Problem 42 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ i^{1002} $$

Step-by-Step Solution

Answer

The result is \(-1 + 0i\).

Step 1: Understand the Powers of i

The imaginary unit \(i\) has a cyclical pattern of powers: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). Therefore, every four powers, the cycle repeats: \(i^5 = i\), \(i^6 = -1\), and so on.

Step 2: Determine the Cycle Position

To find \(i^{1002}\), first determine where 1002 falls in the cycle by calculating the remainder when 1002 is divided by 4. This is because the cycle length is 4. Compute: \(1002 \div 4 = 250 \) remainder \(2\).

Step 3: Find i to the Remainder's Power

Since the remainder is 2, we evaluate \(i^2\) because it represents the position of \(i^{1002}\) in the cycle. From the cyclical pattern, \(i^2 = -1\).

Step 4: Write the Result in a + bi Form

The expression \(i^{1002}\) simplifies to \(-1\), which can be written in the form \(a + bi\) as \(-1 + 0i\).

Key Concepts

Powers of i

The powers of the imaginary unit, denoted by the letter \(i\), reveal an interesting cyclical pattern. The base of this cycle begins with \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). Once it reaches the fourth power, the cycle starts again from \(i^1\).

Understanding this cycle:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\) - completes the cycle

This pattern continues indefinitely, so the power of \(i\) can always be reduced to one of these four values. For example, \(i^5\) is the same as \(i^1\), \(i^6\) is the same as \(i^2\), and so forth. These simplifications make complex number calculations more manageable.

Imaginary Unit

In mathematics, the imaginary unit \(i\) is fundamental for working with complex numbers. \(i\) is defined as the square root of \(-1\), an operation that is not possible using only real numbers. Thus, \(i^2 = -1\), which is distinct from any operation in the realm of real numbers.

By introducing \(i\), mathematicians expanded the number system to include solutions to equations that did not have real solutions before. For instance, the equation \(x^2 + 1 = 0\) has no real number solutions, but with \(i\), the solutions are \(x = i\) and \(x = -i\). This expanded framework allows the interpretation and manipulation of complex numbers.

Cyclical Pattern

The repetition found in the powers of \(i\) is known as the cyclical pattern. This pattern, cycling through a sequence every four powers, is crucial in simplifying expressions with high powers of \(i\). To find the position of a power within this cycle, we calculate the remainder of the exponent divided by 4.

For example, in the original exercise, we need to find \(i^{1002}\). By dividing 1002 by 4, the remainder is 2. This remainder tells us that \(i^{1002} = i^2 = -1\). Recognizing this pattern reduces complex problems to simpler arithmetic, assisting in both theoretical understanding and practical calculations.

Complex Form a + bi

A complex number is expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The term \(a\) represents the real part, while \(bi\) represents the imaginary part.

Consider the expression \(i^{1002}\) from the original exercise. The computed result was \(-1\), which in the complex form is expressed as \(-1 + 0i\). This format is universal for all complex numbers, irrespective of whether the imaginary part \(b\) equals zero. Such representation is essential in complex number arithmetic, providing clarity and consistency in the manipulation of both simple and intricate expressions.