<Dd> n m = m n − m . (∗) (\ displaystyle (\ frac (n) (m)) = (\ frac (m) (n-m)). \ qquad (*)) </Dd> <P> To say that φ is rational means that φ is a fraction n / m where n and m are integers . We may take n / m to be in lowest terms and n and m to be positive . But if n / m is in lowest terms, then the identity labeled (*) above says m / (n − m) is in still lower terms . That is a contradiction that follows from the assumption that φ is rational . </P> <P> Another short proof--perhaps more commonly known--of the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication . If 1 + 5 2 (\ displaystyle \ textstyle (\ frac (1 + (\ sqrt (5))) (2))) is rational, then 2 (1 + 5 2) − 1 = 5 (\ displaystyle \ textstyle 2 \ left ((\ frac (1 + (\ sqrt (5))) (2)) \ right) - 1 = (\ sqrt (5))) is also rational, which is a contradiction if it is already known that the square root of a non-square natural number is irrational . </P> <P> The golden ratio is also an algebraic number and even an algebraic integer . It has minimal polynomial </P>

1 plus the square root of 5 divided by 2