(1+i)/(2-i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Fractions: (1 + i) / (2 - i)

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

Understanding Complex Numbers

Before diving into the simplification, let's refresh our understanding of complex numbers. A complex number is a number 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).

Simplifying Complex Fractions

To simplify a complex fraction, we need to eliminate the complex number from the denominator. We achieve 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.

Let's apply this to our problem:

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

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

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

3. Expand the products:

   [(1 + i)(2 + i)] / [(2 - i)(2 + i)]

   [2 + i + 2i + i²] / [4 - i²]

4. Substitute i² with -1:

   [2 + i + 2i - 1] / [4 + 1]

   [1 + 3i] / [5]

5. Express the result in the form a + bi:

   1/5 + (3/5)i

Conclusion

The simplified form of the complex fraction (1 + i) / (2 - i) is 1/5 + (3/5)i.

By multiplying both the numerator and denominator by the complex conjugate of the denominator, we successfully eliminated the complex number from the denominator and expressed the result in the standard form of a complex number.