Problem 34 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ i^{1002} $$

Step-by-Step Solution

Answer

\(i^{1002} = -1 + 0i\)

Step 1: Identify the properties of i

The imaginary unit \(i\) is defined such that \(i^2 = -1\). Powers of \(i\) follow a specific cycle: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then it repeats every four powers.

Step 2: Find the remainder of 1002 divided by 4

The powers of \(i\) repeat every four terms, so we need to find the remainder when 1002 is divided by 4 to determine which power in the cycle corresponds to \(i^{1002}\). Calculating 1002 divided by 4 gives a quotient of 250 and a remainder of 2.

Step 3: Apply the result of the division to the cycle of i

Since the remainder of 1002 divided by 4 is 2, \(i^{1002}\) corresponds to the same value as \(i^2\) in the cycle of \(i\). Recall from the cycle, \(i^2 = -1\).

Step 4: Write the result in the form of a complex number

The exercise asks for the expression to be written in the form \(a + bi\). Since \(i^{1002} = -1\), this corresponds to the complex number \(-1 + 0i\).

Key Concepts

Imaginary Unit

The concept of the imaginary unit, denoted by the symbol \(i\), is a fundamental building block in complex numbers. Essentially, \(i\) is a mathematical construct that allows us to extend the real number system so that we can include solutions to equations like \(x^2 = -1\), which would not be possible with only real numbers. In real numbers, no square yields a negative result, but the imaginary unit \(i\) is defined to have this property, such that \(i^2 = -1\). This special definition unlocks the ability to work with and understand more complex mathematical concepts involving numbers that don't have real counterparts.
Whenever \(i\) is used in calculations, it's important to keep in mind that it represents an imaginary value that complements real numbers, forming what we call complex numbers.

Powers of i

Understanding the powers of \(i\) is key to grasping its behavior in complex numbers. Powers of the imaginary unit cycle through a predictable pattern every four terms, making them easier to compute without directly multiplying. Starting with \(i^1 = i\), the cycle proceeds as follows:

• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)
• \(i^5 = i\) again, repeating the cycle.

This repeating cycle means that any power of \(i\) can be reduced to one of these four core results, simply by finding where it falls in the cycle. For instance, \(i^{1002}\) is simplified by noticing that dividing 1002 by 4 leaves a remainder of 2. This tells us that \(i^{1002} = i^2 = -1\). This cyclical behavior simplifies working with complex powers in mathematics.

Complex Number Form

Complex numbers are numbers composed of a real part and an imaginary part, written in the form \(a + bi\), where \(a\) and \(b\) are real numbers. The real part, \(a\), can be thought of as any typical number you’d find on the number line, while the imaginary part, \(bi\), involves the imaginary unit \(i\), creating a broader scope of number representation.
For example, in the solution for \(i^{1002}\), we end up with \(-1 + 0i\). Here, \(-1\) is the real part, and while \(0i\) means there's no imaginary component, it shows the number could have an imaginary part. This format is important because it allows for clear communication in calculations involving both parts.
Complex number form is crucial not just in pure mathematics but also in applied fields like engineering and physics, where real and imaginary components often represent different dimensions of a problem, such as amplitude and phase in electrical engineering.