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Chapter 1: Problem 71

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 is false. The corrected statement is: All irrational numbers are complex numbers.

Step by step solution

01

Define the categories of the numbers

Irrational numbers are numbers that cannot be expressed as a ratio of two integers. Examples of irrational numbers are \(\sqrt{2}\), \(\pi\), and \(e\). On the other hand, complex numbers are all numbers that can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers.
02

Analyze the statement about the numbers

'Some irrational numbers are not complex numbers' - This statement is claiming that there exist some numbers which are irrational but not complex.
03

Evaluate the statement

All real numbers (which includes irrational numbers) can be considered as complex numbers with zero imaginary part, i.e, a complex number of the form \(a + 0i\). Therefore, all irrational numbers are indeed also complex numbers.
04

Conclude and correct the statement

It follows that, the given statement is false. The corrected statement that would be true is: 'All irrational numbers are complex numbers', because an irrational number falls within the set of real numbers, which is a subset of the set of complex numbers.

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