(−3+2i)⋅(1−i)

Jasonbradley

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Jun 17, 2024

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Multiplying Complex Numbers: (−3+2i)⋅(1−i)

This article will guide you through the process of multiplying complex numbers, specifically focusing on the expression (−3+2i)⋅(1−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

To multiply complex numbers, we use the distributive property, just like with real numbers. Here's how it works:

1. Distribute: Multiply each term in the first complex number by each term in the second complex number.
2. Simplify: Use the fact that i² = -1 to simplify the resulting expression.
3. Combine: Combine the real terms and the imaginary terms.

Applying the Process

Let's multiply (−3+2i)⋅(1−i):

1. Distribute:

   • (−3)⋅(1) + (−3)⋅(−i) + (2i)⋅(1) + (2i)⋅(−i)

2. Simplify:

   • −3 + 3i + 2i - 2i²
   • −3 + 3i + 2i + 2 (since i² = -1)

3. Combine:

   • (−3 + 2) + (3 + 2)i

Therefore, (−3+2i)⋅(1−i) = −1 + 5i.

Conclusion

We successfully multiplied the complex numbers (−3+2i)⋅(1−i), arriving at the solution −1 + 5i. By applying the distributive property and simplifying using the fact that i² = -1, we can efficiently multiply any two complex numbers. This process is crucial in various mathematical and engineering fields.