No matter what unit and/or equal parts of the abstract unit you use, sqrt(2) has no measure.
It doesn't matter which you unit and/or equal parts of the unit that you choose, you cannot measure the hypotenuse of a right-angled isosceles triangle with it. The applet shows you this because it uses a different part of the unit each time the measure precision is increased. Observe that the abstract unit also does not care about size or type. This is how the Ancient Greeks concluded that the hypotenuse has no measure (is an irrational magnitude).
In geometry, the measure of the hypotenuse is exact:
https://www.academia.edu/69488136/The...
Therefore, in algebra an exact measure is not possible using the abstract unit.
In the following article I discuss this proof with ChatGPT and it reaches the same conclusion which I think is a good sign in terms of its (ChatGPT's) credibility.
https://www.academia.edu/98634012/%CE...
Link to applet:
https://drive.google.com/file/d/1E_7U...