(-5%2B4i).jpeg "(4-3i)(-5+4i)")
Multiplying Complex Numbers: A Step-by-Step Guide
This article will guide you through the process of multiplying two complex numbers: (4 - 3i)(-5 + 4i).
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 (i² = -1).
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property, just like we would with any binomial expression.
-
Distribute the first term: (4 - 3i)(-5 + 4i) = (4)(-5) + (4)(4i) + (-3i)(-5) + (-3i)(4i)
-
Simplify each term: = -20 + 16i + 15i - 12i²
-
Remember i² = -1: = -20 + 16i + 15i - 12(-1)
-
Combine real and imaginary terms: = (-20 + 12) + (16 + 15)i
-
Simplify the final result: = -8 + 31i
Therefore, the product of (4 - 3i) and (-5 + 4i) is -8 + 31i.
Visualizing Complex Multiplication
It's helpful to visualize complex multiplication geometrically. The product of two complex numbers can be represented by the product of their magnitudes and the sum of their angles. This allows us to see how complex multiplication rotates and scales the complex numbers in the complex plane.
Conclusion
Multiplying complex numbers is a straightforward process that follows the same principles as multiplying binomials. By using the distributive property and remembering that i² = -1, we can simplify the expression to obtain the product in the form a + bi. Understanding complex multiplication is crucial for various mathematical applications, including algebra, calculus, and electrical engineering.