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Magazine Chapter 2: Problem 6
\(\left(\frac{1-i}{1+i}\right)^{100}=a+i b\), then (a) \(a=2, b=-1\) (b) \(a=1, b=0\) (c) \(a=0, b=1\) (d) \(a=-1, b=2\)
Short Answer
Expert verified
The answer is (b) \(a=1, b=0\).
Step by step solution
01
Simplify the Expression
Start by simplifying the expression \(\frac{1-i}{1+i}\). Multiply both the numerator and denominator by the conjugate of the denominator, \(1-i\). This results in \(\frac{(1-i)(1-i)}{(1+i)(1-i)}\).
02
Calculate the Denominator
Calculate \((1+i)(1-i)\), which is the difference of squares. This simplifies to \(1^2 - i^2 = 1 - (-1) = 2\).
03
Calculate the Numerator
Expand \((1-i)(1-i)\) using the distributive property: \(1 - 2i + i^2\). Since \(i^2 = -1\), it simplifies to \(1 - 2i - 1 = -2i\).
04
Simplify Fraction
The expression is now simplified to \(-\frac{2i}{2} = -i\).
05
Powers of \(-i\)
Since \(-i = e^{-i\frac{\pi}{2}}\), calculating \((-i)^{100}\) is equivalent to \(e^{-i\frac{100\pi}{2}} = e^{-50\pi i}\).
06
Apply Euler's Formula
Using Euler's formula, \(e^{-50\pi i} = \cos(-50\pi) + i\sin(-50\pi)\). Simplify \(-50\pi\) by noting it is 25 full cycles of \(2\pi\), resulting in \(\cos(0) + i\sin(0)\).
07
Determine Final Values
Since \(\cos(0) = 1\) and \(\sin(0) = 0\), the result is \(1 + 0i\), hence \(a=1\) and \(b=0\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Conjugate
When working with complex numbers, the conjugate plays a crucial role. For a complex number expressed as \(a + bi\), its conjugate is \(a - bi\). The conjugate essentially "flips" the sign of the imaginary part.
This concept becomes incredibly useful when simplifying complex fractions. To eliminate a complex number in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator.
In our exercise, the complex fraction \(\frac{1-i}{1+i}\) was simplified by multiplying with the conjugate. The conjugate of \(1+i\) is \(1-i\). This multiplication helps in converting the denominator into a real number, simplifying the process immensely.
Key points to remember about conjugates include:
This concept becomes incredibly useful when simplifying complex fractions. To eliminate a complex number in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator.
In our exercise, the complex fraction \(\frac{1-i}{1+i}\) was simplified by multiplying with the conjugate. The conjugate of \(1+i\) is \(1-i\). This multiplication helps in converting the denominator into a real number, simplifying the process immensely.
Key points to remember about conjugates include:
- Conjugates always result in a real number in the denominator when multiplied.
- This strategy is crucial for simplifying expressions involving division of complex numbers.
- Conjugates help in various algebraic manipulations and can simplify complex arithmetic operations greatly.
Euler's Formula
Euler's Formula is a fundamental bridge between complex numbers and trigonometry. This powerful formula is expressed as:
\[ e^{ix} = \cos(x) + i\sin(x) \]
Euler's formula allows us to represent complex exponential expressions simply in terms of real and imaginary parts.
In the exercise, we used this formula to convert the expression \(-i\) into its exponential form. By expressing \(-i\) as \(e^{-i\frac{\pi}{2}}\), it became easier to calculate \((-i)^{100}\) by raising it to the power of 100, thus simplifying it to \(e^{-50\pi i}\).
Key insights about Euler's formula include:
\[ e^{ix} = \cos(x) + i\sin(x) \]
Euler's formula allows us to represent complex exponential expressions simply in terms of real and imaginary parts.
In the exercise, we used this formula to convert the expression \(-i\) into its exponential form. By expressing \(-i\) as \(e^{-i\frac{\pi}{2}}\), it became easier to calculate \((-i)^{100}\) by raising it to the power of 100, thus simplifying it to \(e^{-50\pi i}\).
Key insights about Euler's formula include:
- It simplifies exponentiation of complex numbers, especially those involving powers of \(i\).
- It connects exponential expressions to trigonometric functions, which is helpful in various fields like signal processing and physics.
- Euler's formula helps to easily identify rotations in the complex plane through angle \(x\).
Powers of i
Understanding powers of the imaginary unit \(i\) is critical in dealing with complex numbers. The imaginary unit \(i\) has a unique property:
\(i^2 = -1\)
This property leads to a regular cycle as we raise \(i\) to higher powers:
Within the given exercise, the concept of \(-i\), which is equivalent to \(i^3\), played a key role. Recognizing that multiplying \(-i\) results in cycles similar to those of \(i\), we used this cyclic behavior to compute powers of \(-i\).
Important aspects of \(i\) include:
\(i^2 = -1\)
This property leads to a regular cycle as we raise \(i\) to higher powers:
- \(i^1 = i\)
- \(i^2 = -1\)
- \(i^3 = -i\)
- \(i^4 = 1\)
- Repeat...
Within the given exercise, the concept of \(-i\), which is equivalent to \(i^3\), played a key role. Recognizing that multiplying \(-i\) results in cycles similar to those of \(i\), we used this cyclic behavior to compute powers of \(-i\).
Important aspects of \(i\) include:
- Recognizing its cycle helps simplify powers of \(i\) beyond 4 quite easily.
- Understanding the rotation by \(\frac{\pi}{2}\) radii on the complex plane through powers of \(i\).
- Enabling transformations into exponential forms for more complex calculations when combined with Euler's formula.
