Problem 1 - Elementary and Intermediate Algebra

Question

Fill in the blanks. The ______ number \(i\) is defined as \(i=\sqrt{-1} .\) We call \(i^{25}\) a _____________ of i.

Step-by-Step Solution

Answer

The imaginary number is defined as \(i=\sqrt{-1}\). We call \(i^{25}\) a power of \(i\).

Step 1: Understand the Imaginary Unit

The imaginary unit is denoted by the letter \(i\) and defined as the square root of \(-1\), i.e., \(i = \sqrt{-1}\). This unit helps us work with complex numbers, which include real and imaginary parts.

Step 2: Concept of Cycles of Forms

The powers of \(i\) repeat in a cycle every 4 powers, which can be simplified as: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), then repeats.

Step 3: Simplify \(i^{25}\)

To find \(i^{25}\), divide 25 by 4. This gives a quotient of 6 and a remainder of 1. This means \(i^{25} = i^1\), because powers of \(i\) repeat every 4.

Step 4: Conclusion

Therefore, \(i^{25} = i\) and is known as a power of \(i\).

Key Concepts

Complex Numbers

Complex numbers are a fascinating concept in mathematics. They extend the idea of one-dimensional numbers on the number line to a two-dimensional plane. This is similar to x and y coordinates on a graph.

• A complex number is expressed in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
• The letter \(i\) is used to represent the imaginary unit, which we'll discuss further.

This combination of real and imaginary parts allows complex numbers to represent more situations than just real numbers alone. For example, they are widely used in engineering and physics to simplify calculations. Understanding complex numbers can open up a new world of mathematical possibilities.

Imaginary Unit

The imaginary unit, represented by the symbol \(i\), is a core concept when dealing with complex numbers. It's defined as \(i = \sqrt{-1}\).

• This definition might seem strange at first, since in the realm of real numbers, you can't take the square root of a negative number.
• However, the imaginary unit is what allows for representing numbers beyond the familiar real number line.
• Imaginary numbers do not exist on the real number line, but instead on a separate imaginary axis perpendicular to it.

Understanding the imaginary unit is crucial because it helps us explore numbers that are not "real" but still very useful. Through this exploration, the calculations involving complex numbers become feasible.

Powers of i

The powers of \(i\) provide a fascinating and useful characteristic that extends the application of imaginary numbers. They are cyclical, repeating every four terms.

• Let's look at these powers: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\).
• After the fourth power, these values start to repeat: \(i^5 = i\), \(i^6 = -1\), and so on.
• This cyclic nature makes it easier to work with higher powers of \(i\). For example, \(i^{25}\) can be simplified by dividing 25 by 4. The remainder is 1, so \(i^{25} = i^1 = i\).

These cycles provide a powerful tool for simplifying expressions, particularly in equations involving complex numbers. This repetitive pattern helps conclude calculations swiftly and avoids complex computations.