Operations with Complex Numbers
Aligned to N-CN.A.2 — Common Core State Standards for Mathematics.
What this lesson teaches
Add and subtract complex numbers by combining real parts and imaginary parts separately. Multiply by using the distributive property, then replace i^2 with −1.
Worked examples
Example 1
Multiply (2 + 3i)(4 − i). Distribute: 2×4 + 2×(−i) + 3i×4 + 3i×(−i) = 8 − 2i + 12i − 3i^2. Since i^2 = −1, −3i^2 = +3, giving 8 + 3 + 10i = 11 + 10i.
Example 2
Divide (3 + 2i) by (1 − i). Multiply top and bottom by the conjugate of the denominator, 1 + i. Numerator: (3 + 2i)(1 + i) = 3 + 3i + 2i + 2i² = 3 + 5i − 2 = 1 + 5i. Denominator: (1 − i)(1 + i) = 1 − i² = 1 + 1 = 2. The quotient is (1 + 5i)/2 = 0.5 + 2.5i.
Example 3
Common mistake: squaring a complex number by squaring each part. (2 + 3i)² is not 4 + 9i². Treat it as a binomial: (2 + 3i)(2 + 3i) = 4 + 6i + 6i + 9i² = 4 + 12i − 9 = −5 + 12i. The middle term 12i is exactly what the shortcut drops, and it is usually the largest piece of the answer.
Practice questions
- Add (5 − 2i) + (−3 + 6i).
- Subtract (7 + i) − (4 − 3i).
- Multiply (1 + i)(1 − 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
Operaciones con Números Complejos
Suma y resta números complejos combinando por separado las partes reales y las partes imaginarias. Multiplica usando la propiedad distributiva y luego reemplaza i^2 por −1.
Ejemplo 1: Multiplica (2 + 3i)(4 − i). Distribuye: 2×4 + 2×(−i) + 3i×4 + 3i×(−i) = 8 − 2i + 12i − 3i^2. Como i^2 = −1, −3i^2 = +3, resultando 8 + 3 + 10i = 11 + 10i.
Ejemplo 2: Divide (3 + 2i) entre (1 − i). Multiplica arriba y abajo por el conjugado del denominador, 1 + i. Numerador: (3 + 2i)(1 + i) = 3 + 3i + 2i + 2i² = 3 + 5i − 2 = 1 + 5i. Denominador: (1 − i)(1 + i) = 1 − i² = 1 + 1 = 2. El cociente es (1 + 5i)/2 = 0.5 + 2.5i.
Ejemplo 3: Error común: elevar al cuadrado un número complejo elevando cada parte. (2 + 3i)² no es 4 + 9i². Trátalo como binomio: (2 + 3i)(2 + 3i) = 4 + 6i + 6i + 9i² = 4 + 12i − 9 = −5 + 12i. El término central 12i es justo lo que el atajo elimina, y suele ser la parte más grande de la respuesta.
More Number & Quantity lessons
Teach this lesson today
An AI tutor that guides with questions instead of just giving answers — all K–12 core subjects, every child in the family.
Start free — no card required