Real Numbers Composition Explained

The question asks to identify the components of the set of real numbers ($\mathbb{R}$). Understanding the different number sets is key.

Components of Real Numbers

• Rational Numbers ($\mathbb{Q}$): Numbers that can be expressed as a fraction $\frac{p}{q}$, where p and q are integers and $q \neq 0$. Examples include integers (like 5), terminating decimals (like 0.5), and repeating decimals (like 0.333...). Whole numbers are a subset of rational numbers.
• Irrational Numbers ($\mathbb{I}$): Numbers that cannot be expressed as a simple fraction $\frac{p}{q}$. Their decimal representations are non-terminating and non-repeating. Examples include $\pi$ and $\sqrt{2}$.

Definition of Real Numbers

The set of real numbers ($\mathbb{R}$) is defined as the union of the set of rational numbers ($\mathbb{Q}$) and the set of irrational numbers ($\mathbb{I}$).

Mathematically, this is represented as: $ \mathbb{R} = \mathbb{Q} \cup \mathbb{I} $ Therefore, real numbers include both rational and irrational numbers.

Analyzing the Options

• Option 1 is incorrect because real numbers also include irrational numbers.
• Option 2 is incorrect because real numbers also include rational numbers.
• Option 3 is correct as it accurately states that real numbers encompass both rational and irrational numbers.
• Option 4 is incorrect because while whole numbers are part of the real number system, they do not represent the entire set.