Understanding Number Systems and Identifying False Statements

The question asks us to identify which statement among the given options is false regarding different types of numbers: rational, irrational, integers, and real numbers. To answer this, we need to understand the definitions and relationships between these number sets.

Exploring Different Types of Numbers

Let's define the key number types mentioned in the options:

• Rational Numbers: Numbers that can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \ne 0\). Examples include \(\frac{1}{2}, -3, 0.75\).
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction \(\frac{p}{q}\) of integers. Their decimal representations are non-terminating and non-repeating. Examples include \(\sqrt{2}, \pi\), \(e\).
• Integers: Whole numbers (positive, negative, or zero). Examples include \(-3, -2, -1, 0, 1, 2, 3\). Integers are a subset of rational numbers because any integer \(n\) can be written as \(\frac{n}{1}\).
• Real Numbers: The set of all rational numbers and all irrational numbers combined. Real numbers can be represented on the number line.

Analyzing Each Statement

Now let's evaluate each statement given in the options:

Statement 1: \(\sqrt{2}\) is a rational number

• We need to determine if \(\sqrt{2}\) can be written as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \ne 0\).
• It is a well-known mathematical proof that \(\sqrt{2}\) cannot be expressed in this form. Its decimal representation is \(1.41421356...)\ which is non-terminating and non-repeating.
• Therefore, \(\sqrt{2}\) is an irrational number, not a rational number.
• This statement is false.

Statement 2: Every irrational number is a real number

• The set of real numbers is defined as the union of the set of rational numbers and the set of irrational numbers.
• By definition, every irrational number is included in the set of real numbers.
• This statement is true.

Statement 3: Every integer is a real number

• Integers are a subset of rational numbers (as any integer \(n\) can be written as \(\frac{n}{1}\)).
• Rational numbers are a subset of real numbers.
• Since integers are rational and rational numbers are real, every integer must also be a real number.
• This statement is true.

Statement 4: Every rational number is a real number

• As mentioned earlier, the set of real numbers is the union of rational and irrational numbers.
• This means that the set of rational numbers is a subset of the set of real numbers.
• Therefore, every rational number is included in the set of real numbers.
• This statement is true.

Based on the analysis, only the statement "\(\sqrt{2}\) is a rational number" is false.

Revision Table: Summary of Number Types

Additional Information on Number Systems

The different sets of numbers form a hierarchy, where each set is contained within the next larger set. Understanding this hierarchy helps in classifying numbers.

• Natural Numbers (\(\mathbb{N}\)): \{1, 2, 3, ...\}
• Whole Numbers (\(\mathbb{W}\)): \{0, 1, 2, 3, ...\} (includes 0)
• Integers (\(\mathbb{Z}\)): \{..., -2, -1, 0, 1, 2, ...\} (includes negative whole numbers)
• Rational Numbers (\(\mathbb{Q}\)): Numbers \(\frac{p}{q}\) (includes integers, fractions, terminating/repeating decimals)
• Real Numbers (\(\mathbb{R}\)): All rational and irrational numbers (includes all numbers on the number line)

The set of irrational numbers is disjoint from the set of rational numbers, meaning they have no elements in common. However, both sets together form the set of real numbers.

The relationships can be summarized as: \(\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\). Irrational numbers are part of \(\mathbb{R}\) but not \(\mathbb{Q}\).