Problem 39 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and write the result in standard form. $$4 i(8+5 i)$$

Step-by-Step Solution

Answer

The result of the operation is \(-20 + 32i\).

Step 1: Distribute the real number

Distribute the real number (4i) across each term in the complex number \(8+5i\).

Step 2: Perform the multiplication

The multiplication will look like this: \(4i*8 + 4i*(5i) = 32i + 20i^2\).

Step 3: Simplify the result

Since \(i^2 = -1\), replace \(i^2\) in the equation. This results in \(32i - 20\). The standard form for a complex number is \(a+bi\), where a is the real part and b is the imaginary part. So you should rearrange the expression to: \(-20 + 32i\).

Key Concepts

Imaginary Unit

The concept of imaginary numbers can seem perplexing at first, but it is essential in the realm of complex numbers. The imaginary unit, denoted by \(i\), is defined by the property \(i^2 = -1\). This means that \(i\) is the square root of \(-1\). That's why it's called 'imaginary' – it doesn't correspond to a real number on the number line.
\(i\) allows for a broader number system where equations that include negative square roots have solutions. It extends the real numbers into what is known as complex numbers, making it easier to handle and solve equations that were otherwise unsolvable within the real numbers alone.
To see \(i\) in action, consider squaring the imaginative component of complex numbers:

• If you square \(i\),: \(i \times i = i^2 = -1\).
• When you see higher powers of \(i\), keep in mind the cycle: \(i^3 = -i\), \(i^4 = 1\), and this cycle repeats.

Understanding \(i\) is your first step into the exciting world of complex analysis.

Multiplication of Complex Numbers

Multiplication of complex numbers is a key operation that you'll often perform. It involves both the real and imaginary parts of the numbers involved. Let's dissect how it works through the given example.
The task involves multiplying \(4i\) by the complex number \(8+5i\). This is done by distributing \(4i\) to each part of \(8+5i\). The steps translate mathematically to:

• Multiply the real part: \(4i \times 8 = 32i\).
• Multiply the imaginary part: \(4i \times 5i = 20i^2\). While doing this, recall that \(i^2 = -1\).

After performing these operations, you combine like terms and apply the property of \(i^2\), resulting in \(32i - 20\).
Complex number multiplication not only strengthens algebraic skills but also helps with polynomial roots and other advanced topics.

Standard Form of Complex Numbers

The standard form of complex numbers is a way of expressing complex numbers for easy comprehension and calculation. It is written in the form \(a + bi\), where:

• \(a\) is the real part.
• \(b\) is the coefficient of the imaginary part \(i\).

In our example, after multiplication and simplification, the complex number is \(32i - 20\). To write this in standard form, adjust it to \(-20 + 32i\). The real part is \(-20\), and the imaginary part is \(32i\).
Presenting complex numbers in this standard form makes it easier to add, subtract, and analyze them. This format is especially useful for graphing and understanding the geometric meaning of complex number operations, such as transformations and rotations in the complex plane.