(2-3i)(4+5i)

2 min read Jun 16, 2024
(2-3i)(4+5i)

Multiplying Complex Numbers: A Step-by-Step Guide

This article will guide you through the process of multiplying two complex numbers: (2 - 3i)(4 + 5i).

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.

Multiplying Complex Numbers

To multiply complex numbers, we use the distributive property (also known as FOIL method), just like multiplying binomials:

  1. Multiply the first terms: (2)(4) = 8
  2. Multiply the outer terms: (2)(5i) = 10i
  3. Multiply the inner terms: (-3i)(4) = -12i
  4. Multiply the last terms: (-3i)(5i) = -15i²

Now, remember that i² = -1. So, we can substitute -1 for i² in the last term.

This gives us: 8 + 10i - 12i - 15(-1)

Simplifying the Result

Combining the real and imaginary terms, we get:

8 + 15 + 10i - 12i = 23 - 2i

Final Answer

Therefore, the product of the complex numbers (2 - 3i) and (4 + 5i) is 23 - 2i.

Key Takeaway

Multiplying complex numbers involves treating them like binomials and utilizing the distributive property. Remember to simplify the terms by substituting -1 for i² in the final steps.

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