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.

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## 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

## FAQs

**Q1: What is the density property of rational numbers?**

Answer: The density property of Q (set of rational numbers) states that between any two rational numbers x and y (where x<y) there exist infinitely many rational numbers.