Describe the relationship between rational numbers and whole numbers

Answer 1

Rational numbers and whole numbers are both subsets of the real number system, but they have distinct definitions and characteristics that outline their relationship.

1. Definitions:

   • Whole Numbers: Whole numbers are the set of non-negative integers, which includes 0, 1, 2, 3, and so on. In mathematical terms, the set can be represented as $W=0,1,2,3,\dots$.
   • Rational Numbers: Rational numbers include all numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. In formal notation, a rational number can be expressed as $\frac{a}{b}$, where $a$ and $b$ are integers, and $b\ne 0$.

2. Inclusion:

   • Every whole number is a rational number. This is because any whole number $n$ can be expressed as $\frac{n}{1}$. Thus, the set of whole numbers (W) is a subset of the set of rational numbers (Q). In set notation, we can express this relationship as $W\subseteq Q$.

3. Examples:

   • Examples of whole numbers: 0, 1, 2, 3, ...
   • Examples of rational numbers: $\frac{1}{2},-\frac{3}{4},2,0,\frac{0}{1}$ (where 2 and 0 are whole numbers).

4. Decimal Representation:

   • Rational numbers can also be expressed in decimal form. They may terminate (e.g., $\frac{1}{4}=0.25$) or repeat (e.g., $\frac{1}{3}=0.333...$). Whole numbers, when expressed in decimal form, are simply represented by themselves (e.g., 0, 1, 2, ...).

5. Conclusion:

   • The relationship between rational numbers and whole numbers is that all whole numbers are rational, but not all rational numbers are whole numbers. Whole numbers are a specific, limited category within the broader category of rational numbers.