Question

Find one or more values of each of the following complex expressions and compare with a computer solution. $$\left(\frac{1+i}{1-i}\right)^{2718}$$

Short answer

-1

Step-by-step solution

- Simplify the Fraction

Start by simplifying the fraction \(\frac{1+i}{1-i}\). To do this, multiply the numerator and the denominator by the conjugate of the denominator, which is \(1+i\). The expression becomes \(\frac{(1+i)(1+i)}{(1-i)(1+i)}= \frac{1 + 2i + i^2}{1 + i - i(1 - i^2)} \).

- Simplify Further

Since \(i^2 = -1\), simplify the expression to get \( \frac{1 + 2i - 1}{1 - (-1)} = \frac{2i}{2} = i \).

- Evaluate the Power

Now that the simplified expression is \(i\), compute \(i^{2718}\). Recall that \(i^{4k} = 1\), \(i^{4k+1} = i\), \(i^{4k+2} = -1\), and \(i^{4k+3} = -i\) for any integer \(k\). To find \(i^{2718}\), note that \(2718 \text{ mod } 4 = 2\), so \(i^{2718} = i^{4\bullet679+2} = (i^4)^{679} \bullet i^2 = 1^{679} \bullet (-1) = -1 \).

- Compare with Computer Solution

Verify the result \(-1\) using a calculator or computer algebra system to confirm the manual calculation.

Key concepts

Complex Expressions

Complex expressions involve numbers in the form of a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit (satisfying \(i^2 = -1\)).
To simplify complex expressions, often you will need to use operations like addition, subtraction, multiplication, and division.
When dealing with division, it's essential to multiply both the numerator and the denominator by the conjugate of the denominator to rationalize it.
For example, consider the expression \(\frac{1+i}{1-i}\):

• Multiply the numerator and the denominator by the conjugate of the denominator, which is \(1+i\).
• Simplify the resulting expression by expanding and combining like terms.

Practicing these steps can help you handle more complex operations involving complex numbers.

Powers of i

Understanding the powers of the imaginary unit 'i' is crucial when working with complex numbers. The cyclical nature of its powers simplifies many computations:

• Recall that \(i^4 = 1\).
• Hence, the powers of 'i' cycle every four terms: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\).

For example, in the given problem, you need to compute \(i^{2718}\).

• First, determine the remainder when 2718 is divided by 4 (i.e., \(2718 \mod 4\)).
• Since \(2718 \mod 4 = 2\), it follows that \(i^{2718} = i^2\).
• Therefore, \(i^{2718} = -1\).

Using this method can simplify exponentiation for any power of 'i'.

Simplifying Fractions

In many problems involving complex numbers, you will come across complex fractions. Simplifying these fractions often involves:

Multiplying the numerator and the denominator by the conjugate of the denominator to rationalize it.
For instance, to simplify \(\frac{1+i}{1-i}\), follow these steps:

• Identify the conjugate of the denominator, which is \(1+i\).
• Multiply both the numerator and the denominator by \(1+i\).
• Expand both the numerator and the denominator: \( (1+i)(1+i) = 1 + 2i + i^2\) and \( (1-i)(1+i) = 1 - i^2\).
• Simplify these expressions using \(i^2 = -1\): numerator becomes \(1 + 2i - 1 = 2i\); denominator becomes \(1 - (-1) = 2\).
• As a result, the fraction \(\frac{1+i}{1-i}\) simplifies to \(\frac{2i}{2} = i\).

Approaching complex fractions systematically in this way will make even challenging problems more straightforward to solve.