Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. (a) \(i^{43}\) (b) \(i^{-30}\)

Step-by-Step Solution

Answer

(a) 0 - i; (b) -1 + 0i.

Step 1: Understanding Powers of i

The imaginary unit, denoted as \(i\), is defined such that \(i^2 = -1\). Notice that the powers of \(i\) follow a repeating cycle: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). After \(i^4\), the cycle repeats, so \(i^5 = i\), \(i^6 = -1\), and so on.

Step 1: Calculating i^{43}

Since the cycle of powers of \(i\) is 4, we can simplify \(i^{43}\) by finding the remainder when 43 is divided by 4, which helps us identify the equivalent power within the cycle. Dividing, we find \(43 \div 4 = 10\) with a remainder of 3. Thus, \(i^{43} = i^3 = -i\). So, \(a = 0\) and \(b = -1\), and the expression is in the form \(a + bi = 0 - i\).

Step 2: Calculating i^{-30}

To handle negative exponents, recall that \(i^{-n} = \frac{1}{i^n}\). We need to find the equivalent positive power of \(i\) that corresponds to \(-30\). First, find \(30 \mod 4\), which is 2, meaning \(i^{30} = i^2 = -1\). Thus, \(i^{-30} = \frac{1}{i^{30}} = \frac{1}{-1} = -1\). Here, \(a = -1\) and \(b = 0\), so the expression is in the form \(a + bi = -1 + 0i\).

Key Concepts

Imaginary Unit

The imaginary unit, often represented by the symbol \(i\), is a fundamental concept in the study of complex numbers. It is defined as the square root of \(-1\). As such, this unit forms the basis for imaginary numbers, which extend the number system beyond real numbers to accommodate solutions to equations that involve the square root of negative numbers. Here's what you need to remember about \(i\):

• \(i\) is specifically defined such that \(i^2 = -1\).
• This imaginary unit helps expression of numbers that cannot be captured by just real numbers.
• It is crucial for solving certain types of polynomial equations and for expressing any complex number \( z \) in the form \( a + bi \), where \( a \) and \( b \) are real numbers.

Understanding \(i\) can be quite intuitive once you realize it's a tool to handle negatives inside square roots, resulting in a broader, more flexible number system.

Powers of i

The powers of the imaginary unit \(i\) repeat in a cycle. This is very useful in simplifying complex expressions that involve high powers of \(i\). Let's go through this cycle to understand it better:

• \(i^1 = i\) - This is the imaginary unit itself.
• \(i^2 = -1\) - By definition, the square of the imaginary unit is \(-1\).
• \(i^3 = -i\) - Multiplying \(i^2\) by \(i\) gives us \(-i\).
• \(i^4 = 1\) - Multiplying \(i^3\) by \(i\) returns us to 1, completing the cycle.

This cycle repeats every four exponents. So, for example, to determine \(i^{43}\), you would divide 43 by 4 and focus on the remainder, which in this case is 3, equating \(i^{43}\) to \(i^3 = -i\). Understanding these cycles helps greatly in reducing complicated powers of \(i\) to simpler terms, making calculations way easier.

Complex Expressions

Complex expressions involve both real and imaginary components. A complex number is typically written as \( a + bi \), where \( a \) is the real part and \( b \) is the coefficient of the imaginary part \( i \). When working with complex expressions, here's what you should focus on:

• Real part, \(a\): The term without the imaginary unit. For example, in \(-1 + 0i\), \(-1\) is the real part.
• Imaginary part, \(b\): The coefficient of the imaginary unit \(i\). For example, in \(0 - i\), \(-1\) is the imaginary part even though it's usually written as \(0 - 1i\).
• Combining like terms: Often, when adding or subtracting complex numbers, you will combine the real parts and imaginary parts separately to simplify expressions.

Complex expressions often become simpler when using the properties of \(i\), especially by taking advantage of the cyclic nature of its powers, such as calculating \(i^{-30}\) as shown in the original exercise. These strategies are integral in understanding and simplifying complex expressions efficiently.