Problem 6 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

\(\left(\frac{1-i}{1+i}\right)^{100}=a+i b\), then (a) \(a=2, b=-1\) (b) \(a=1, b=0\) (c) \(a=0, b=1\) (d) \(a=-1, b=2\)

Step-by-Step Solution

Answer

The answer is (b) \(a=1, b=0\).

Step 1: 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 \(\frac{(1-i)(1-i)}{(1+i)(1-i)}\).

Step 2: Calculate the Denominator

Calculate \((1+i)(1-i)\), which is the difference of squares. This simplifies to \(1^2 - i^2 = 1 - (-1) = 2\).

Step 3: Calculate the Numerator

Expand \((1-i)(1-i)\) using the distributive property: \(1 - 2i + i^2\). Since \(i^2 = -1\), it simplifies to \(1 - 2i - 1 = -2i\).

Step 4: Simplify Fraction

The expression is now simplified to \(-\frac{2i}{2} = -i\).

Step 5: Powers of \(-i\)

Since \(-i = e^{-i\frac{\pi}{2}}\), calculating \((-i)^{100}\) is equivalent to \(e^{-i\frac{100\pi}{2}} = e^{-50\pi i}\).

Step 6: 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 \(\cos(0) + i\sin(0)\).

Step 7: Determine Final Values

Since \(\cos(0) = 1\) and \(\sin(0) = 0\), the result is \(1 + 0i\), hence \(a=1\) and \(b=0\).

Key Concepts

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:

• 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:

• 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:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)
• Repeat...

These results repeat every four cycles, making calculations with \(i\) highly predictable.

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.