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Chapter 3: Problem 4

Express the number in terms of i. $$\sqrt{-100}$$

Short Answer

Expert verified
10i

Step by step solution

01

- Understand the problem

Recognize that the problem involves finding the square root of a negative number.
02

- Identify the imaginary unit

Recall that the imaginary unit is defined as: \[ i = \sqrt{-1} \]
03

- Break down the radical expression

Rewrite the original expression using the properties of square roots: \[ \sqrt{-100} = \sqrt{100} \cdot \sqrt{-1} \]
04

- Simplify the square roots

Calculate the square root of 100 and use the definition of the imaginary unit: \[ \sqrt{100} = 10 \] and \[ \sqrt{-1} = i \]
05

- Combine the results

Multiply the results from Step 4: \[ 10 \cdot i = 10i \]

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Key Concepts

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

Square Roots
Square roots are a fundamental concept in mathematics. They represent a number which, when multiplied by itself, produces the original number. For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\). However, calculating square roots of negative numbers requires special consideration. Normally, square roots of negative numbers do not exist within the set of real numbers.
Instead, we use a special number called the imaginary unit (\
Complex Numbers
Complex numbers extend the idea of real numbers. They are made up of two parts: a real part and an imaginary part. They have the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Here, \(i\) stands for the imaginary unit, which is defined as \(i = \sqrt{-1}\). Complex numbers are crucial for various applications in engineering and physics.
In the context of our example, expressing \sqrt{-100}\ as \10i\ shows that \10\ is the real coefficient and \i\ is the imaginary unit. This combination makes the number a complex number.
Imaginary Unit
The imaginary unit \(i\) is a mathematical concept that allows us to extend the real number system to include solutions to equations that don't have real solutions. The defining property of \(i\) is that \(i^2 = -1\). This means that multiplying \(i\) by itself yields \-1\. Therefore, the square root of \-1\ is defined as \(i\). In every complex number, \(i\) plays a critical role in connecting the imaginary part with the real part of a complex number. For example, \sqrt{-100} = 10i\. This usage clarifies that although \-100\ is a negative number, it has a square root within the realm of complex numbers.

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Most popular questions from this chapter

Fill in the blank with the correct term. Some of the given choices will not be used. distance formula, midpoint formula, function, relation, \(x\) -intercept, y-intercept, perpendicular, parallel ,horizontal lines, vertical lines,symmetric with respect to the \(x\) -axis, symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, increasing, decreasing, constant \(\mathrm{A}(\mathrm{n})\)___ is a correspondence such that each member of the domain corresponds to exactly one member of the range.

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Solve. $$\sqrt{x+5}+\sqrt{x+2}=3$$
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