Definition:Binomial (Euclidean)

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

Then $a + b$ is a binomial iff $\frac a b \not \in \Q$ and $\left({ \frac a b }\right) ^2 \in \Q$ and $a + b \not \in \Q$.


 * 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.

(: Book X: Proposition $36$)

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