%E2%8B%85(%E2%88%923%2B2i).jpeg "(3−4i)⋅(−3+2i)")
Multiplying Complex Numbers: (3−4i)⋅(−3+2i)
This article will guide you through the process of multiplying the complex numbers (3−4i) and (−3+2i).
Understanding Complex Numbers
Before diving into the multiplication, let's briefly understand complex numbers. A complex number is a number 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).
Multiplication Process
To multiply complex numbers, we can use the distributive property (or FOIL method) just like with binomials in algebra:
-
Expand the product: (3−4i)⋅(−3+2i) = 3(−3) + 3(2i) - 4i(−3) - 4i(2i)
-
Simplify the terms: = -9 + 6i + 12i - 8i²
-
Substitute i² = -1: = -9 + 6i + 12i - 8(-1)
-
Combine real and imaginary terms: = -9 + 8 + 6i + 12i
-
Final result: = -1 + 18i
Therefore, the product of (3−4i) and (−3+2i) is -1 + 18i.
Visualizing the Result
The multiplication of complex numbers can be visualized geometrically. Each complex number can be represented as a point in the complex plane (where the x-axis represents real numbers and the y-axis represents imaginary numbers). The product of two complex numbers can be understood in terms of rotation and scaling in this plane.
Conclusion
Multiplying complex numbers involves the same basic principles as multiplying binomials in algebra. By applying the distributive property and using the fact that i² = -1, we can arrive at a simplified result in the form of another complex number. Understanding complex numbers and their operations is crucial in various fields like electrical engineering, physics, and mathematics.