Definition:Binomial (Euclidean)

Definition
Let $a$ and $b$ be two positive real numbers.

Then $a + b$ is a binomial iff the following conditions are met:
 * $(1): \quad \dfrac a b \notin \Q$
 * $(2): \quad \left({\dfrac a b}\right)^2 \in \Q$
 * $(3): \quad a + b \notin \Q$


 * If two rational straight lines which are commensurable in square only are added together and the sum is irrational, let it be called a binomial.

Note
The terms first binomial, second binomial, third binomial, fourth binomial, fifth binomial, sixth binomial are rarely used (if there is any usage) nowadays.