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Chapter 2: Problem 77

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Some irrational numbers are not complex numbers.

Short Answer

Expert verified
The statement 'Some irrational numbers are not complex numbers' is false. The correct statement should be: 'All irrational numbers are complex numbers.'

Step by step solution

01

Understand the Definitions

Start by understanding what irrational and complex numbers are. An irrational number is a real number that cannot be represented as a simple fraction. This means it has a decimal expansion that neither terminates nor repeats. Complex numbers, on the other hand, are numbers of the form a + bi, where a and b are real numbers, and i is the square root of -1.
02

Analyze the Statement

The statement says 'Some irrational numbers are not complex numbers', which suggests that there are irrational numbers that don't belong to the set of complex numbers. However, according to the definition of complex numbers, every real number (including irrational numbers) is a complex number where the b (imaginary part) is zero.
03

Evaluate the Statement

So, based on above analysis of definitions of irrational and complex numbers, the statement 'Some irrational numbers are not complex numbers' is false because every real number, including irrational, is indeed a complex number with zero imaginary part.

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Most popular questions from this chapter

What is a rational inequality?

Use a graphing utility to graph \(y=\frac{1}{x^{2}}, y=\frac{1}{x^{4}},\) and \(y=\frac{1}{x^{6}}\) in the same viewing rectangle. For even values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{1-\frac{3}{x+2}}{1+\frac{1}{x-2}}$$

Use inspection to describe each inequality's solution set. Do not solve any of the inequalities. $$\frac{1}{(x-2)^{2}}>0$$

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{3}+1}{x^{2}+2 x}$$
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