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Chapter 1: Problem 9

The imaginary number \(i\) is defined so that \(i=\sqrt{-1}\) and \(i^{2}=\) _____ .

Short Answer

Expert verified
i^2 = -1

Step by step solution

01

Understand the definition of the imaginary unit

The imaginary unit is defined as the square root of -1. This means: \(i = \sqrt{-1}\)
02

Square the imaginary unit

To find what \(i^2\) equals, square both sides of the equation from Step 1: \(i^2 = (\sqrt{-1})^2\)
03

Simplify the expression

We know that squaring the square root of a number will give us the number itself. Therefore, \(i^2 = -1\)

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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 is a concept in mathematics that helps us when dealing with the square root of negative numbers, which are not possible in the realm of real numbers. The imaginary unit is denoted by the letter i. It is defined as follows:

\(i = \sqrt{-1}\)

This definition allows us to work with numbers that we previously couldn't handle. For example, using the imaginary unit, we can say:

\sqrt{-4} = 2i\

Here, i acts as a sort of placeholder to show that we're dealing with the square root of a negative number.
square root
The square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical notation, if \(x = \sqrt{y}\), then \ x^2 = y\.

For example, \2^2 = 4\, so \sqrt{4} = 2\. But what if we need to find the square root of a negative number? In the realm of real numbers, this is not possible. That's where the imaginary unit comes in. With the imaginary unit, we can take the square root of negative numbers:

\(\sqrt{-1} = i\)

Therefore, to find the square root of any negative number, we use the imaginary unit i.
complex numbers
Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form:

\(a + bi\)

Where a is the real part and b is the imaginary part. A complex number allows us to include the imaginary unit in calculations. For example:

\(3 + 4i\)

In this complex number, 3 is the real part and 4i is the imaginary part. Complex numbers are very useful in various fields of science and engineering, as they can represent phenomena that real numbers cannot.
algebra
Algebra is a branch of mathematics that deals with the properties and manipulation of symbols and numbers. It allows us to solve equations and understand relationships between quantities. When it comes to imaginary numbers and complex numbers, algebra provides tools to work with them just like with real numbers.

Some key algebraic operations involving imaginary and complex numbers include:
  • Addition: \( (a + bi) + (c + di) = (a+c) + (b+d)i \)
  • Subtraction: \( (a + bi) - (c + di) = (a-c) + (b-d)i \)
  • Multiplication: \( (a + bi)(c + di) = (ac - bd) + (ad + bc)i \)


Using algebraic rules and properties, solving equations involving these numbers becomes manageable and seamless.

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Most popular questions from this chapter

Write the solution set. a. \(18=4-|y+7|\) b. \(18 \leq 4-|y+7|\) c. \(18 \geq 4-|y+7|\)

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Write the solution set. a. \(|k+4|=0\) b. \(|k+4|<0\) c. \(|k+4| \leq 0\) d. \(|k+4|>0\) e. \(|k+4| \geq 0\)

Write a three-part inequality to represent the given statement. A small plane's average speed \(s\) is at least \(220 \mathrm{mph}\) but not more than \(410 \mathrm{mph}\).

Solve the inequality, and write the solution set in interval notation. \(-11 \leq 5-|2 p+4|\)
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