(1−5i)⋅(3−2i)

Jasonbradley

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Jun 16, 2024

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Multiplying Complex Numbers: (1 - 5i) * (3 - 2i)

This article will demonstrate how to multiply two complex numbers, specifically (1 - 5i) * (3 - 2i). We will use the distributive property, similar to how we multiply binomials.

Understanding Complex Numbers

A complex number is a number of the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit. The imaginary unit 'i' is defined as the square root of -1 (i² = -1).

Multiplication Process

1. Distribute: Apply the distributive property, multiplying each term in the first complex number by each term in the second complex number.

   (1 - 5i) * (3 - 2i) = (1 * 3) + (1 * -2i) + (-5i * 3) + (-5i * -2i)

2. Simplify: Combine like terms and remember that i² = -1.

   = 3 - 2i - 15i + 10i² = 3 - 17i + 10(-1) = 3 - 17i - 10

3. Combine Real and Imaginary Components: Group the real terms and the imaginary terms.

   = (3 - 10) - 17i = -7 - 17i

Conclusion

Therefore, the product of (1 - 5i) and (3 - 2i) is -7 - 17i.