Question

Are there any real numbers that are both rational and irrational? Are there any real numbers that are neither? Explain your reasoning.

Short answer

No real numbers are both rational and irrational, and no real numbers are neither.

Step-by-step solution

- Definitions

First, define what rational and irrational numbers are. Rational numbers can be expressed as a fraction \(\frac{a}{b}\) where both \(a\) and \(b\) are integers and \(b eq 0\). Irrational numbers cannot be expressed as a fraction of two integers.

- Examine the Possibility of Intersection

Since a number cannot be both expressible as a fraction and not expressible as a fraction at the same time, there are no real numbers that can be both rational and irrational.

- Real Number System

All real numbers are either rational or irrational since every real number must fall into one of these two categories by definition. Therefore, there are no real numbers that are neither rational nor irrational.

- Conclusion

From the definitions and properties of rational and irrational numbers, conclude that a real number cannot be both or neither.

Key concepts

Rational Numbers

Rational numbers are numbers that can be written as a fraction \(\frac{a}{b}\) where both \(a\) and \(b\) are integers and \(b eq 0\). This includes numbers like 1/2, -3/4, and even whole numbers and integers like 5 and -7 because they can be written as fractions (e.g., 5/1 and -7/1).
Some important properties of rational numbers:

• They can be either positive, negative, or zero.
• The decimal expansion of a rational number is either finite or repeating. For example, 1/4 = 0.25 (finite) and 1/3 = 0.333... (repeating).

Rational numbers are a significant part of the number system because they are able to represent most quantities and measurements in everyday life.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a fraction of two integers. They include numbers like \(\frac{\text{π}}\) (Pi), \(\frac{e}\) (the base of the natural logarithm), and \( \sqrt{2}\) (the square root of 2).
Some key characteristics of irrational numbers:

• Their decimal expansion goes on forever without repeating.
• They fill the gaps between rational numbers on the number line, making the real number system continuous.
• Though they cannot be written as simple fractions, irrational numbers are crucial in various fields of mathematics, science, and engineering.

These numbers often arise from geometrical contexts, such as the length of a diagonal in a square, and are essential for understanding the complete set of real numbers.

Number Classification

In the study of real numbers, it is important to understand how they are categorized. All real numbers fall into one of two categories: rational or irrational.
Here's a quick summary of how they are classified:

• Real Numbers: All the numbers on the number line, including both rational and irrational numbers.
• Rational Numbers: Can be written as fractions \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\). Example: 7, -2, 0.5, 3/4.
• Irrational Numbers: Cannot be written as fractions of two integers. They have non-repeating, infinite decimal expansions. Example: \(\frac{\text{π}}\), \( \sqrt{3} \), \( \sqrt{2} \).

By definition, every real number must be categorized as either rational or irrational. It is impossible for a number to be both rational and irrational simultaneously. Furthermore, there are no real numbers that are neither rational nor irrational since those two categories encompass all real numbers. This classification helps in understanding and solving various mathematical problems.