Density Property of Rational Numbers: Statement, Proof, Examples

The set of rational numbers is dense in the set of real numbers, that is, Q is dense in R. This is regarded as the density property of rational numbers. Before we proof this property, let us recall what are rational numbers.

A number of the form p/q with q≠0 (p, q integers) is called a rational number. The set of all such numbers is denoted by the symbol Q.

ℝ: The set of real numbers.

Statement of Density Property

Between any two rational numbers, there exists another rational number, in fact infinitely many rational numbers.

Proof of Density Property

Let us consider two rational numbers x and y with x<y. To prove the density property, we need to show that ∃ r∈ Q such that

x < r < y.

As x< y, we have that

x+y < y+y.

⇒ ½ (x+y) < ½ × 2y

⇒ $\dfrac{x+y}{2}$ < y …(I)

On the other hand, x< y implies that

x+x < x+y.

⇒ ½ × 2x < ½ (x+y)

⇒ x < $\dfrac{x+y}{2}$ …(II)

From (I) and (II), we conclude that the rational number (x+y)/2 lies between x and y. This proves that between any two rational numbers, there exists another rational number.

Applying the same argument, we can say that there is a rational number between x and (x+y)/2. This way we get an infinite number of rational numbers between x and y. That is, between any two rational numbers there exist infinitely many rational numbers. So the set Q of rational numbers is dense; and this is called the density property of rational numbers.

See Also

• Completeness Axiom of Real Numbers
• Supremum & Infimum of a Set
• Archimedean Property

Examples

Consider two rational numbers 1/3 and 1/2.

See that $\dfrac{1/3 + 1/2}{2}=\dfrac{5}{12}$ lies between 1/3 and 1/2. That is, 1/3 < 5/12 < 1/2.

This way the number (1/3 + 5/12)/2 = 9/24 lies between 1/3 and 5/12. Again, the number (5/12 + 1/2)/2 = 11/24 lies between 5/12 and 1/2.

Therefore, the numbers 9/24, 5/12, 11/24 lie between the numbers 1/3 and 1/2. In a similar fashion, we can get infinitely many rational numbers between 1/3 and 1/2. This is an example of the density property of rational numbers.

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