Problem 33 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ i^{100} $$

Step-by-Step Solution

Answer

The expression \( i^{100} \) equals 1, which is written as \( 1 + 0i \).

Step 1: Understand the Properties of i

The imaginary unit i is defined such that \( i^2 = -1 \). From this, we can derive the cycle of powers of i: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). This cycle then repeats every four powers.

Step 2: Determine the Power Cycle Position

Since the powers of i repeat every four numbers, we calculate the remainder when 100 is divided by 4, as this will determine which power in the cycle \( i^1, i^2, i^3, i^4 \) the expression corresponds to. Perform the division: \( 100 \div 4 = 25 \) with a remainder of 0.

Step 3: Evaluate Based on Remainder

A remainder of 0 means that \( i^{100} \) is equivalent to \( i^4 \), which equals 1 as per the properties of i. Therefore, \( i^{100} = 1 \).

Step 4: Write in the Form a + bi

Express the result in the form \( a + bi \), where a and b are real numbers. Since \( i^{100} = 1 \), this can be written as \( 1 + 0i \).

Key Concepts

Complex Numbers

Complex numbers are all about combining real and imaginary numbers into a single entity. A complex number is typically written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.

This is an interesting way to extend our traditional number systems. Within complex numbers, you might encounter expressions that seem impossible with just real numbers, like the square root of a negative number, which is ordinarily undefined. The introduction of the imaginary unit \( i \)—defined such that \( i^2 = -1 \)—solves this problem!

Essentially, the imaginary part \( bi \) helps us calculate and understand phenomena beyond the scope of real numbers. This foundational concept allows mathematicians, engineers, and scientists to convey complex concepts more effectively. It's fascinating because nearly every real-world problem can be described by real numbers, and complex numbers extend that reach to a new dimension that handles the suppositional scenarios.

Powers of i

The powers of the imaginary unit \( i \) create a repeating cycle, which is really useful for simplifying expressions that include imaginary numbers. Here is how it plays out:

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

After \( i^4 \), the cycle starts over. So essentially, every fourth power of \( i \) results in 1.

This property becomes very handy for evaluating large powers of \( i \), such as \( i^{100} \). The key to simplifying them is to determine where they fall within this 4-step cycle. Understanding this pattern allows you to quickly break down expressions involving complex numbers.

For example, in the case of \( i^{100} \), knowing that the cycle repeats every four steps helps you compute that \( i^{100} = i^4 \), which evaluates to 1.

Remainder Theorem

When working with powers of \( i \), the remainder theorem helps tremendously to find where in the power sequence your number lands. The remainder theorem revolves around finding the remainder of a division, which tells you which power of \( i \) you are dealing with.

Imagine you need to calculate powers of \( i \) for a very large exponent. Instead of multiplying \( i \) repeatedly, you can use the remainder of that number when divided by 4.

• If the remainder is 0, the power of \( i \) equals 1 (as in \( i^4 \)).
• If the remainder is 1, it equals \( i \) (as in \( i^1 \)).
• If the remainder is 2, it equals \( -1 \) (as in \( i^2 \)).
• If the remainder is 3, it equals \( -i \) (as in \( i^3 \)).

For example, with \( i^{100} \), you divide 100 by 4 and get a remainder of 0. This means \( i^{100} = i^4 \), thus the answer is 1. This makes the remainder theorem a powerful tool for evaluations involving complex numbers and powers of \( i \).