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1. Find some more rational numbers that are close to √2.

User Raycohen
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Answer:

Let A be the set of all rational numbers p such that p2<2, and let B be the set of all rational numbers p such that p2>2. It can be shown that A has no largest element and that B has no smallest element by associating to each p>0 the number

q=p−p2−2p+2

and then proving that p∈A⇒q∈A and p∈B⇒q∈B.

Explanation:

We want to find a rational number q such that

2–√<q<p.

One idea is try to lower p, like

q=k−1kp<p

with a suitable rational k. If you write the condition 2–√<q, you get

2–√<k−1kp

and obtain

k≥pp−2–√.

Any positive rational number k above that bound gives you a good q. In order to get a rational number, you try to get rid of the 2–√ by rationalization

pp−2–√=p(p+2–√)p2−2

and 2>2–√, so

k=p(p+2)p2−2>p(p+2–√)p2−2.

Substituting it in q, you obtain

q=k−1kp=p−pk=p−p2−2p+2.

1. Find some more rational numbers that are close to √2.-example-1
User Samczsun
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