simplify the squared term. Remember that i² = –1, so write the answer as –15 – 14i + 8(–1) = –15 – 14i – 8 = –23 – 14i.

Practice questions

1. Compute: (2 + 3i) + (–3 + 4i)

2. Compute: (–6 + i) – (3 – 2i)

3. Compute: (–8 + 6i) – 4(–1 + i)

4. Compute: (–3 + 2i)(4 – 9i)

Following are answers to the practice questions:

1. The answer is –1 + 7i.

   Group the real parts and imaginary parts and simplify: 2 – 3 + 3i + 4i = –1 + 7i.

2. The answer is –9 + 3i.

   Distribute the negative sign over the second complex number; then group the real parts and imaginary parts and simplify: –6 + i – 3 + 2i = –6 – 3 + i + 2i = –9 + 3i.

3. The answer is –4 + 2i.

   Distribute the –4 over the second complex number; then group the real parts and imaginary parts and simplify: –8 + 6i + 4 – 4i = –8 + 4 + 6i – 4i = –4 + 2i.

4. The answer is 6 + 35i.

   Multiply the two complex numbers together using FOIL. Replace the i² with –1. Then group the real parts and imaginary parts and simplify: –12 + 27i + 8i – 18i² = –12 + 27i + 8i – 18(–1) = –12 + 18 + 27i + 8i = 6 + 35i.