(1+i)/(1-i)

Jasonbradley

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Jun 16, 2024

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

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

Understanding Complex Numbers

Before diving into the simplification, let's quickly recap what complex numbers are:

• 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 (1 + i) / (1 - i), we use a technique called multiplying by the conjugate:

1. Identify the conjugate: The conjugate of (1 - i) is (1 + i).
2. Multiply the numerator and denominator by the conjugate: (1 + i) / (1 - i) * (1 + i) / (1 + i)
3. Expand the numerator and denominator: (1 + i + i + i²) / (1 + i - i - i²)
4. Simplify using i² = -1: (1 + 2i - 1) / (1 + 1)
5. Combine like terms: (2i) / (2)
6. Simplify the fraction: i

Conclusion

Therefore, the simplified form of (1 + i) / (1 - i) is i. This process demonstrates how to manipulate complex numbers effectively and efficiently, ultimately arriving at a simplified form.