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Chapter 2: Problem 9

In Exercises 1 to 10 , write the complex number in standard form. $$8-\sqrt{-18}$$

Short Answer

Expert verified
The complex number in standard form is \(8 - 3\sqrt{2}i\).

Step by step solution

01

Translate \(\sqrt{-1}\) to \(i\)

The square root of -1 is equal to \(i\). Thus, \(\sqrt{-18}\) can be written as \(\sqrt{18} \cdot \sqrt{-1}\), or \(3\sqrt{2}i\).
02

Use combining like terms to put the expression in standard form

Substitute \(3\sqrt{2}i\) back to the given expression, and combine like terms to simplify. This makes the expression become \(8 - 3\sqrt{2}i\).

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Key Concepts

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

Standard Form
In mathematics, especially when dealing with complex numbers, it's important to express numbers in a consistent and recognizable format. The standard form for a complex number is written as \(a + bi\). This makes identifying and manipulating complex numbers easier. Here, \(a\) is the real part, and \(bi\) is the imaginary part, where \(b\) is a real number and \(i\) is the imaginary unit. This standard form is
  • Clear and straightforward
  • Makes calculation and comparison simple
  • Consistent for notation across mathematical disciplines
For instance, when given an expression like \(8 - \sqrt{-18}\), reorganizing it into \(a + bi\) form helps in not only recognizing its components but also in operations like addition, subtraction, or even plotting on the complex plane.
Imaginary Unit
The imaginary unit, denoted as \(i\), is a cornerstone of complex numbers. It is defined by the equation \(i^2 = -1\). This property allows mathematicians and engineers to work with square roots of negative numbers, something that is impossible with real numbers alone. Here's a breakdown of what the imaginary unit makes possible:
  • Allows computation with negative roots through the expression of \(i\)
  • Facilitates transformations and operations in complex problem-solving scenarios
  • Empowers the representation of numbers required in advanced fields such as electrical engineering and quantum physics
For example, translating \(\sqrt{-18}\) in the exercise involves acknowledging \(i\) by first isolating \(\sqrt{-1}\) resulting in \(3\sqrt{2}i\), where \(3\sqrt{2}\) is the magnitude, and \(i\) signifies the imaginary component.
Simplification
Simplification of expressions, particularly when dealing with complex numbers, entails reducing them to their most straightforward form without altering the value. This involves combining like terms and organizing them to reveal their simplest state. Let's look at the key steps:
  • Identify and separate real and imaginary parts
  • Use known identities and properties for reduction, such as \(\sqrt{-1} = i\)
  • Merge components into a form that's easy to read and calculate with, like \(8 - 3\sqrt{2}i\)
During simplification, the goal is to express complex numbers in a form where both components are distinguished, making it easier to undertake operations such as addition, multiplication, or further simplification if needed. The exercise's reduction from \(8 - \sqrt{-18}\) to \(8 - 3\sqrt{2}i\) is a prime example of maintaining clarity and operability through proper simplification.

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