Finding an irrational number between which given pair of numbers supports the idea that irrational numbers are dense in real numbers?

Step-by-step explanation:

"Dense" here means that there are infinite irrational numbers between two rational numbers. Also, there are infinite rational numbers between two rational numbers. That's the meaning of dense. Actually, that can be apply to all real numbers, there always is gonna be a number between other two.

But, to demonstrate that irrationals are dense, we have to based on an interval with rational limits, because the theorem about dense sets is about rationals, and the dense irrational set is a deduction from it. That's why the best option is 2, because that's an interval with rational limits.

We can say that a set is dense if between any two elements of that set, we can find another element of the same set.

We will see that the correct option is the first one, 3.14 and pi.

Now, if we want to say that the set of irrational numbers is dense in real numbers, then we need to, for two given real numbers, find an irrational number between them.

We know that irrational numbers are dense (between two irrational numbers we can find another irrational number) so to expand this to the real set (the union of irrational numbers and rational numbers) we should be able to find an irrational number between a rational number and an irrational number (or between an irrational number and a rational number)

Then the correct option will be the one that has one of each, which is the first one.

Finding an irrational number between these two means that the irrational numbers are dense in the real set.

(Note that the third option also has an irrational number and a rational number, but the first one, e^2, is larger than the second one √54, so it is written incorrectly, and this is why we did not take that option).

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