Problem 2 - Advanced Engineering Mathematics

Question

Write the given number in the form \(a+i b\). $$ 3 i^{5}-i^{4}+7 i^{3}-10 i^{2}-9 $$

Step-by-Step Solution

Answer

The expression is \(-4i\).

Step 1: Apply Powers of i

Recall that the powers of the imaginary unit, \(i\), repeat in cycles of four: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\). Use this cycle to simplify each term in the expression.

Step 2: Simplify i^5

Since \(i^5 = i^{4+1} = i^4 \cdot i^1 = 1 \cdot i = i\), replace the term \(3i^5\) with \(3i\).

Step 3: Simplify i^4

Since \(i^4 = 1\), replace the term \(-i^4\) with \(-1\).

Step 4: Simplify i^3

Since \(i^3 = -i\), replace the term \(7i^3\) with \(-7i\).

Step 5: Simplify i^2

Since \(i^2 = -1\), replace the term \(-10i^2\) with \(10\).

Step 6: Combine Real Parts

Now combine all real numbers: \(-1 + 10 - 9 = 0\). This is the real part of the expression.

Step 7: Combine Imaginary Parts

Combine the imaginary parts: \(3i - 7i = -4i\). This is the imaginary part of the expression.

Step 8: Write in a + bi Form

The expression is rewritten in the form \(a + bi\). Based on the previous steps, the expression becomes \(0 - 4i\) or simply \(-4i\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined as the square root of negative one, so \(i^2 = -1\). This was introduced to extend the concept of numbers beyond the real number line to tackle equations where solutions do not exist within the real numbers alone.

When we discuss the imaginary unit, we delve into numbers that cannot be expressed on the conventional number line. Instead, they are expressed in the complex plane. The imaginary part is expressed using multiples of \(i\), allowing for a broader range of mathematical solutions and applications.

• \(i = \sqrt{-1}\)
• \(i^2 = -1\)

These facts about \(i\) are essential building blocks when working with complex numbers, providing the basis for understanding more complex operations.

Complex Number Simplification

Simplification of complex numbers often involves expressing them in their simplest form, \(a + bi\), where \(a\) and \(b\) are real numbers. Simplifying these numbers involves handling both real and imaginary parts separately and ensuring they are presented clearly.

One key task in simplification is applying the properties of the imaginary unit \(i\). For instance, in our original problem, the terms involving powers of \(i\) were addressed individually:

• Converted powers of \(i\) to their simplest form using the rules of \(i\): \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\).
• Handling coefficients multiplied with these powers, converting terms like \(3i^5\) to \(3i\).

Finally, the simplification culminates in combining like terms. Real numbers are summed up together, while imaginary terms are also combined. Thus, we achieve the expression in its standard form.

Powers of i

The powers of the imaginary unit \(i\) follow a repetitive and predictable cycle. Understanding this cycle is crucial to simplifying complex expressions that include high powers of \(i\). The cycle is:

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

This cycle repeats every four powers, so any higher power of \(i\) can be simplified using these results. For example, \(i^5\) can be thought of as \(i^{4+1} = i^4 \times i = 1 \times i = i\).

Recognizing this cycle allows us to break down the expression quickly and determine what each power of \(i\) truly represents, simplifying calculations considerably and preventing errors in algebraic manipulation.