Problem 89 - Precalculus Student Solutions Manual 5th

Question

Simplify each power of i. $$i^{23}$$

Step-by-Step Solution

Answer

i^{23} = -i

Step 1: Understand Powers of i

First, recognize the cycle of the powers of i. The powers of the imaginary unit i repeat every 4 steps: • i^1 = i • i^2 = -1 • i^3 = -i • i^4 = 1

Step 2: Find the Remainder

To simplify higher powers of i, divide the exponent by 4 and find the remainder. For example, for i^{23}, compute 23 ÷ 4. The quotient is 5 with a remainder of 3.

Step 3: Apply the Remainder to the Cycle

Use the remainder to determine the simplified form. Since 23 mod 4 is 3, i^{23} is equivalent to i^3 in the cycle. From the cycle, i^3 = -i.

Step 4: Write the Simplified Result

Therefore, i^{23} simplifies to -i.

Key Concepts

imaginary unit

In mathematics, we often encounter the concept of the imaginary unit, denoted as i. The imaginary unit is defined by the property \( i^2 = -1 \). This means that i is a solution to the equation \( x^2 + 1 = 0 \). Imaginary units allow us to extend the real number system to include complex numbers, enabling solutions to equations that we couldn't solve with real numbers alone.
Think of the imaginary unit as a new dimension in which numbers can exist. While real numbers lie on a horizontal number line, complex numbers, which include imaginary parts, lie on a plane known as the complex plane.

modulo operation

The modulo operation, expressed as 'mod,' is a mathematical tool used to find the remainder of a division of one number by another. It plays a crucial role in simplifying high powers of numbers, especially when dealing with cycles, such as the powers of i.
To perform a modulo operation, you divide the number by another number (the modulus) and find the remainder. For example, to simplify \( i^{23} \), you divide 23 by 4 (since the powers of i repeat every 4 steps) and get a remainder of 3. Therefore, \( 23 \text{ mod } 4 = 3 \). This remainder tells us that \( i^{23} \) behaves like \( i^3 \) in the cycle, simplifying our calculations significantly.

complex numbers

Complex numbers are an extension of real numbers and consist of a real part and an imaginary part, typically written in the form \( a + bi \), where a and b are real numbers and i is the imaginary unit.
These numbers can be added, subtracted, multiplied, and divided just like real numbers, except we also follow the rule that \( i^2 = -1 \). In the complex plane, the real part is represented on the horizontal axis, and the imaginary part on the vertical axis, providing a richer, two-dimensional structure for numbers.
Complex numbers are incredibly useful in fields like engineering, physics, and applied mathematics. They help solve problems involving electrical circuits, fluid dynamics, quantum physics, and more.