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Chapter 8: Problem 94

Explain why \(\sqrt{-1}\) is not a real number.

Short Answer

Expert verified
The square root of -1, denoted as \(\sqrt{-1}\), is not a real number because the square of any real number is always non-negative and there does not exist a real number whose square generates a negative result. However, \(\sqrt{-1}\) exists in the set of complex numbers and is represented by the imaginary unit 'i'.

Step by step solution

01

Understanding Real Numbers

Real numbers are those numbers which can be found on the number line. This includes both the rational and irrational numbers. In more formal words, any number which can be expressed as a ratio where both the numerator and the denominator are integers is known as a real number.
02

Understanding the Square Root Concept

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because if 3 is multiplied by 3, it produces 9, which is the original number.
03

Understanding why Square Root of a Negative number is not a Real Number

In context of real numbers, when we take the square of any real number, whether positive or negative, the result is always a positive number. So, it is impossible to find a real number which when squared gives a negative result.
04

Scrutinizing the Square Root of -1

As established above, there does not exist a real number whose square is negative. Therefore, \(\sqrt{-1}\) cannot be a real number. Instead, \(\sqrt{-1}\) is defined as 'i', an imaginary unit in the set of complex numbers, where 'i' stands for 'imaginary'.

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Most popular questions from this chapter

In Exercises \(75-82\), rationalize each denominator. Simplify, if possible $$\frac{2 x+4-2 h}{\sqrt{x+2-h}}$$

Make Sense? In Exercises \(90-93,\) determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I use the fact that 1 is the multiplicative identity to both rationalize denominators and rewrite rational expressions with a common denominator.

Without using a calculator, simplify the expressions completely. $$25^{\frac{1}{4}} \cdot 25^{-\frac{3}{4}}$$

Simplify by first writing the expression in radical form. If applicable, use a calculator to verify your answer. $$(-64)^{-\frac{2}{3}}$$

What is the meaning of \(a^{-\frac{m}{n}} ?\) Give an example.
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