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

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Fractions: (3 + 4i) / (2 - i)

This article will guide you through the process of simplifying the complex fraction (3 + 4i) / (2 - 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.

The Process of Simplification

To simplify a complex fraction, we need to get rid of the imaginary number in the denominator. We achieve this by multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi.

Here's how we simplify (3 + 4i) / (2 - i):

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

2. Multiply both the numerator and denominator by the conjugate: (3 + 4i) / (2 - i) * (2 + i) / (2 + i)

3. Expand the numerator and denominator: (3 * 2) + (3 * i) + (4i * 2) + (4i * i) / (2 * 2) + (2 * i) + (-i * 2) + (-i * i)

4. Simplify the terms: 6 + 3i + 8i - 4 / 4 + 2i - 2i + 1

5. Combine real and imaginary terms: (6 - 4) + (3 + 8)i / 5

6. Final Result: (2 + 11i) / 5

Therefore, the simplified form of (3 + 4i) / (2 - i) is (2 + 11i) / 5.

Summary

Simplifying complex fractions involves multiplying both the numerator and denominator by the conjugate of the denominator. This eliminates the imaginary term from the denominator, resulting in a simpler expression.