Question

Provide an example of application of the density property of real numbers.

Answer 1

The density property of real numbers says that there is always a real number between two other real numbers.

Between the number 1 and 2, some examples of this are 1.252,
`√(2)`, 1.5,
`(\pi )/(2)`, etc.

An application of this is the intermediate value theorem, which states that in a continuous function on interval whose domain contains the interval

[a, b], then any given value between f(a) and f(b) exists at some point within the interval.

We can say that if I am on a swing, and a function gives my height, there is an infinite amount of real numbers between my minimum height and maximum height. If I swing 7 feet off the ground, and return to 1 foot height, at some point I was 4 feet in the air,
`√(2)` feet in the air, or
`\pi` feet in the air.

Hope this helps,

Jeron

:- )