(-2-5i).jpeg "(-9+4i)(-2-5i)")
Multiplying Complex Numbers: A Step-by-Step Guide
This article will guide you through the process of multiplying two complex numbers: (-9 + 4i)(-2 - 5i). We'll break down the steps and explain the concepts involved.
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.
The Multiplication Process
To multiply complex numbers, we use the distributive property (also known as FOIL - First, Outer, Inner, Last) similar to how we multiply binomials.
Let's break down the multiplication of (-9 + 4i)(-2 - 5i):
- Multiply the First terms: (-9)(-2) = 18
- Multiply the Outer terms: (-9)(-5i) = 45i
- Multiply the Inner terms: (4i)(-2) = -8i
- Multiply the Last terms: (4i)(-5i) = -20i²
Now, we have: 18 + 45i - 8i - 20i²
Remember that i² = -1. Substitute this in the expression:
18 + 45i - 8i - 20(-1)
Simplify by combining like terms:
(18 + 20) + (45 - 8)i
This gives us the final result: 38 + 37i
Conclusion
Therefore, the product of (-9 + 4i) and (-2 - 5i) is 38 + 37i. This example illustrates the process of multiplying complex numbers. Remember the distributive property and the key identity i² = -1, and you'll be able to multiply complex numbers confidently.