(1+i)^3/(1-i)^3

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Numbers: (1+i)³ / (1-i)³

This article explores the simplification of the complex number expression (1+i)³ / (1-i)³. We'll employ the fundamental properties of complex numbers and algebraic manipulation to arrive at a simplified form.

Understanding Complex Numbers

Complex numbers are expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying i² = -1.

Simplifying the Expression

1. Expanding the Cubes: We begin by expanding the cubes in both numerator and denominator using the binomial theorem or simply multiplying out the terms:

   • (1+i)³ = (1+i)(1+i)(1+i) = (1 + 2i + i²) (1+i) = (2i)(1+i) = 2i + 2i² = -2 + 2i
   • (1-i)³ = (1-i)(1-i)(1-i) = (1 - 2i + i²) (1-i) = (-2i)(1-i) = -2i + 2i² = -2 - 2i

2. Substituting and Simplifying: Now, substitute the expanded values back into the original expression:

   • (1+i)³ / (1-i)³ = (-2 + 2i) / (-2 - 2i)

3. Rationalizing the Denominator: To eliminate the complex number in the denominator, we multiply both numerator and denominator by the complex conjugate of the denominator:

   • (-2 + 2i) / (-2 - 2i) * (-2 + 2i) / (-2 + 2i)

4. Expanding and Simplifying:

   • (4 - 4i - 4i + 4i²) / (4 - 4i + 4i - 4i²) = (4 - 8i - 4) / (4 + 4) = -8i / 8

5. Final Result: Finally, we obtain the simplified form of the expression:

   • (1+i)³ / (1-i)³ = -i

Conclusion

By employing algebraic manipulations and the properties of complex numbers, we have successfully simplified the expression (1+i)³ / (1-i)³ to -i. This process demonstrates the importance of understanding complex number operations and their applications in various mathematical contexts.