Question

Make a table that shows the powers of \(i\) from \(i^1\) to \(i^8\) in the first row and the simplified forms of these powers in the second row. Describe the pattern you observe in the table. Verify the pattern continues by evaluating the next four powers of \(i\).

Short answer

The powers of \(i\) repeat every four terms. The pattern is: \(i, -1, -i, 1\). So, \(i^1\) to \(i^4\) are \(i, -1, -i, 1\) and \(i^5\) to \(i^8\) are \(i, -1, -i, 1\) and \(i^9\) to \(i^{12}\) are also \(i, -1, -i, 1\).

Step-by-step solution

Understand the Imaginary Unit \(i\)

The imaginary unit \(i\) is defined as \(i = \sqrt{-1}\). With this, we can compute the powers of \(i\).

Calculate the Powers of \(i\)

Start calculating the powers. Starting with \(i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1\). Then, \(i^5 = i * i^4 = i, i^6 = i^2 * i^4 = -1, i^7 = i^3 * i^4 = -i, i^8 = (i^4)^2 = 1\).

Observing the Pattern

Observe the pattern that after every fourth power, the powers of \(i\) repeat. So powers 1 to 4 correspond to powers 5 to 8, and so on. The pattern is \(i, -1, -i, 1\).

Continue and Confirm the Pattern

According to the pattern, \(i^9 = i, i^{10} = -1, i^{11} = -i, i^{12} = 1\). This indeed matches with the pattern.

Key concepts

Imaginary Numbers

Imaginary numbers are crucial in extending our number system beyond the real numbers. They are essential for solving equations where taking the square root of a negative number is necessary. The basic unit of imaginary numbers is denoted by the letter \( i \), which is defined as \( i = \sqrt{-1} \). When dealing with imaginary numbers, we often explore their properties and how they interact with real numbers:

• Imaginary numbers are typically expressed as a real part and an imaginary part. For example, in the expression \( a + bi \), \( a \) is the real part and \( bi \) is the imaginary part.
• The arithmetic of imaginary numbers involves combining like terms, similar to working with variables.

Understanding imaginary numbers opens up a whole new world of mathematics and is foundational for more advanced topics such as complex numbers.

Cyclic Pattern

One fascinating trait of powers of the imaginary unit \( i \) is the cyclic pattern that emerges. When you calculate the powers of \( i \), like other cyclical sequences, it repeats itself in a predictable manner:

• Start with \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \).
• Notice how after every four powers, the sequence starts over. For instance, \( i^5 \) is again \( i \), \( i^6 = -1 \) mirrors \( i^2 \), and so on.

The cyclic nature simplifies calculations significantly, as knowing that every fourth power returns to \( i \) allows one to determine large powers quickly by thinking of them in terms of cycles of four. This pattern is crucial for computational efficiency in mathematical applications.

Complex Numbers

Complex numbers are a natural extension of imaginary numbers and consist of two parts: a real part and an imaginary part. The standard form of a complex number is \( a + bi \), where \( a \) is the real portion, and \( bi \) is the imaginary portion:

• Complex numbers can be represented graphically on a complex plane, where the x-axis represents the real component, and the y-axis represents the imaginary component.
• Operations such as addition, subtraction, multiplication, and division are extensions of operations for real numbers, adhering to similar rules but accounting for the imaginary component.

Complex numbers are vital in many fields, including engineering and physics, as they offer a way to solve problems involving two-dimensional quantities.

Mathematical Powers

Understanding mathematical powers is fundamental in grasping the powers of the imaginary unit \( i \). Powers involve repeated multiplication and are denoted as \( a^n \) where \( a \) is the base, and \( n \) is the exponent:

• The cycle of powers of \( i \) is a special case but reflects broader concepts of periodicity and sequences in mathematics.
• When dealing with powers, recognizing patterns or cycles can simplify complex calculations.
• These insights into powers are not only useful for imaginary numbers but also applicable to real-world applications like signal processing and quantum physics.

Conceptualizing powers beyond mere arithmetic allows for engagement with deeper aspects of mathematical theory, making seemingly complex problems more manageable.