%E2%8B%85(%E2%88%924%2B2i).jpeg "(1−4i)⋅(−4+2i)")
Multiplying Complex Numbers: (1-4i) * (-4+2i)
This article explores the multiplication of two complex numbers: (1-4i) and (-4+2i). We will go through the steps and provide the final result.
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 follow the distributive property similar to multiplying binomials.
Step 1: Expand the expression: (1-4i) * (-4+2i) = 1 * (-4) + 1 * (2i) + (-4i) * (-4) + (-4i) * (2i)
Step 2: Simplify the expression: = -4 + 2i + 16i - 8i²
Step 3: Remember that i² = -1. Substitute this value: = -4 + 2i + 16i - 8(-1)
Step 4: Combine real and imaginary terms: = (-4 + 8) + (2 + 16)i
Step 5: Simplify the expression: = 4 + 18i
Conclusion
Therefore, the product of (1-4i) and (-4+2i) is 4 + 18i. This result is also a complex number, demonstrating that the multiplication of complex numbers results in another complex number.