%E2%8B%85(4%2Bi).jpeg "(1−2i)⋅(4+i)")
Multiplying Complex Numbers: (1-2i)⋅(4+i)
This article will guide you through the process of multiplying two complex numbers: (1-2i) and (4+i).
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 (i.e., i² = -1).
Multiplying Complex Numbers
Multiplying complex numbers is similar to multiplying binomials, using the distributive property. We multiply each term in the first complex number by each term in the second complex number.
The Calculation
Let's multiply (1-2i) and (4+i):
(1 - 2i) ⋅ (4 + i) = 1(4) + 1(i) - 2i(4) - 2i(i)
Expanding the terms:
= 4 + i - 8i - 2i²
Remember that i² = -1. Substitute this value:
= 4 + i - 8i - 2(-1)
Simplify:
= 4 + i - 8i + 2
Combine the real and imaginary terms:
= (4 + 2) + (1 - 8)i
= 6 - 7i
Conclusion
Therefore, the product of (1 - 2i) and (4 + i) is 6 - 7i.