(1+2i)(3-2i)

Jasonbradley

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Jun 16, 2024

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Multiplying Complex Numbers: (1 + 2i)(3 - 2i)

This article will walk through the process of multiplying two complex numbers: (1 + 2i) and (3 - 2i).

Understanding Complex Numbers

Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1 (i² = -1).

Multiplication Process

To multiply complex numbers, we use the distributive property (often referred to as FOIL - First, Outer, Inner, Last) just like we do with binomials in algebra.

Here's how we multiply (1 + 2i)(3 - 2i):

1. First: (1)(3) = 3
2. Outer: (1)(-2i) = -2i
3. Inner: (2i)(3) = 6i
4. Last: (2i)(-2i) = -4i²

Now we combine the terms:

3 - 2i + 6i - 4i²

Recall that i² = -1. Substituting this into our expression, we get:

3 - 2i + 6i - 4(-1)

Simplifying further:

3 - 2i + 6i + 4

Combining the real and imaginary terms:

(3 + 4) + (-2 + 6)i

This simplifies to:

7 + 4i

Conclusion

Therefore, the product of (1 + 2i) and (3 - 2i) is 7 + 4i. This demonstrates how to multiply complex numbers using the distributive property and the knowledge that i² = -1.