%E2%8B%85(%E2%88%921%2Bi).jpeg "(−1−3i)⋅(−1+i)")
Multiplying Complex Numbers: A Step-by-Step Example
This article will demonstrate how to multiply two complex numbers, specifically (−1−3i)⋅(−1+i).
Understanding Complex Numbers
Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as √-1.
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property (also known as FOIL method) just like we do with binomials.
1. Expand the product:
(-1 - 3i) ⋅ (-1 + i) = (-1)⋅(-1) + (-1)⋅(i) + (-3i)⋅(-1) + (-3i)⋅(i)
2. Simplify:
1 + i + 3i + 3i²
3. Substitute i² with -1:
1 + i + 3i + 3(-1)
4. Combine real and imaginary terms:
(1 - 3) + (1 + 3)i
5. Final Result:
(-1−3i)⋅(−1+i) = -2 + 4i
Conclusion
Therefore, the product of (-1 - 3i) and (-1 + i) is -2 + 4i. This process demonstrates how to multiply complex numbers using the distributive property and understanding the properties of the imaginary unit 'i'.