Problem 13 - Advanced Engineering Mathematics

Question

In Problems 1-26, write the given number in the form \(a+i b\). $$ \frac{2}{i} $$

Step-by-Step Solution

Answer

\(0 + 2i\)

Step 1: Understand the Problem

We need to rewrite the expression \(\frac{2}{i}\) in the form \(a + ib\), where \(a\) and \(b\) are real numbers. The number \(i\) is the imaginary unit, where \(i^2 = -1\).

Step 2: Multiply by the Conjugate

To eliminate the imaginary unit \(i\) from the denominator, multiply both the numerator and the denominator of \(\frac{2}{i}\) by \(-i\), the conjugate of \(i\). This results in: $$ \frac{2 \times (-i)}{i \times (-i)} = \frac{-2i}{i^2}. $$

Step 3: Simplify the Expression

Simplify \(\frac{-2i}{i^2}\), knowing that \(i^2 = -1\). We have: $$ \frac{-2i}{-1} = 2i. $$

Step 4: Write in the Form \(a + ib\)

The expression \(2i\) can be written in the form \(a + ib\) by setting \(a = 0\) and \(b = 2\). Thus, \(2i = 0 + 2i\).

Key Concepts

Imaginary Unit

When dealing with complex numbers, the imaginary unit is a crucial concept to grasp. It is denoted by the letter \(i\), and it holds the unique property that when squared, \(i^2 = -1\). This is not something that can happen with real numbers, making \(i\) fundamental for expanding the number system to include complex numbers. If you're working with complex expressions, the imaginary unit often appears in equations and manipulations, as seen in expressions like \(2i\). Here, \(i\) helps represent numbers that are not found on the traditional real number line.
Understanding the imaginary unit is key to exploring complex number arithmetic. It often appears in polynomial equations that have no real solutions, and helps to express solutions in a broader mathematical context.

Complex Conjugate

The concept of a complex conjugate is vital when working with division in complex numbers. A complex conjugate of a number in the form \(a + ib\) is \(a - ib\). It is essentially the mirror image of the number across the horizontal axis in the complex plane. Complex conjugates are handy, particularly in eliminating imaginary units from denominators.
For instance, to simplify \(\frac{2}{i}\), we multiply both the numerator and the denominator by \(-i\), resulting in \(\frac{-2i}{i^2}\). This technique ensures that the denominator is a real number since \(i \times (-i) = i^2 = -1\). Using the complex conjugate leverages properties of imaginary units to simplify complex fractions by removing the \(i\) from the denominator.

Real and Imaginary Parts

Every complex number consists of two parts: a real part and an imaginary part. The number is usually expressed in the form \(a + ib\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary unit \(i\). Understanding this form helps to categorize and work with complex numbers easily.
In the expression \(2i\), it might seem like there's no real part at first glance. However, it's implicitly understood that the real part is zero, i.e., \(2i = 0 + 2i\). All parts are accounted for despite not being prominently visible. When solving problems involving complex numbers, identifying these parts is crucial. This helps in performing operations such as addition, subtraction, and in finding magnitudes or conjugates of complex numbers.