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Chapter 9: Problem 19

Find each product. $$ (2 i)(i) $$

Short Answer

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Step by step solution

01

Understand Complex Numbers

Recall that in complex numbers, the imaginary unit is represented by \(i\), where \(i = \sqrt{-1}\). Therefore, \(i^2 = -1\).
02

Multiply the Terms

Given the expression \( (2i)(i) \), apply the multiplication rule for complex numbers: \( 2i \times i \).
03

Simplify the Expression

Multiply the coefficients and the imaginary units: \2 \times i^2\. Since \(i^2 = -1\), replace \(i^2\) with \(-1\): \2 \times (-1) = -2\.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Imaginary Unit
In the realm of complex numbers, we encounter the concept of the imaginary unit, denoted as \(i\). This special number is defined by the property \(i = \sqrt{-1}\), which sounds strange at first. Normally, the square root of a negative number does not exist within the set of real numbers. That's where the imaginary unit comes into play.
The imaginary unit allows us to work with these numbers in a consistent way. Any multiple of \(i\) is considered an imaginary number. For example, \(2i\) and \(5i\) are both imaginary numbers. These numbers are crucial for solving equations that do not have a solution in the set of real numbers alone.
i^2 = -1
A key property of the imaginary unit \(i\) is that when you square it, you get -1. That is, \(i^2 = -1\).
This property comes directly from the definition of the imaginary unit: \(i = \sqrt{-1}\). Here's why: if you take \(\text{i}\) and multiply it by itself, you have \(i \times i = (\sqrt{-1}) \times (\sqrt{-1})\). When you multiply two square roots together, you just get the number under the root: \(\sqrt{-1} \times \sqrt{-1} = -1\).

This property is very useful when working with complex expressions, as it allows for the simplification of otherwise complex-looking terms.
Simplifying Complex Expressions
When simplifying complex expressions, we often need to apply basic arithmetic rules along with the special rules for the imaginary unit. Let's look at an example to make this clear.
Suppose we need to find the product \((2i)(i)\).

We can break this into simple steps:
  • Step 1: Multiply the coefficients and imaginary units separately. Here, we have coefficients of 2 and 1. So,\(2 \times 1 = 2\).
  • Step 2: Multiply the imaginary units. \(i \times i\) equals \(i^2\).
  • Step 3:Replace \(i^2\) with \(-1\) (as we know \(i^2 = -1\)).
  • Step 4: Combine the results. This gives us \(2 \times (-1) = -2\).
Therefore, \((2i)(i) = -2\). As you see, understanding the basic properties of \(i\) makes simplifying complex expressions straightforward.

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Most popular questions from this chapter

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