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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

1 Answer

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Final Answer:

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.

User Mythriel
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