Problem 5 - Elementary and Intermediate Algebra

Question

Fill in the blanks. a. \(i=\) b. \(i^{2}=\) c. \(i^{3}=\) d. \(i^{4}=\) e. In general, the powers of \(i\) cycle through ________ possible outcomes.

Step-by-Step Solution

Answer

a. \( i \), b. \( -1 \), c. \( -i \), d. \( 1 \), e. four

Step 1: Definition of i

In mathematics, the imaginary unit \( i \) is defined as \( i = \sqrt{-1} \). This means that \( i \times i \) or \( i^2 \) is equal to \( -1 \). Thus, \( i = \sqrt{-1} \).

Step 2: Square of i

Using the definition of \( i \), we calculate \( i^{2} \): \( i^{2} = i \times i = \sqrt{-1} \times \sqrt{-1} = -1 \). Thus, \( i^{2} = -1 \).

Step 3: Cube of i

To find \( i^{3} \), which is \( i \times i^{2} \): \( i^{3} = i \times (-1) = -i \). Thus, \( i^{3} = -i \).

Step 4: Fourth power of i

For \( i^{4} \), use \( i^2 \) again: \( i^{4} = i^{2} \times i^{2} = (-1) \times (-1) = 1 \). Thus, \( i^{4} = 1 \).

Step 5: Recognize the Cycle

Notice that the powers of \( i \) repeat every four exponents: \( i, -1, -i, 1 \). Therefore, in general, the powers of \( i \) cycle through four possible outcomes: \( i, -1, -i, \text{and } 1 \).

Key Concepts

Complex Numbers

Complex numbers are a rich and fascinating part of mathematics. They have the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is an imaginary unit defined as \( i = \sqrt{-1} \).
This form allows complex numbers to incorporate both a real part \( a \) and an imaginary part \( bi \).

• The real part \( a \) is simply a standard real number.
• The imaginary part \( bi \) contains \( i \), signifying its placement in the imaginary dimension.

Although the concept of imaginary numbers might sound abstract, they are incredibly useful in fields like engineering and physics.
× The interplay between the components of complex numbers enables calculations that would otherwise be impossible with real numbers alone.

Powers of i

Understanding the powers of \( i \) involves recognizing its cyclical patterns.
Since \( i = \sqrt{-1} \), this unit has special multiplication properties.

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

Each calculation builds on the previous power and reveals the natural cyclical behavior of the powers of \( i \).
Once the fourth power is reached, the cycle restarts from the beginning of this sequence.
You can predict any power of \( i \) by identifying its position in this cycle, which tremendously simplifies complex calculations.

Mathematical Cycles

Mathematical cycles are patterns that repeat at regular intervals.
For the powers of \( i \), the cycle length is four, which means we see the same four results repeating endlessly. This pattern occurs because of how multiplication works with the imaginary unit \( i \).

• Begin with \( i \)
• Multiply by \( i \) to get \( -1 \)
• Multiply by \( i \) again to get \( -i \)
• Multiply by \( i \) one more time to return to 1

From here, the sequence restarts, showing the complete cycle length of four.
Using these cycles can optimize problem-solving, especially when dealing with larger powers, where the cyclical pattern allows simplification by traveling back to the beginning of the cycle for easy computation.