(4%2B5i).jpeg "(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:
- Multiply the first terms: (2)(4) = 8
- Multiply the outer terms: (2)(5i) = 10i
- Multiply the inner terms: (-3i)(4) = -12i
- 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.