describe the set of numbers that are real numbers but can not be expressed as one integer divided by another

Answer 1

The set of numbers that are real numbers but cannot be expressed as one integer divided by another is known as the set of irrational numbers.

Irrational numbers are defined as numbers that cannot be expressed as the quotient (or fraction) of two integers \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \). In other words, an irrational number cannot be written in the form of a simple fraction.

Examples of irrational numbers include:

• The square root of any non-perfect square (e.g., \( \sqrt{2}, \sqrt{3}, \sqrt{5}, \ldots \))
• The mathematical constant \( \pi \) (approximately 3.14159...)
• The mathematical constant \( e \) (approximately 2.71828...)

These numbers have non-repeating, non-terminating decimal expansions, meaning that their decimal representation goes on forever without repeating a pattern.

In summary, the set of real numbers that cannot be expressed as one integer divided by another consists of all irrational numbers.

Answer 2

to which sets of real numbers does zero belong

Answer 3

Zero (0) belongs to several sets of real numbers, including:

1. Natural Numbers: Traditionally, the set of natural numbers (denoted as \(\mathbb{N}\)) includes the positive integers starting from 1 (1, 2, 3, ...). However, in some definitions, particularly in computer science and set theory, zero is included. So, in those cases, it can be counted as a natural number.

2. Whole Numbers: The set of whole numbers (which can be denoted as \(\mathbb{W}\)) includes all natural numbers and zero. This set looks like this: {0, 1, 2, 3, ...}.

3. Integers: Zero is part of the set of integers (denoted as \(\mathbb{Z}\)), which includes all positive and negative whole numbers, as well as zero: {..., -3, -2, -1, 0, 1, 2, 3, ...}.

4. Rational Numbers: Zero is also included in the set of rational numbers (denoted as \(\mathbb{Q}\)), which consists of numbers that can be expressed as the quotient of two integers. For example, \(0\) can be expressed as \(\frac{0}{1}\).

5. Real Numbers: Zero belongs to the set of real numbers (denoted as \(\mathbb{R}\)), which includes all rational and irrational numbers.

In summary, zero belongs to the sets of natural numbers (in some definitions), whole numbers, integers, rational numbers, and real numbers.