I have a question from an A-level revision book:
Find an irrational number which lies between $\frac34$ and $\frac78$.
What is the correct method for doing this? Here is my method:
- Square numerators and denominators of the (in this case: both) fractions.
- Find LCD (Lowest Common Denominator) for denominators and convert all fractions to this LCD.
- The numerators of the fractions are now perfect squares. Write a new fraction with a numerator that is not a perfect square and between the original fractions' numerators.
- Put the new fraction inside a square root. The new fraction (inside a square root sign) is now a surd, and an irrational number (also an irrational fraction).
Using this method, $\sqrt{\frac{37}{64}}$ would be one such irrational number between $\frac34$ and $\frac78$.
Please let me know if I've made any glaring lapses in logic. If I'm correct, a fraction with an irrational (surd) numerator is itself an irrational number. And non-perfect square numbers are all irrational. Therefore a fraction with an irrational numerator is irrational. (Pardon me if I'm not using all the correct terminology yet.)
Many thanks in advance
PS. I want to check I understand these relatively basic concepts correctly. (I apologize in advance if I have made a schoolboy error or stupid mistake. I'm trying to take the A-level mathematics exams in nearly 4 months, the reason is a rather long story, and I'm refreshing all my mathematics which I did at school. I did well at Math GCSE, although it was the intermediate not higher tier, and I'm considering blitzing all the A-level maths in a little over 3 months to take exams starting mid-May.)
Best Answer
You are making this more complicated than it needs to be. If $p,q$ are any two distinct rational numbers, then $$p + \frac {q-p}{\sqrt 2}$$ is an irrational number between $p$ and $q$.