&quot;Is it conceivable that different irrational numbers have different &quot;frequencies&quot;, such that high-frequency numbers exhibit their expected averages very quickly, whereas low-frequency numbers may require billions of digits to exhibit the expected averages?&quot;<p>Yes, and obviously so. Take a (base 10) normal irrational and divide it by 10^1000000000000. The first trillion or so digits of the result will be 0, but the result is still irrational (because duh) and (base 10) normal because the frequencies of the digits have to converge in the infinite limit and giving one digit a large finite head start won&#x27;t change its overall proportion in the limit. QED.