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Chapter 0: Problem 53

If \(a\) is a positive number, is \(-\sqrt{-a}\) positive, negative, or not a real number?

Short Answer

Expert verified
Not a real number

Step by step solution

01

Define the given variable

The exercise states that \(a\) is a positive number. Let's denote \(a > 0\).
02

Evaluate the expression inside the square root

The expression inside the square root is \(-a\). Since \(a\) is positive, \(-a\) is a negative number.
03

Consider the square root of a negative number

The square root of a negative number is not a real number. Thus, \(\sqrt{-a}\) is not a real number.
04

Determine the sign of the expression

Since \(\sqrt{-a}\) is not a real number, the expression \(-\sqrt{-a}\) is also not a real number. It cannot be classified as positive or negative in real numbers.

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Key Concepts

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

square roots
Square roots answer the question: what number, when multiplied by itself, gives the original number? For example, \(\text{ \( \sqrt{9} = 3 \) }\). In this case, 3 is the square root of 9, because 3 multiplied by 3 equals 9. The square root symbol \(\text{ \( \sqrt{} \) }\) represents this concept. When dealing with negative numbers under the square root, we face special challenges.
negative numbers
Negative numbers are numbers less than zero. They are typically represented with a minus sign (\text{ \( - \) }) in front. For example, -5 is a negative number. In this exercise, we have -\text{ \(a \) }, since \text{ \(a \) } is positive, -\text{ \(a \) } is negative. Trying to take the square root of a negative number in the context of real numbers doesn't work because no real number squared gives a negative result.
real numbers
Real numbers include all positive numbers, negative numbers, and zero. They can be whole numbers, fractions, or decimals. Real numbers are represented on a number line. However, some problems arise when dealing with the square roots of negative numbers within real numbers. The main point to understand is that the square root of any negative number is not a real number.
complex numbers
Since the square root of a negative number isn't a real number, we need complex numbers. Complex numbers include a real part and an imaginary part. They are written in the form \text{ \( a + bi \) }, where \text{ \( a \) } is the real part, and \text{ \( bi \) } is the imaginary part. Here, \text{ \( i \) } represents the imaginary unit, defined as \(\text{ \( \sqrt{-1} \) }\). For instance, \(\text{ \( \sqrt{-a} = \sqrt{a}i \) }\). Thus, \(\text{ \( -\sqrt{-a} \) }\) is \(\text{ \( -\sqrt{a}i \) }\) and is not a real number.

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Most popular questions from this chapter

Simplify each inequality if needed. Then determine whether the statement is true or false. $$ -7<4+1 $$

Solve each problem. The amount, in billions of dollars, that Americans spent on their pets from 2003 to 2016 can be approximated by substituting a given year for \(x\) in the following expression. $$2.493 x-4962$$ (Data from American Pet Products Manufacturers Association.) Find the amount spent in each year. Round answers to the nearest tenth. (a) 2003 (b) 2010 (c) 2016 (d) How did the amount Americans spent on their pets change from 2003 to \(2016 ?\)

Complete the table of fraction, decimal, and percent equivalents. $$\begin{array}{|c|c|c|}\hline \begin{array}{c}\text { Fraction in Lowest Terms } \\\\\text { (or Whole Number) }\end{array} & \text { Decimal } & \text { Percent } \\\\\hline \frac{1}{10}& & \\\\\hline\end{array}$$

The distributive property can be used to mentally perform calculations. $$\begin{array}{l}38 \cdot 17+38 \cdot 3 \\\=38(17+3) \\\=38(20) \\\=760\end{array} $$Use the distributive property to calculate each value mentally. $$ 96 \cdot 19+4 \cdot 19 $$

Evaluate each expression for \(a=-3, b=64,\) and \(c=6 .\) $$ 3 a+\sqrt{b} $$
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