How do you add and multiply complex numbers?

Adding Complex Numbers

• Write complex numbers as a + bi and c + di.
• Add real parts: a + c.
• Add imaginary parts: b + d.
• Result: (a + c) + (b + d)i.

Example:

• (3 + 2i) + (1 + 5i) = (3 + 1) + (2 + 5)i = 4 + 7i.

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Multiplying Complex Numbers

• Use distributive property (like binomials):

(a + bi) × (c + di).

• Expand:

ac + adi + bci + bdi².

• Remember i² = -1.
• Combine like terms:
• Real: ac - bd
• Imaginary: (ad + bc)i
• Result: (ac - bd) + (ad + bc)i.

Example:

• (2 + 3i) × (4 + i)
• = (2 × 4) + (2 × i) + (3i × 4) + (3i × i)
• = 8 + 2i + 12i + 3i²
• Replace i² with -1:

= 8 + 14i + 3(-1)

• = 8 + 14i - 3
• = 5 + 14i

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Takeaways:

• Add or subtract real and imaginary parts separately.
• Multiply using distributive property, and replace i² with -1.