Problem 76 - A Graphical Approach to College Algebra

Question

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$i^{19}$$

Step-by-Step Solution

Answer

\( i^{19} = -i \)

Step 1: Understanding Powers of i

Recall that the imaginary unit i has a cyclical pattern in its powers. The pattern repeats every four terms as follows: 1. \( i^1 = i \)2. \( i^2 = -1 \)3. \( i^3 = -i \)4. \( i^4 = 1 \)and then repeats again from \( i^5 = i \) and so on.

Step 2: Find the Remainder

To simplify \( i^{19} \), you need to determine its position in the cycle of four. Divide 19 by 4 to find the remainder: \[ 19 \div 4 = 4 ext{ remainder } 3 \]So, the remainder is 3.

Step 3: Determine Equivalent Power in the Cycle

The remainder from the division tells us which power in the cycle \( i^{19} \) corresponds to. Since the remainder is 3, it corresponds to \( i^3 \).

Step 4: Simplify Using the Cycle

According to the cycle, \( i^3 = -i \). Therefore, \( i^{19} = -i \).

Key Concepts

Imaginary Unit

The imaginary unit, represented as \(i\), is a fundamental concept in mathematics, particularly in complex numbers. The defining property of \(i\) is that it is the square root of \(-1\).
This means:

• \(i^2 = -1\)
• \(i^1 = i\)

Understanding \(i\) is crucial because it allows us to handle square roots of negative numbers, which cannot be addressed by real numbers alone. This expands the number system to include complex numbers, greatly enriching mathematical theory and applications.

Cyclical Pattern

The powers of the imaginary unit \(i\) follow a predictable cycle that repeats every four terms. This pattern is essential for simplifying powers of \(i\). Observing the first few powers, we see:

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

After \( i^4 = 1 \), the powers repeat: \( i^5 = i \), \( i^6 = -1 \), \( i^7 = -i \), and so forth. This cyclical pattern is incredibly useful, as it allows us to simplify large powers of \(i\) by relating them back to these four basic results.

Remainder Division

To simplify an expression like \(i^{19}\), we use remainder division. This step helps us identify the position of the power in the cycle of four. By dividing the exponent by 4, we find the remainder:

• \(19 \div 4 = 4\) with a remainder of 3

This remainder indicates which power of \(i\) from our cycle corresponds to the given power—in this case, \(i^{19}\) corresponds to \(i^3\). Hence, simplifying powers of \(i\) often hinges on effectively using remainder division to decode the cycle.

Complex Numbers

Complex numbers are numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit. They extend the real numbers and include all possible combinations of real and imaginary parts.
Some key points include:

• The real part is \(a\), and the imaginary part is \(b\), multiplied by \(i\).
• Complex numbers can be added, subtracted, multiplied, and divided, similar to real numbers.
• They have profound applications in fields such as engineering, physics, and computer science.

Understanding complex numbers allows you to solve equations that have no real solutions, thereby broadening your mathematical toolkit.