Problem 34 - Precalculus Mathematics for Calculus

Question

Evaluate the expression and write the result in the form \(a+b i\) $$i^{1002}$$

Step-by-Step Solution

Answer

The expression evaluates to \(-1 + 0i\).

Step 1: Understand the Pattern of Powers of i

The imaginary unit, denoted as \(i\), has a repeating cycle of powers: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). After this, the pattern repeats every four powers. Recognizing that \(i^{4k} = 1\) for any integer \(k\) is crucial for simplifying higher powers.

Step 2: Determine the Remainder in Modulo 4

Given that the powers of \(i\) repeat every four terms, we compute \(1002 \mod 4\). Dividing 1002 by 4, we perform: 1002 ÷ 4 = 250 remainder 2. Thus, \(1002 \equiv 2 \pmod{4}\).

Step 3: Use the Remainder to Find the Expression

From Step 2, we know that \(1002 \equiv 2 \pmod{4}\). This means \(i^{1002} = i^2\) because the powers repeat every four terms and have the same remainder 2. From the pattern in Step 1, we see that \(i^2 = -1\).

Step 4: Express the Result in the Form a + bi

The result \(-1\) can be written in the form \(a+bi\) by considering \(a = -1\) and \(b = 0\). Thus, \(i^{1002} = -1 + 0i\).

Key Concepts

Imaginary Unit

The imaginary unit is often denoted by the symbol \(i\), and it is a fundamental concept in complex numbers. The defining property of \(i\) is that it is the square root of \(-1\). This means when \(i\) is squared, it results in \(-1\):\[ i^2 = -1 \]Understanding \(i\) is crucial because it forms the basis of all complex numbers, which are numbers composed of a real part and an imaginary part. Here’s why \(i\) is special:

• It enables us to extend the real number system to include solutions to equations like \(x^2 + 1 = 0\).
• It helps in modeling periodic phenomena, such as electrical currents and waves, effectively.

By allowing the square root of negative numbers, \(i\) makes it possible to work with complex numbers which have numerous applications in engineering and physics.

Powers of i

The powers of the imaginary unit \(i\) repeat in a predictable cycle of four. This cycle can be a great shortcut for simplifying problems involving powers of \(i\). The cycle is as follows:

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

Once you reach \(i^4\), the sequence starts over again. So, for any positive integer power of \(i\), you can find the corresponding power within these four:- Identify the power.- Divide the power by 4 to find the remainder.- Use the remainder to find the equivalent power.For example, to evaluate \(i^{1002}\), compute \(1002 \div 4\) which gives a remainder of 2,meaning \(i^{1002} = i^2 = -1\). Using this cycle makes working with high powers of \(i\) straightforward.

Modulo Arithmetic

Modulo arithmetic is a mathematical technique used to find the remainder of a division operation. It is particularly useful in problems involving powers, including those with the imaginary unit \(i\). The concept is expressed as:\[ a \equiv b \pmod{n} \]This means that when \(a\) is divided by \(n\), it leaves a remainder of \(b\).In the context of powers of \(i\), modulo arithmetic simplifies calculations by focusing on the remainder rather than the quotient. Since the powers of \(i\) repeat every four terms, using modulo 4 arithmetic helps determine the position within the repeating cycle. For instance, to compute \(i^{1002}\), determine \(1002 \mod 4\), which results in a remainder of 2. This means \(i^{1002} = i^2\), taking advantage of the repeating cycle to ease computations.

Complex Number Notation

Complex numbers are typically expressed in the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. This notation allows for simplified operations on both real and imaginary components together. Here's what each part represents:

• \(a\): The real component, representing real numbers on the number line.
• \(b\): The imaginary component, which is a multiple of the imaginary unit \(i\).

In solving expressions such as \(i^{1002}\), the solution often needs to be presented in this standard form. For example, when finding \(i^{1002} = -1\), this can be rewritten as \(-1 + 0i\). Here:

• The real part \(a = -1\).
• The imaginary part \(b = 0\), indicating there is no imaginary component in the final expression.

This form makes it easy to interpret the result within the context of complex numbers.