Problem 7 - Advanced Engineering Mathematics

Question

In Problems 1-26, write the given number in the form \(a+i b\). $$ i(5+7 i) $$

Step-by-Step Solution

Answer

The number is \(-7 + 5i\).

Step 1: Distribute the Imaginary Unit

To solve the problem, we need to distribute the imaginary unit \(i\) through the expression \(5 + 7i\). We start by multiplying \(i\) by each term inside the parentheses: \(i \times 5\) and \(i \times 7i\).

Step 2: Multiply and Simplify

First, multiply the real number: \(i \times 5 = 5i\). Then, multiply the imaginary part: \(i \times 7i = 7i^2\). Recall that \(i^2 = -1\). Substitute \(-1\) for \(i^2\), giving us \(7(-1) = -7\).

Step 3: Combine Terms

The expression simplifies to \(-7 + 5i\). In this form, the real part \(a\) is \(-7\), and the imaginary part \(b\) is \(5\).

Step 4: Write the Result in Standard Form

Finally, write the number in the form \(a + ib\) as \(-7 + 5i\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted by the letter \(i\), is a foundational concept in the study of complex numbers. It's defined as the square root of \(-1\), which is a number that doesn't exist in the realm of real numbers. This concept opened new possibilities in mathematics, allowing us to work with numbers in a two-dimensional number plane.

When you square the imaginary unit, you get \(i^2 = -1\). This property of \(i\) is crucial for performing operations with complex numbers, particularly when you're dealing with multiplication.

By understanding \(i\), you can perform operations that incorporate both real and imaginary numbers, paving the way for exploring a vast domain of numbers beyond just the real line. This further shows how \(i\) bridges the real numbers to complex number arithmetic.

Complex Number Multiplication

Multiplying complex numbers requires you to understand how to distribute and simplify expressions involving \(i\). In the exercise, we multiply the imaginary unit \(i\) by the complex number \(5 + 7i\).

Here's how it works step by step:

• First, distribute \(i\) to each term inside the parenthesis:
• Multiply \(i\) by \(5\) resulting in \(5i\).
• Then, multiply \(i\) by \(7i\) to get \(7i^2\).

Remember that \(i^2 = -1\). This identity allows us to simplify \(7i^2\) to \(7(-1)\), which equals \(-7\).

Ultimately, this multiplication results in the complex number written in standard form: \(-7 + 5i\). The real part from the multiplication is \(-7\), and the imaginary part is \(5\).

This operation showcases how multiplication affects both real and imaginary components and keeps the operations within the realm of complex numbers.

Imaginary and Real Parts

In a complex number, such as \(a + ib\), there are two parts you need to recognize: the real part \(a\), and the imaginary part \(b\). These are separate components, where\(
\)

• The real part, \(a\), is a real number.
• The imaginary part, \(b\), is effectively a real number multiplied by \(i\), which gives the term its imaginary nature.

For the complex number resulting from our exercise, \(-7 + 5i\), the real part is \(-7\), and the imaginary part is \(5\).

This concept is critical because any complex number can be analyzed or transformed by observing these two parts. It's similar to using x and y axes to locate a point in a coordinate system in geometry.

Understanding how to interpret these parts helps in graphically representing complex numbers, performing arithmetic operations, and solving equations where complex numbers are involved.