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

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression: (1-i)² / (1+i)²

This article explores the simplification of the complex expression (1-i)² / (1+i)². We will utilize the fundamental properties of complex numbers to arrive at a concise and simplified result.

Understanding Complex Numbers

Before diving into the simplification, let's briefly recap some key points about complex numbers:

• Imaginary Unit (i): Defined as the square root of -1, denoted by 'i'.
• Complex Number: A number of the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit.

Simplifying the Expression

1. Expanding the Squares:

   We start by expanding the squares in the numerator and denominator:

   (1-i)² = (1-i)(1-i) = 1 - 2i + i² = 1 - 2i - 1 = -2i

   (1+i)² = (1+i)(1+i) = 1 + 2i + i² = 1 + 2i - 1 = 2i

2. Substituting the Expanded Forms:

   Now we can substitute the expanded forms back into the original expression:

   (-2i) / (2i)

3. Simplifying the Expression:

   Finally, we simplify by canceling out the common factor of 2i:

   (-2i) / (2i) = -1

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

Conclusion

This exercise demonstrates how to effectively simplify complex expressions involving complex numbers. By understanding the fundamental properties of complex numbers and applying basic algebraic operations, we can achieve a concise and simplified result.