Q1: Can √7 ever be rounded to a fraction that works exactly?
A1: No. You can create rational approximations that come arbitrarily close, such as 37/14 or 127/48, but squaring any fraction will always leave a small discrepancy. No integer numerator squared can ever equal exactly seven times an integer denominator squared.
Q2: Is the square root of 7 considered a normal number?
A2: Mathematicians widely believe √7 is normal in base 10, meaning every digit from 0 through 9 appears with equal statistical frequency (10% of the time) across its infinite decimal expansion. However, like the digits of π and √2, this remains an unproven conjecture in number theory.
Q3: Why doesn't the proof for √7 work for √9?
A3: The proof by contradiction relies on 7 being a prime number. If you attempt the same proof on √9, the equation becomes a² = 9b². The number 9 is composite (3²). Stating that 9 divides a² does not force 9 to divide a; it only guarantees that 3 divides a. The cycle of deduction breaks, which allows √9 to cleanly equal the rational integer 3.
