(2+i)/(1-4i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Fractions: (2 + i) / (1 - 4i)

This article will guide you through the process of simplifying the complex fraction (2 + i) / (1 - 4i).

Understanding Complex Numbers

Before diving into the simplification, let's recall some key concepts about 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.
• Real part: The real part of a complex number is the term without 'i'. In this case, the real part is '2' in (2 + i) and '1' in (1 - 4i).
• Imaginary part: The imaginary part of a complex number is the term with 'i'. In this case, the imaginary part is '1' in (2 + i) and '-4' in (1 - 4i).

Simplifying the Fraction

To simplify the fraction (2 + i) / (1 - 4i), we need to get rid of the imaginary term in the denominator. We can achieve this by multiplying both the numerator and denominator by the conjugate of the denominator.

The conjugate of (1 - 4i) is (1 + 4i). The conjugate of a complex number is simply obtained by changing the sign of the imaginary part.

Here's the step-by-step process:

1. Multiply numerator and denominator by the conjugate: (2 + i) / (1 - 4i) * (1 + 4i) / (1 + 4i)

2. Expand the numerator and denominator: (2 + 8i + i + 4i²) / (1 + 4i - 4i - 16i²)

3. Simplify using i² = -1: (2 + 9i - 4) / (1 - 16(-1))

4. Combine real and imaginary terms: (-2 + 9i) / 17

5. Separate the real and imaginary parts: -2/17 + 9/17i

Final Answer

Therefore, the simplified form of (2 + i) / (1 - 4i) is -2/17 + 9/17i.