Problem 71 - Algebra and Trigonometry

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. A modified correct statement would be: 'All irrational numbers are complex numbers.'

Step 1: Understand the definitions

The first step is to understand the definitions of an irrational number and a complex number. An irrational number is any real number that cannot be expressed as a simple fraction. That means it cannot be written as a ratio, such as 7/5. These numbers, when written in decimal form, go on forever without repeating. Complex numbers, on the other hand, are numbers that consist of a real part and an imaginary part. They are typically written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\).

Step 2: Evaluate the statement

Now that we understand the definitions, we can evaluate the statement 'Some irrational numbers are not complex numbers'. Since all real numbers (including irrational numbers) can be thought of as complex numbers where the imaginary part \(b\) is zero (i.e., of the form \(a + 0i\)), the statement is not correct. All irrational numbers are in fact complex numbers. But there are complex numbers that are not irrational numbers (those with a non-zero imaginary part).

Step 3: Modify the statement to make it true

To make the original statement true, it could be changed to: 'All irrational numbers are complex numbers.'