%E2%8B%85(%E2%88%922%2B2i).jpeg "(3−3i)⋅(−2+2i)")
Multiplying Complex Numbers: (3−3i)⋅(−2+2i)
This article will guide you through the multiplication of two complex numbers: (3−3i)⋅(−2+2i).
Understanding Complex Numbers
Complex numbers are numbers that extend the real number system by including the imaginary unit i, where i² = -1. They are expressed in the form a + bi, where a and b are real numbers.
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property, just like we do with real numbers. This means we multiply each term in the first complex number by each term in the second complex number.
Step 1: Distribute
(3−3i)⋅(−2+2i) = (3)(-2) + (3)(2i) + (-3i)(-2) + (-3i)(2i)
Step 2: Simplify
= -6 + 6i + 6i - 6i²
Step 3: Substitute i² with -1
= -6 + 6i + 6i - 6(-1)
Step 4: Combine Real and Imaginary Terms
= -6 + 6 + 6i + 6i
= 12i
Conclusion
Therefore, the product of (3−3i)⋅(−2+2i) is 12i. This example demonstrates how to multiply complex numbers and highlights the importance of understanding the properties of the imaginary unit i.