Question

Find an irrational number between the given pair of numbers to support the idea that irrational numbers are dense in real numbers.

a. 3.1415 and π
b. 3.3333 and 1/3
c. e^2 and √54
d. √64/2 and √16

Answer 1

The average of two irrational numbers,
`e^2` and √54, given by
`(√54 + e^2)/2`, is irrational, supporting the density of irrational numbers between them. Thus the option c is correct c.
`e^2` and √54

Step-by-step explanation:

To find an irrational number between
`e^2` and √54, we can consider the average of these two values. Let's denote this average as x. Thus, x = (
`e^2` + √54)/2. Now, we need to show that x is irrational.

Assume, for the sake of contradiction, that x is rational. This implies that both
`e^2` and √54 are also rational, as the sum and division of rational numbers result in rational numbers. However, we know that
`e^2` is irrational, and √54 is also irrational (since 54 is not a perfect square). This contradicts our assumption that x is rational, proving that x must be irrational.

In summary, the irrational number (√54 +
`e^2)/2` lies between
`e^2` and √54, supporting the idea that irrational numbers are dense in real numbers.