(1+2i)/(2-3i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Numbers: (1 + 2i) / (2 - 3i)

This article will guide you through the process of simplifying the complex number expression (1 + 2i) / (2 - 3i).

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).

Simplifying the Expression

To simplify the expression (1 + 2i) / (2 - 3i), we need to get rid of the imaginary unit 'i' in the denominator. We do this by multiplying both the numerator and denominator by the complex conjugate of the denominator.

The complex conjugate of a complex number a + bi is a - bi.

1. Find the complex conjugate of the denominator: The complex conjugate of (2 - 3i) is (2 + 3i).

2. Multiply the numerator and denominator by the complex conjugate:

   (1 + 2i) / (2 - 3i) * (2 + 3i) / (2 + 3i)

3. Expand the numerator and denominator:

   (1 * 2 + 1 * 3i + 2i * 2 + 2i * 3i) / (2 * 2 + 2 * 3i - 3i * 2 - 3i * 3i)

4. Simplify using i² = -1:

   (2 + 3i + 4i - 6) / (4 + 6i - 6i + 9)

5. Combine like terms:

   (-4 + 7i) / (13)

6. Write the result in standard complex number form:

   -4/13 + 7/13i

Conclusion

Therefore, the simplified form of (1 + 2i) / (2 - 3i) is -4/13 + 7/13i. This process demonstrates how to manipulate complex numbers by eliminating the imaginary unit from the denominator, resulting in a standard complex number form.