Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-34: Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. (a) \(i^{66}\) (b) \(i^{-55}\)

Step-by-Step Solution

Answer

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

Step 1: Understand powers of i

The imaginary unit is denoted by \(i\), which is defined as \(i = \sqrt{-1}\). For imaginary powers, we have: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\), after which the powers repeat every four terms.

Step 2: Simplify i^{66}

To simplify \(i^{66}\), use the repeating pattern cycle length of 4. Find the remainder of 66 divided by 4: \[ 66 \div 4 = 16 \,\text{remainder}\, 2 \].This means \(i^{66} = i^2 = -1\), so it can be expressed in the form \(a + bi\) as \(-1 + 0i\).

Step 3: Simplify i^{-55} using inverse and cycles

To simplify \(i^{-55}\), first observe that \(i^{-1} = \frac{1}{i} = -i\) when rationalizing the denominator. Then, factor the exponent:\(-55 = -4 \times 13 + 3\), indicating \(i^{-55} = (i^{-4})^{13} \times i^{-3} = 1 \times (-i) = -i\).Express \(-i\) in the form \(a + bi\) as \(0 - 1i\).

Key Concepts

Imaginary Unit

Complex numbers have a special component called the imaginary unit, represented by the symbol \(i\). The imaginary unit is defined mathematically where \(i = \sqrt{-1}\). This interesting property significantly extends the realm of numbers, allowing us to include solutions for equations involving square roots of negative numbers.

The concept is simple but very powerful. With \(i\), expressions like square roots of negative numbers become manageable. For example, \(\sqrt{-4}\) can be simplified to \(2i\).

Remembering how \(i\) operates is essential:

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

These key expressions lay the foundation for working with complex numbers in various mathematical operations and equations.

Powers of i

When dealing with powers of \(i\), it helps to remember that its powers cycle every four terms. This cyclical pattern is crucial for simplifying expressions containing \(i\) raised to large exponents.

Observing the cycle:

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

For any integer \(n\), the power \(i^n\) can be found using the remainder of \(n\) divided by 4. Thus, expressions such as \(i^{66}\) can be simplified by determining the remainder of 66 divided by 4, which yields 2, resulting in \(i^{66} = i^2 = -1\).

Similarly, the expression \(i^{-55}\) can be tackled by understanding that a negative power of \(i\) involves the reciprocal. Here, simplifying using the cycle, we factor \(-55\) as \(-4 \times 13 + 3\), leading to \(i^{-55} = -i\). Understanding these patterns simplify computations involving the imaginary unit.

Simplifying Complex Expressions

Simplifying complex expressions often involves rewriting them in the form \(a + bi\), where both \(a\) and \(b\) are real numbers. This standard form of complex numbers makes them easy to work with and compare.

Let's look at how this applies to our previous examples:

• \(i^{66} = -1\), expressed as \(-1 + 0i\)
• \(i^{-55} = -i\), expressed as \(0 - 1i\)

Each part of these expressions has a real component \(a\) and an imaginary component \(b\), even if one part is zero.

By simplifying in this way, we can perform additional operations like addition, subtraction, multiplication, and division on complex numbers easily. This breakdown is crucial in fields ranging from engineering to physics, where complex numbers frequently appear.