Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. (a) \(i^{73}\) (b) \(i^{-46}\)

Step-by-Step Solution

Answer

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

Step 1: Understanding powers of i

The imaginary unit \(i\) is defined such that \(i^2 = -1\). It follows that \(i^3 = i^2 \cdot i = -i\), \(i^4 = i^2 \cdot i^2 = 1\), and \(i^5 = i^4 \cdot i = i\). Thus, the powers of \(i\) cycle every 4: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then it repeats.

Step 2: Simplifying i^73

Since the powers of \(i\) cycle every 4, we need to find the remainder of 73 when divided by 4 to determine which power corresponds to \(i^{73}\). Calculating the remainder: \(73 \div 4 = 18\) remainder \(1\). Therefore, \(i^{73} = i^1 = i\).

Step 3: Writing i^73 in a+bi form

Since \(i^{73} = i\), the expression can be directly written in the form \(a + bi\) as \(0 + 1i\), where \(a = 0\) and \(b = 1\).

Step 4: Simplifying i^{-46}

To simplify \(i^{-46}\), first determine \(i^{46}\) using the cycle. Calculate \(46 \div 4 = 11\) remainder \(2\). Therefore, \(i^{46} = i^2 = -1\). Using the property \(i^{-n} = \frac{1}{i^n}\), we have \(i^{-46} = \frac{1}{i^{46}} = \frac{1}{-1} = -1\).

Step 5: Writing i^{-46} in a+bi form

Since \(i^{-46} = -1\), we can write it in the form \(a + bi\) as \(-1 + 0i\), where \(a = -1\) and \(b = 0\).

Key Concepts

Powers of Imaginary Unit

The imaginary unit, represented as \(i\), is a fundamental concept in mathematics, especially when dealing with complex numbers. It is defined such that \(i^2 = -1\). From this basic definition, we can derive the powers of \(i\). Let's take a look:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = i^2 \cdot i = -i\)
• \(i^4 = i^2 \cdot i^2 = 1\)
• \(i^5 = i^4 \cdot i = i\)

As you can see, after \(i^5\), the cycle repeats with \(i\) as the result. Knowing the basic powers helps us simplify more complex expressions involving \(i\) by recognizing patterns.

Cyclic Nature of Powers of i

One interesting property of the imaginary unit \(i\) is its cyclic nature.

• The powers of \(i\) cycle every four steps: \(i, -1, -i, 1\).
• This means that when handling a power of \(i\), you can find the equivalent smaller one by noting the remainder when dividing by 4.

For example, to simplify \(i^{73}\):

• Divide 73 by 4 and get the remainder, which is 1.
• This remainder tells us that \(i^{73} = i^1 = i\).

Understanding the cyclic nature helps you quickly and easily find equivalent powers, making it easier to work with large exponents.

Imaginary Unit in a+bi Form

Complex numbers are generally expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Simplifying powers of \(i\) can allow us to express results in this form. Let's reflect on our exercise:

• Taking \(i^{73}\), we found it simplifies to \(i\), which is directly written as \(0 + 1i\).
• Similarly, for \(i^{-46}\), recognizing the cycle, we determine it simplifies to \(-1\), expressible as \(-1 + 0i\).

This form of representation is essential not only in simplifying complex expressions but also in performing operations like addition, subtraction, and multiplication of complex numbers. By expressing complex numbers in \(a + bi\) form, we bring a systematic approach to tackling problems involving complex numbers.