Square Root of 7 to 100 Decimal Places: Exact Digits and Step-by-Step Proof
By Marcus Vance
To establish that the decimal expansion of √7 is non-repeating and non-terminating, mathematicians use proof by contradiction. This formal argument demonstrates that assuming √7 can be written as an irreducible ratio of two integers forces a logical impossibility.
Assume that √7 is rational. If true, two co-prime integers a and b (where b ≠ 0 and the greatest common divisor gcd(a, b) = 1) must exist such that:
√7 = a / b
Squaring both sides of this equation isolates the radical:
7 = a² / b², which rearranges to a² = 7b²
Because 7 divides the right side of the equality, 7 must also divide a². By Euclid's lemma, if a prime number divides a square integer, it must divide the base integer itself. Therefore, 7 divides a.
Since 7 divides a, we can substitute a with an integer multiple: a = 7k, where k is an integer. Inserting this back into the rearranged equation produces:
(7k)² = 7b²
49k² = 7b²
Dividing both sides by 7 leaves:
7k² = b²
This reveals that 7 also divides b². Applying Euclid’s lemma a second time confirms that 7 divides b.
A fatal contradiction appears: both a and b share a common factor of 7. This violates our initial requirement that a and b share no factors other than 1. The initial assumption that √7 is rational collapses. Consequently, √7 is an irrational number possessing an endless, non-repeating decimal expansion.
Marcus Vance is a cybersecurity auditor and technology writer dedicated to educating the public about online safety, data privacy regulations, enterprise security, and emerging cyber threats.
