Problem 71 - College Algebra

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 'some irrational numbers are not complex numbers' is false. The corrected statement is 'All irrational numbers are complex numbers.'

Step 1: Understanding terminology

Irrational numbers are real numbers that cannot be expressed as a ratio of two integers. Examples include \(\sqrt{2}\) and \(π\). Complex numbers are numbers that can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is an imaginary unit with the property \(i^2 = -1\). Given that both \(a\) and \(b\) in \(a+bi\) can be real numbers, it is correct to say that all irrational numbers can be expressed as complex numbers.

Step 2: Evaluate the statement

The statement provided in the exercise was 'some irrational numbers are not complex numbers.' Considering that all irrational numbers can be expressed as complex numbers (as defined in the previous step), the statement is not accurate.

Step 3: Corrections to make the statement true

To correct the statement, one could rephrase it as 'All irrational numbers are complex numbers.' This assertion respects the nature of irrational and complex numbers as defined in step 1, making it accurate.