Square Root of 3 is irrational (Proven by 2 methods)

Every number in existence belongs to one of two opposing camps: the dependable, well-behaved rational numbers that pack neatly into whole-number fractions, or the wild, untamable irrational rebels whose decimals spill into infinity without repeating. For centuries, thinkers hit a wall trying to neatly pin down the square root of three. When standard decimal calculation turns into an endless chase toward infinity, how can you definitively prove a number is irrational without sitting at a desk testing every integer combination until the end of time?

You don’t look for an endless pattern—you trap the number in its own logical paradox. By launching a full-scale mathematical investigation using a classic proof by contradiction, we assume $\sqrt{3}$ is a clean, fully reduced fraction, only to watch that assumption self-destruct as both the numerator and denominator are forced to share a factor of 3. And if that isn’t enough, we put $\sqrt{3}$ into the interrogation room of the Rational Root Theorem to systematically eliminate every possible suspect, shutting down the rational disguise from two completely different angles.