Problem 71 - College Algebra Essentials

Question

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.

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.