Question

Evaluate each power of \(i\). $$i^{10}$$

Short answer

\(i^{10} = -1\)

Step-by-step solution

Understanding the Powers of i

Recall that the imaginary unit, denoted by \(i\), has the property that \(i^2 = -1\). Further powers of \(i\) are cyclical in nature, following a pattern: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then it repeats. Any power of \(i\) can be reduced using this cycle.

Reducing the Power Modulo 4

Since the powers of \(i\) repeat every four powers, we can reduce the power modulo 4 to find an equivalent power that is easier to evaluate. For \(i^{10}\), we compute \(10 \mod 4 = 2\). This means that \(i^{10}\) is equivalent to \(i^2\).

Evaluating the Power

Knowing that \(i^2 = -1\), we can now evaluate \(i^{10}\) as \(i^{10} = i^2 = -1\).

Key concepts

Imaginary Unit

In mathematics, the concept of an imaginary unit, represented as i, is utilized to extend the real numbers and create the complex numbers. i is defined by its fundamental property that i² = -1. This is not possible within the realm of real numbers, hence the term 'imaginary'.

It is crucial to comprehend that i is not a tangible quantity but is indispensable in various fields of engineering and physics, particularly when dealing with phenomena that have both magnitude and direction, like alternating current in electrical engineering.

When working with the imaginary unit, it simplifies many problems – a seemingly complex equation can often be broken down to basic arithmetic by knowing the properties of i. As such, it's a stepping stone to mastering more intricate areas of mathematics.

Modulo Operation

The modulo operation, often presented as mod, finds the remainder of the division of one number by another. For instance, 7 mod 3 would equal 1, because 3 goes into 7 twice with a remainder of 1.

This operation is not just useful in arithmetic, but is a key concept in computer science, where it helps in algorithms that need to wrap around a certain value–like the hands of a clock. Applying modulo to the powers of i, we can determine the equivalent minimal power that exhibits the same property, thus simplifying computations. In the context of complex numbers, after every fourth power, the values repeat in a cycle, which is why modulo 4 is specifically used.

Cyclical Patterns in Powers

Powers of i show a recurring or cyclical pattern, which is a fascinating characteristic when studying exponents in complex numbers. After i⁴, the cycle starts anew: i⁵ = i¹, i⁶ = i², and so on. This pattern recurs because i⁴ is 1, and multiplying 1 by i will start us back at the beginning of the cycle.

Understanding this repetition allows for the simplification of seemingly daunting expressions to much simpler forms. It empowers students to tackle higher powers of i efficiently by recognizing that they only need to consider the power of i modulo 4—a powerful tool when diving deeper into the study of complex numbers.