Between Every Two Real Numbers Lies Another Real Number: A Proven Property

Imagining the real number line, one might question whether there is always a third real number between any two given numbers, no matter how close they are. Through rigorous mathematical proof, this property, known as the density of real numbers, is established. This concept explores how the real number system is dense and uncountable, ensuring that there is always an infinite number of real numbers between any two given points.

Proof of the Density of Real Numbers

Let us take any two real numbers a and b such that a b. Our goal is to show that there exists at least one real number c such that a c b. A simple and effective choice for c is the average of a and b: frac{a b}{2}

To prove that c is between a and b, we need to verify two things: that c a and that c b.

Show that c a

Starting with the fact a b, we can write: a a b Adding a to both sides yields:

2a a b

Dividing both sides by 2, we get:

frac{a b}{2} a

Show that c b

Similarly, since a b, we can write:

b b b

Adding b to both sides yields:

a b 2b

Dividing both sides by 2, we get:

frac{a b}{2} b

Thus, we have shown that a c b. This confirms the existence of at least one real number c between any two distinct real numbers a and b.

Infinite Real Numbers Between Any Two Distinct Real Numbers

This property can be further generalized. For instance, we can choose c a - frac{b - a}{3} or c a epsilon for any small epsilon 0 such that c b. This demonstrates the density of the real numbers and confirms that there are an infinite number of such real numbers between any two distinct real numbers A and B.

Exploring CoPrime Pairs

To further illustrate the density of real numbers, consider any two real numbers A and B with A B. For each co-prime pair of integers (X, Y) satisfying 0 X Y, we can find a real number between A and B given by:

A frac{X}{Y}(B - A)

Since there are an infinite number of such co-prime pairs, there are an infinite number of real numbers between A and B that can be expressed in this form.

So, through rigorous proofs and concrete examples, we can confidently claim that the set of real numbers is indeed dense, ensuring the existence of an infinite number of real numbers between any two given points.