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Chapter 2: Problem 19

Write the complex number in standard form.$$-10 i+i^{2}$$.

Short Answer

Expert verified
The given complex number in standard form is \(-1 - 10i\).

Step by step solution

01

Understand the properties of \(i\)

First of all, it should be understood that the imaginary unit \(i\) is defined as \(\sqrt{-1}\). Hence, \(i^{2} = -1\).
02

Substitute \(i^{2}\) by its value

Replace \(i^{2}\) in the expression by \(-1\). The equation therefore transforms from \(-10i + i^{2}\) to \(-10i - 1\).
03

Write the expression in standard form

The complex number in standard form, \(a + bi\), is written with the real part first followed by the imaginary part. Rewrite the expression to \(-1 - 10i\).

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Key Concepts

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

Imaginary Unit
The imaginary unit, denoted as \(i\), is a fundamental concept in the world of complex numbers. It provides a way to work with numbers that cannot be represented on the traditional number line. By definition, the imaginary unit represents the square root of \(-1\).
This unique property allows \(i\) to enable the expansion of real numbers into the set of complex numbers, where every number has a real part and an imaginary part.
Complex numbers take the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. Understanding \(i\) is crucial because it sets the foundation for performing mathematical operations involving complex numbers.
Standard Form
Complex numbers are best understood when represented in a specific way called the standard form.
This representation clearly shows both the real and imaginary components of the number.
The standard form of a complex number is \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
  • The real part of the complex number is any real number, positive or negative, whole, or fraction.
  • The imaginary part also involves a real number but is always multiplied by the imaginary unit \(i\).
When writing a complex number in standard form, make sure to keep the real term before the imaginary term.
In exercises, like transforming \(-10i + i^2\) into standard form, the real part should always be placed first, leading to \(-1 - 10i\) as the correct answer.
Properties of i
To work effectively with complex numbers, it is essential to know the properties of the imaginary unit \(i\).
The most basic property is \(i^2 = -1\). This fundamental relationship stems directly from the definition \(i = \sqrt{-1}\).
These properties allow us to simplify expressions and transform complex numbers in order to perform algebraic operations more easily.
  • Knowing \(i^2 = -1\) lets us replace any power of \(i^2\) in calculations.
  • For higher powers, we can use the cyclical pattern: \(i^3 = -i\) and \(i^4 = 1\).
  • This cycle repeats every four powers \(i^5 = i\) and so forth, making it predictable and manageable.
Understanding these properties not only helps in solving exercises but also builds a foundation for more complex algebraic and geometric explorations involving complex numbers.

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