Problem 61 - Algebra for College Students

Question

Find each of the products and express the answers in the standard form of a complex number. $$(5 i)(4 i)$$

Step-by-Step Solution

Answer

-20

Step 1: Apply the Multiplication of Complex Numbers

To find the product of \((5i)(4i)\), first multiply the coefficients, which are 5 and 4. This gives: \(5 \times 4 = 20\).

Step 2: Multiply the Imaginary Units

Now, multiply the imaginary units \(i \times i\). Recall that \(i^2 = -1\). So, \(i \times i = i^2 = -1\).

Step 3: Calculate the Product

Combine the results from Steps 1 and 2: \( 20 \times i^2 = 20 \times (-1) = -20 \).

Step 4: Express as a Standard Form Complex Number

The product \(-20\) is already in the standard form of a complex number, as its imaginary part is 0 (i.e., \(-20 + 0i\)).

Key Concepts

Standard Form of a Complex Number

The standard form of a complex number is quite straightforward. It takes the form of \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) is known as the real part, and \(b\) is the coefficient of the imaginary part.
However, when it's just a real number, like \(-20\), it can still be in standard form as \(-20 + 0i\).
Even though the imaginary part is zero, it is acceptable because you can write it this way without changing the value. Whenever you deal with complex numbers, put them in standard form to clearly express both their real and imaginary parts. This makes calculations, comparisons, and graphs more seamless. It's a great way to maintain clarity in your math work!

Imaginary Unit

The imaginary unit is fundamental when working with complex numbers. It’s symbolized by \(i\) and has a distinctive property: \(i^2 = -1\). This concept helps us handle the square roots of negative numbers with ease.
In mathematics, the term 'imaginary' might sound like it's not real or practical, but it's an essential part of complex number theory.
Understanding the imaginary unit is crucial because it forms the basis for writing and manipulating complex numbers.

• The imaginary unit allows us to extend the real number system.
• Operations with \(i\) follow specific rules, like \(i^2 = -1\).
• When multiplying imaginary units together, remember this rule to simplify correctly.

The imaginary unit is a small but mighty component, bringing depth and breadth to number systems. It’s an exciting part of learning math that opens up new possibilities.

Multiplication of Complex Numbers

Multiplying complex numbers may seem tricky, but you only need a few straightforward steps. When multiplying numbers like \((5i)(4i)\), follow these guidelines:

• First, multiply the coefficients: 5 and 4 to get 20.
• Next, handle the imaginary parts: \(i \times i \). Since \(i^2 = -1\), replace \(i^2\) with \(-1\).
• Finally, combine results: Multiply 20 by \(-1\) to get \(-20\).

This systematic approach makes complex-number multiplication manageable.
Always write the answer in standard form for full clarity. This ensures you are considering both the real and potential imaginary parts.
Practice these steps, and soon multiplying complex numbers will feel like second nature!