Hard Year 11 Irrational and Rational Number problem

Question

Suppose that $a$ and $b$ are positive irrational numbers, where $a < b$. Choose any positive integer $n$ such that $1/n < b − a$, and let $p$ be the greatest integer such that $p/n < a$:

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a) Prove that the rational number $\frac{p + 1}{n}$ lies between $a$ and $b$.

b) If $a = \frac{1}{\sqrt{1001}}$, and $b = \frac{1}{\sqrt{1000}}$, find the least possible value of $n$ and the corresponding value of $p$.

c) Hence use part a) to construct a rational number between $\frac{1}{\sqrt{1001}}$ and $\frac{1}{\sqrt{1000}}$.

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I am currently not learned in mathematical induction so im not really sure how to solve this. was thinking if there was another way of solving this question.

This question was taken from the Cambridge Maths Extension 1 Year 11 Book. Would like an intuitive explanation that a high school student in year 11 like me can understand since even my teachers cant seem to solve this problem lol. Sorry if this is a hassle of your time.

Answer 1

By definition, $(p+1)/n\ge a$. However, $a$ is irrational, so equality cannot hold.

Next you want to show that $(p+1)/n<b$; you have $$ \frac{p+1}{n}=\frac{p}{n}+\frac{1}{n}<a+(b-a) $$ Note that it's never said (nor it is generally possible) that $p>0$, but this is not a problem.

For the second part, follow the rules: since $$ \frac{1}{b-a}=\sqrt{1000\cdot1001}\bigl(\sqrt{1001}+\sqrt{1000}\,\bigr) $$ you can use a calculator to show that $n=63293$. It turns out that $an\approx2000.5$, so $p=2000$.

I can't see how to avoid using a calculator.

Answer 2

Hints: for part (a), you need to use that \begin{align} \frac{1}{n} &< b - a \\[4pt] \frac{p}{n} &< a \, . \end{align}

For part (b), we are told that \begin{align} b-a &= \frac{1}{\sqrt{1000}}-\frac{1}{\sqrt{1001}} \\[4pt] &= \frac{\sqrt{1001}-\sqrt{1000}}{\sqrt{1000}\sqrt{1001}} \end{align} We need to find the smallest value of $n$ such that $$ \frac{1}{n}<\frac{\sqrt{1001}-\sqrt{1000}}{\sqrt{1000}\sqrt{1001}} $$ Rearrange this inequality to find the answer. Part (c) should be straightforward: use the values of $p$ and $n$ that you found in part (b) to get the answer.