(1+i)/(2-5i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Fractions: (1 + i) / (2 - 5i)

In mathematics, 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.

When working with complex numbers, it's common to encounter fractions where both the numerator and denominator are complex numbers. To simplify these fractions, we use a technique called conjugate multiplication.

The Conjugate

The conjugate of a complex number a + bi is a - bi. Multiplying a complex number by its conjugate results in a real number.

Simplifying (1 + i) / (2 - 5i)

1. Identify the conjugate: The conjugate of (2 - 5i) is (2 + 5i).

2. Multiply both numerator and denominator by the conjugate:

   (1 + i) / (2 - 5i) * (2 + 5i) / (2 + 5i)

3. Expand:

   (1 + i)(2 + 5i) / (2 - 5i)(2 + 5i) = (2 + 5i + 2i + 5i²) / (4 + 10i - 10i - 25i²)

4. Simplify using i² = -1:

   (2 + 7i - 5) / (4 + 25) = (-3 + 7i) / 29

5. Express in standard form:

   (-3/29) + (7/29)i

Therefore, the simplified form of (1 + i) / (2 - 5i) is (-3/29) + (7/29)i.