Generally, any square root that doesn't reduce to a rational number is irrational. Some examples:
• √2 is irrational
• √4 = 2 is rational
• √(1/2) = 1/√2 = √2/2 is irrational
• √(1/9) = 1/√9 = 1/3 is rational
Numbers with non-terminating decimal expansions are rational if there is a repeating pattern after the decimal. The pattern in 4.737737773… is pretty obvious (one 7, one 3, two 7s, one 3, three 7s, one 3, etc) but it does not repeat, the number of 7s just keeps going up. Compare this to 4.737373… or 4.773773773… - these two numbers are rational in contrast.
