Prove that there is a digit that appears infinitely often in the decimal expansion of $\sqrt{7}$. [closed]

Prove that there is a digit that appears infinitely often in the decimal expansion of $$\sqrt{7}$$.

closed as off-topic by Connor Harris, Don Thousand, ArsenBerk, Toby Mak, ShaileshOct 20 '18 at 2:54

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• Hi and welcome to MSE. What are your thoughts on the problem? What have you tried? – Surb Oct 19 '18 at 20:49

I'd prove it by absurdum saying that if there are no digits that appear infinitely often, then each one of the 10 digits appears at most a finite number of times. So the decimal expansion of $$\sqrt(7)$$ would be finite and this would be a contradiction, because if it was finite, then I can show that there is a rational number that has the same expansion. This leads to say that $$\sqrt(7)$$ is rational.

• What do you mean by “if there are no digits that appear infinitely often, then each one of the 10 digits appears at most once”? – JavaMan Oct 20 '18 at 11:29
• Sorry, my mistake. Thanks for pointing it out. Edit: I've corrected it. I'm sorry, but when I wrote the answer the time for me was 11.40 PM, and I was getting pretty tired.. – LuxGiammi Oct 20 '18 at 12:59

Hint: $$\sqrt 7$$ is irrational. What would happen if all digits appeared only finitely many times? Moreover, this shows that at least $$2$$ digits must appear infinity often.

• Still need to prove that $\sqrt{7}$ is irrational – Surb Oct 19 '18 at 20:51
• @Surd: My answer is a hint, not a complete solution. It also depends on the starting assumptions of the problem. – JavaMan Oct 19 '18 at 20:52
• @surb in my answer I've taken it for known, because not only the proof is simple, but also because the problem does not ask to prove it. – LuxGiammi Oct 19 '18 at 20:54
• 1) I upvoted Javaman's answer because his justification is fair. 2) "the problem does not ask to prove it" this is certainly a very (very very) bad reason. The problem doesn't ask to prove that there is a digit appearing infinitely many times in the expansion of $\sqrt{p}$ for prime $p$ as well... But I guess you'd agree this fact can't be used here. – Surb Oct 19 '18 at 20:58
• Of course I agree on the last point, but when you prove a theorem you don't prove all the theorems or lemmas you need to prove that theorem and take them for true (or already proved elsewhere). I haven' seen a single answer on MSE about, for instance, limits evaluation, that proves that the limit of a sum, under certain hypothesis, is the sum of limits. That's what I meant in saying that "the problem does not ask to prove it". The proof of irrationality of $sqrt(7)$ in my opinion was out of scope of the question. Of course, I absolutely agree on the fact that if the question does not ask – LuxGiammi Oct 20 '18 at 6:19

Suppose not, that every digit $$i \in \{0, \dotsc, 9\}$$ appears a finite number of times, say $$a_i$$. Then The decimal expansion of $$\sqrt{7}$$ would have exactly $$a_0 + \dotsb + a_9$$ digits, but it is infinite instead.

• Come on...... :) – Hugo Oct 19 '18 at 20:51
• Sorry I am new at these topics, I don't understand exactly, can you explain more specific ? Thank you. – user606126 Oct 19 '18 at 21:14