Problem 41 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ i^{100} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Powers of i

The imaginary unit, denoted as \( i \), has specific powers due to the property \( i^2 = -1 \). Therefore, higher powers of \( i \) cycle every four: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). After \( i^4 \), the cycle repeats.

Step 2: Simplify the Exponent

The exponent given is 100. To simplify this, divide 100 by 4 and find the remainder: \( 100 \div 4 = 25 \) with a remainder of \( 0 \). This means \( i^{100} = i^{4 \times 25} \).

Step 3: Apply the Cycle of i

Since the remainder is 0, \( i^{100} = i^{4 \times 25} = (i^4)^{25} \). We know that \( i^4 = 1 \), so \( (i^4)^{25} = 1^{25} = 1 \).

Step 4: Convert to a + bi Form

The result found is 1, which can be written in the form \( a + bi \) as \(1 + 0i\). This indicates that there is no imaginary part.

Key Concepts

Powers of Imaginary Unit

When dealing with the legendary imaginary unit, symbolized by \( i \), it's essential to understand its powers. These powers cycle through a predictable pattern, largely due to the fundamental property that \( i^2 = -1 \). This leads to a simple periodic cycle every four powers:

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

The beauty here is in the repetitive nature of these values. Each additional power of \( i \) after \( i^4 \) merely returns to a previous value in this loop. So, for any integer exponent of \( i \), you can reduce it by dividing the exponent by 4 and examining the remainder. The remainder determines the power of \( i \) that corresponds to the result. Thus, for \( i^{100} \), since 100 divided by 4 has a remainder of zero, it corresponds to \( i^4 \), which is 1.

Imaginary Numbers

Imaginary numbers might sound like something out of a fantasy novel, but they play a crucial role in mathematics, especially in complex number theory. An imaginary number is effectively a real number multiplied by the imaginary unit \( i \). While real numbers are plotted on a traditional number line, imaginary numbers exist along a perpendicular line, forming a two-dimensional number plane called the complex plane.
This distinct, imaginary axis enables the representation of numbers that extend beyond real limitations. By supplementing equations with the imaginary unit \( i \) (where \( i = \sqrt{-1} \)), we can solve problems involving negative square roots—situations impossible within the realm of real numbers alone. Hence, any number with an \( i \) component can be considered 'imaginary'.
For example, the number \( 2i \) is an imaginary number, as it is obtained by taking twice the imaginary unit \( i \). Sometimes imaginary numbers can exist alone, while other times, as part of a complex number, which we'll touch upon next.

Complex Number Notation

Complex numbers combine both real and imaginary components. The standard notation for a complex number is \( a + bi \), where:

• \( a \) represents the real part.
• \( b \) represents the imaginary part.
• \( i \) is the imaginary unit, satisfying \( i^2 = -1 \).

An essential aspect of complex numbers is that the real and imaginary parts can both play their roles independently but also combine to provide powerful solutions that are unavailable through real numbers alone. For example, consider the complex number \( 3 + 4i \). Here, 3 is the real part, and \( 4i \) is the imaginary part.
The usage of complex number notation allows for the incorporation of real and imaginary elements into equations, enabling us to solve otherwise elusive mathematical objects or expressions. Consequently, if a complex number has no imaginary component (i.e., \( b = 0 \)), it simply reduces to a real number, as we saw with the example of \( i^{100} \), which equals \( 1 + 0i \). This understanding helps in seamlessly moving between real numbers, imaginary numbers, and complex numbers in the realm of mathematics.