%5E2016%2F(1-i)%5E2014.jpeg "(1+i)^2016/(1-i)^2014")
Simplifying Complex Expressions: (1 + i)^2016 / (1 - i)^2014
This article will explore the simplification of the complex expression (1 + i)^2016 / (1 - i)^2014. We will utilize the properties of complex numbers and the polar form to reach a concise solution.
Understanding Complex Numbers
Complex numbers are numbers of 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).
Polar Form
The polar form of a complex number represents it in terms of its magnitude (or modulus) and its angle (or argument).
Let z = a + bi be a complex number. Its magnitude is denoted by |z| and its angle by θ. Then, we can express z in polar form as:
z = |z| (cos θ + i sin θ)
Simplifying the Expression
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Expressing in Polar Form:
- (1 + i): |1 + i| = √2 and θ = 45° (or π/4 radians). Therefore, 1 + i = √2 (cos 45° + i sin 45°).
- (1 - i): |1 - i| = √2 and θ = -45° (or -π/4 radians). Therefore, 1 - i = √2 (cos -45° + i sin -45°).
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Using De Moivre's Theorem:
De Moivre's theorem states that for any complex number z = r(cos θ + i sin θ) and any integer n,
(z)^n = r^n (cos nθ + i sin nθ).Applying this to our expression:
(1 + i)^2016 = (√2)^2016 (cos 2016 * 45° + i sin 2016 * 45°) (1 - i)^2014 = (√2)^2014 (cos 2014 * -45° + i sin 2014 * -45°)
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Simplifying the Expression:
(1 + i)^2016 / (1 - i)^2014 = [(√2)^2016 (cos 2016 * 45° + i sin 2016 * 45°)] / [(√2)^2014 (cos 2014 * -45° + i sin 2014 * -45°)]
Since (√2)^2016 / (√2)^2014 = 2, we have:
= 2 (cos 90480° + i sin 90480°) / (cos -90360° + i sin -90360°)
= 2 (cos 480° + i sin 480°) / (cos -360° + i sin -360°)
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Final Simplification:
Since the cosine and sine functions have a period of 360°, cos 480° = cos 120° and sin 480° = sin 120°. Similarly, cos -360° = cos 0° and sin -360° = sin 0°.
Therefore,
(1 + i)^2016 / (1 - i)^2014 = 2 (cos 120° + i sin 120°) / (cos 0° + i sin 0°)
= 2 (-1/2 + i√3/2) / (1 + 0i)
= -1 + i√3
Conclusion
By utilizing the polar form of complex numbers and De Moivre's theorem, we successfully simplified the complex expression (1 + i)^2016 / (1 - i)^2014 to -1 + i√3. This demonstrates how understanding the properties of complex numbers and their representations can greatly simplify complex mathematical expressions.