(1+i)^4(1+1/i)^4

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: (1 + i)^4 (1 + 1/i)^4

This article will explore the simplification of the complex expression (1 + i)^4 (1 + 1/i)^4. We will utilize the properties of complex numbers and De Moivre's Theorem to achieve a concise solution.

Understanding the Problem

We have a product of two complex numbers raised to the power of 4. Our goal is to simplify this expression into a form that is easier to understand and work with.

Solving the Expression

1. Simplifying the Terms:

   • (1 + 1/i): To simplify this, we multiply both numerator and denominator by i:

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

2. Using De Moivre's Theorem:

   De Moivre's Theorem states: (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ). To apply this, we need to express our complex numbers in polar form (magnitude and angle).

   • (1 + i):

     • Magnitude: √(1² + 1²) = √2
     • Angle: tan⁻¹(1/1) = 45° or π/4 radians.
     • Polar Form: √2 (cos π/4 + i sin π/4)

   • (1 + i)/i:

     • Magnitude: √(1² + 1²) / |i| = √2 / 1 = √2
     • Angle: tan⁻¹(1/1) - 90° = -45° or -π/4 radians.
     • Polar Form: √2 (cos -π/4 + i sin -π/4)

3. Applying De Moivre's Theorem:

   • (1 + i)^4 = [√2 (cos π/4 + i sin π/4)]^4 = 2² (cos π + i sin π)
   • [(1 + i)/i]^4 = [√2 (cos -π/4 + i sin -π/4)]^4 = 2² (cos -π + i sin -π)

4. Simplifying:

   • (1 + i)^4 = 4(-1 + 0i) = -4
   • [(1 + i)/i]^4 = 4(-1 + 0i) = -4

5. Final Calculation:

   (1 + i)^4 (1 + 1/i)^4 = (-4) * (-4) = 16

Conclusion

We have successfully simplified the complex expression (1 + i)^4 (1 + 1/i)^4 to the real number 16. This process involved understanding complex number properties, applying De Moivre's Theorem, and simplifying the results. This example illustrates the power and efficiency of working with complex numbers in polar form.