Question

Write two rational numbers between √4 and √5


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Answer 1

Answer:

Consider an invertible function f:(4,5)→R . Certain functions such as the n -th power, for n≥2 will do. Let's pick n=2 . Then f(x)=x2 and f−1(x)=(√x) .

Now f((4,5))=(16,25) . Pick two different integers b1,2∈{17,…,24}⊆(16,25) . By construction f−1((16,25))=(4,5) , so f−1(b1,2)∈(4,5) .

Now the question is, are these numbers irrational. Well, see if you can find any non-integral rational number whose square is an integer. (Hint: consider x=pq, p,q∈Z and q/| p , and see if there's any reason why x2∈Z .) If you can't find any such number, then doesn't it follow that b1,2−−−√≠Q ?

If so, then you have your numbers.

Of course, you have a lot of other numbers between 4 and 5 from which to choose. See if you can construct other functions that would give you irrational numbers between 4 and 5 using the same sort of approach.

Step-by-step explanation:

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