(-1+4i)(9-4i)

Jasonbradley

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Jun 16, 2024

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Multiplying Complex Numbers: (-1 + 4i)(9 - 4i)

This article will guide you through the process of multiplying two complex numbers, (-1 + 4i) and (9 - 4i).

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 (i² = -1).

Multiplication Process

To multiply complex numbers, we use the distributive property, just like with real numbers:

1. Expand the product: (-1 + 4i)(9 - 4i) = (-1 * 9) + (-1 * -4i) + (4i * 9) + (4i * -4i)

2. Simplify: = -9 + 4i + 36i - 16i²

3. Substitute i² with -1: = -9 + 4i + 36i - 16(-1)

4. Combine real and imaginary terms: = (-9 + 16) + (4 + 36)i

5. Final Result: = 7 + 40i

Therefore, the product of (-1 + 4i) and (9 - 4i) is 7 + 40i.

Conclusion

Multiplying complex numbers involves applying the distributive property and substituting i² with -1. The result is another complex number, expressed in the form a + bi. This process is essential for various applications in mathematics, physics, and engineering.