In a MathOverflow comment on the question of "What is the most useful non-existing object of your field?", an answer is given
A number which is less than 1 and greater than 1.
Which elicited a highly upvoted reply
Integers (strictly) between 0 and 1 form the basis of transcendental number theory.
What does this mean? Is there a different, more relaxed, definition of an integer used for transcendental number theory?
Many proofs in transcendental number theory reach a contradiction by coming up with a quantity which on the one hand is a positive integer, and on the other hand is less than one.
A particular simple example is the proof that $e$ is irrational. Suppose $e = p/q$. Then for $r \geq q$, $$r!e = \sum_{n \leq r} \frac{r!}{n!} + \frac{1}{r+1} \left(1 + \frac{1}{r+2} + \frac{1}{(r+2)(r+3)} + \cdots \right).$$ Now consider the quantity $$ \frac{1}{r+1} \left(1 + \frac{1}{r+2} + \frac{1}{(r+2)(r+3)} + \cdots \right). $$ On the one hand, it's a positive integer. On the other, as $r\to\infty$ it tends to 0.
In other proofs we might come up with (the analog of) some explicit $r$ for which the quantity is less than $1$, rather than showing that it tends to $0$.