(2+5i)⋅(5−3i)

Jasonbradley

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Jun 16, 2024

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Multiplying Complex Numbers: (2 + 5i) ⋅ (5 - 3i)

This article will guide you through the process of multiplying the complex numbers (2 + 5i) and (5 - 3i). We will use the distributive property, similar to how we multiply binomials in algebra.

Understanding Complex Numbers

Before we begin, let's refresh our understanding of 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 (i.e., i² = -1).

The Multiplication Process

1. Distribute: We multiply each term of the first complex number by each term of the second complex number.

   • (2 + 5i) ⋅ (5 - 3i) = (2 * 5) + (2 * -3i) + (5i * 5) + (5i * -3i)

2. Simplify: Perform the multiplications and combine like terms.

   • 10 - 6i + 25i - 15i²

3. Substitute i² with -1: Remember that i² = -1. Substitute this value into the expression.

   • 10 - 6i + 25i - 15(-1)

4. Simplify Further: Combine the real and imaginary terms.

   • (10 + 15) + (-6 + 25)i

5. Final Result: The product of (2 + 5i) and (5 - 3i) is 25 + 19i.

Conclusion

We successfully multiplied the two complex numbers (2 + 5i) and (5 - 3i) using the distributive property. The product is the complex number 25 + 19i. Understanding how to multiply complex numbers is crucial in various mathematical and engineering applications.