Problem 60 - Algebra and Trigonometry

Question

Explain how to multiply complex numbers and give an example.

Step-by-Step Solution

Answer

To multiply two complex numbers \(a + bi\) and \(c + di\), distribute each part of the first complex number to each part of the second, simplify and don't forget that \(i^2 = -1\). For the two complex numbers \(3 + 4i\) and \(1 + 2i\), the product is \(-5 + 10i\).

Step 1: Understand complex numbers

Complex numbers have a real part and an imaginary part, typically written as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

Step 2: Multiply complex numbers

When multiplying complex numbers, distribute each part of the first complex number to each part of the second complex number. For instance, given two complex numbers \(a + bi\) and \(c + di\), their product would be \((a + bi) * (c + di) = ac + adi + bci - bd\). And simplifying this expression will yield \(ac - bd + (ad + bc)i\).

Step 3: Give an example

Consider two complex numbers \(3 + 4i\) and \(1 + 2i\). Multiplying them as per the above rules gives \(3*(1) + 3*(2i) + 4i*(1) + 4i*2i = 3 + 6i + 4i - 8 = -5 + 10i\).