Unreasonable refers to the violation of a qualitative expectation. For example, When I told my daughter that she would have to go to the doctor, she behaved unreasonably by throwing a temper-tantrum.
Irrational refers to violation of a quantitative or mathematical expectation. For example, Even though war was imminent, the President decided irrationally to disband the army.
There is a lot of crossover between irrational and unreasonable, but at the extremes, those are the differences.
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Find the difference between the two numbers, then add an irrational number between zero and one, divided by this difference, to the lower number. Such an irrational number might be pi/10, (square root of 2) / 2, etc.
The difference can be rational or irrational.5 + sqrt(3) and 2 + sqrt(3) are both irrational numbers but their difference is[5 + sqrt(3)] - [2 + sqrt(3)] = 3, which is rational.
No. sqrt(3) - sqrt(2) is irrational.
Synonyms would be 'unhinged', 'mad', 'loopy', 'impulsive', 'unreasonable', 'unsound'.
There are several you could use: absurd, crazy, unreasonable, foolish, distraught, and others.