Adding and Multiplying Complex Numbers
Aligned to N-CN.A.2 — Common Core State Standards for Mathematics.
What this lesson teaches
Treat the imaginary unit i like a variable, but replace i^2 with −1. Add complex numbers by combining real and imaginary parts; multiply by distributing (FOIL).
Worked examples
Example 1
Multiply (2 + 3i)(1 − 4i) = 2 − 8i + 3i − 12i^2 = 2 − 5i − 12(−1) = 2 − 5i + 12 = 14 − 5i.
Example 2
Add and then multiply: (5 − 2i) + (−3 + 7i) = 2 + 5i, combining real parts and imaginary parts separately. Now multiply (5 − 2i)(5 + 2i) = 25 + 10i − 10i − 4i² = 25 + 4 = 29. A complex number times its conjugate always gives a real number.
Example 3
Common mistake: treating i as if i² were +1. In (3 + i)(3 + i) = 9 + 3i + 3i + i², the last term is i² = −1, so the answer is 8 + 6i, not 10 + 6i. Every product of two imaginary parts flips sign. Write i² explicitly, then replace it with −1 as a separate step, and the sign stops getting lost.
Practice questions
- Add (3 + 2i) + (5 − 7i).
- Multiply (1 + i)(1 − i).
- Simplify (2 − 3i)(4 + i).
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Sumar y multiplicar números complejos
Trata la unidad imaginaria i como una variable, pero reemplaza i^2 por −1. Suma números complejos combinando las partes reales e imaginarias; multiplica distribuyendo (FOIL).
Ejemplo 1: Multiplica (2 + 3i)(1 − 4i) = 2 − 8i + 3i − 12i^2 = 2 − 5i − 12(−1) = 2 − 5i + 12 = 14 − 5i.
Ejemplo 2: Suma y luego multiplica: (5 − 2i) + (−3 + 7i) = 2 + 5i, combinando por separado las partes reales y las imaginarias. Ahora multiplica (5 − 2i)(5 + 2i) = 25 + 10i − 10i − 4i² = 25 + 4 = 29. Un número complejo por su conjugado siempre da un número real.
Ejemplo 3: Error común: tratar a i como si i² fuera +1. En (3 + i)(3 + i) = 9 + 3i + 3i + i², el último término es i² = −1, así que la respuesta es 8 + 6i, no 10 + 6i. Todo producto de dos partes imaginarias cambia de signo. Escribe i² explícitamente y luego reemplázalo por −1 en un paso aparte; así el signo deja de perderse.
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