# Why does Sage return negative number when evaluating 181.0%360 sage: print 181%360  , 181.0%360

181 -179.000000000000

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So that this code works

sage: a = 181.0
sage: b = 360
sage: n = (a / b).round()
sage: b * n + (a % b) == a
True


From the sage docstring (which you can read by typing a.__mod__?)

Return the value of "left - n*right", rounded according to the rounding mode of the parent, where "n" is the integer quotient of "x" divided by "y". The integer "n" is rounded toward the nearest integer (ties rounded to even).

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My question wasn't HOW Sage evaluates 181.0%360, the question is WHY is it done this way. Pure Python returns 181 , Wolfram Alpha returns 181. when entering 181. mod 360 - Sage behaves different - what for?

Exactly for the reason I said: so that the output is consistent with the output of a/b, the rounding mode, and the mathematical definition of Euclidean division (which in truth does not make much sense for floats).

In Python rounding is done to the lowest integer, so it makes sense to return 181, indeed

>>> 360 * int(181.0 / 360) + (181.0 % 360)
181.0


In Mathematica, rounding is done to the closest integer, and indeed

360*round(181.0 / 360) + (181.0 mod 360)


answers 541 on WolframAlpha. Oups! Guess Mathematica does not really care for mathematical consistency.