Problem 3 - Advanced Engineering Mathematics

Question

Write the given number in the form \(a+i b\). $$ i^{8} $$

Step-by-Step Solution

Answer

\(i^8 = 1\,\) which is \(1 + 0i\).

Step 1: Understand the powers of i

The imaginary unit \(i\) has a cyclical pattern for its powers. Recall that: - \(i^1 = i\)- \(i^2 = -1\)- \(i^3 = -i\)- \(i^4 = 1\)This pattern repeats for every set of four powers.

Step 2: Calculate \(i^8\)

To find \(i^8\), notice that 8 is a multiple of 4. Since the pattern repeats every four numbers, we know that \(i^4 = 1\). Thus, \(i^8 = (i^4)^2 = 1^2 = 1\).

Step 3: Write \(i^8\) in the form \(a + ib\)

Since \(i^8 = 1\), it can be expressed in the form \(a + ib\), where \(a = 1\) and \(b = 0\). Thus, \(i^8 = 1 + i \times 0\).

Key Concepts

Imaginary Unit

The concept of the imaginary unit, denoted as \(i\), stems from the need to find roots of equations that have no real solutions, such as \(x^2 = -1\). In real numbers, this equation doesn't hold, but the introduction of \(i\) allows for a solution. Simply put, the imaginary unit is defined as \(i = \sqrt{-1}\).
Using \(i\), we can express complex numbers, which have both real and imaginary parts. Imaginary numbers complement the real number system, enabling mathematicians and engineers to solve a wide range of problems.
It’s crucial to differentiate the imaginary unit from real numbers, keeping in mind that the presence of \(i\) indicates an imaginary part of a complex number.

Powers of i

The powers of \(i\) exhibit a repeating pattern every four exponentiations. This cyclical behavior stems from the fundamental properties of \(i\):

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

Beyond the fourth power, these values start repeating: \(i^5 = i\), \(i^6 = -1\), and so on.
This cyclical nature makes it simple to calculate higher powers of \(i\). For example, \(i^8 = (i^4)^2\), which equals \(1^2 = 1\). Recognizing this pattern is crucial for simplifying complex numbers and helps in solving equations involving \(i\).

Complex Number Representation

Complex numbers are expressed in the form \(a + ib\), where \(a\) and \(b\) are real numbers. In this format, \(a\) represents the real part and \(b\) denotes the imaginary part.
To write a pure real number as a complex number, you simply set \(b = 0\), making it appear as \(a + 0i\). Conversely, a purely imaginary number looks like \(0 + bi\).
Understanding this representation helps in performing operations, such as addition and multiplication of complex numbers. When calculating powers as shown with \(i^8 = 1\), we express \(1\) as \(1 + 0i\), adhering to the standard complex number form. This format brings out clarity when dealing with real and imaginary sums in mathematics.