(1+i)(1-i)^-1

Jasonbradley

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Jun 16, 2024

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Simplifying the Complex Expression: (1+i)(1-i)^-1

This article will guide you through simplifying the complex expression (1+i)(1-i)^-1.

Understanding the Fundamentals

Before diving into the simplification, let's refresh our understanding of complex numbers and their operations:

• Complex Number: 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 (i.e., i² = -1).
• Inverse: The inverse of a complex number (a + bi) is denoted by (a + bi)^-1 and is calculated as 1/(a + bi).

Simplifying the Expression

Let's break down the simplification process step-by-step:

1. Simplify the inverse:

   • (1 - i)^-1 = 1/(1 - i)
   • Multiply the numerator and denominator by the conjugate of (1 - i), which is (1 + i):
   • (1/(1 - i)) * ((1 + i)/(1 + i)) = (1 + i) / (1² - i²)
   • Simplify using i² = -1: (1 + i) / (1 + 1) = (1 + i) / 2

2. Multiply by the simplified inverse:

   • (1 + i) * ((1 + i) / 2) = (1 + i)² / 2

3. Expand the square:

   • (1 + i)² / 2 = (1 + 2i + i²) / 2

4. Simplify using i² = -1:

   • (1 + 2i - 1) / 2 = 2i / 2 = i

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

Conclusion

Simplifying complex expressions often involves applying the properties of complex numbers, including conjugates and the imaginary unit 'i'. By following the steps outlined above, you can effectively simplify complex expressions like (1+i)(1-i)^-1 and arrive at a more manageable form.