1 | initial version |

Each Sage object might behave differently with respect to the modulo operator %. Namely in this case this is because how this is coded in rationals numbers, namely x % y if x is rational and y integer will do the following

```
def modulo_for_rational(x,y):
x = ZZ(x)
n = x.numerator() % y
d = x.denominator() % y
d = d.inverse_mod(y)
return (n * d) % y
```

Note that all the modulo in the above code are between integers. Hence the rationale is: in x % y with x a rational and y an integer should be considered as a computation in ZZ / y ZZ.

This is indeed inconsistent with the behavior of % when x is a floating point number... where the definition is of x % y is the unique real number in [-y/2, y/2) of the form x + n y with n integer.

```
sage: 3.2 % 1
0.200000000000000
sage: 3.7 % 1
-0.300000000000000
```

PS: you can look at the complete source code of the modulo operator with

```
sage: Rational.__mod__??
```

2 | No.2 Revision |

Each Sage object might behave differently with respect to the modulo operator %. Namely in your situation this ~~case this ~~is because ~~how this is coded in rationals numbers, namely ~~of the implementation within rational numbers. Namely x % y if x is rational and y integer will do the following

```
def modulo_for_rational(x,y):
x = ZZ(x)
n = x.numerator() % y
d = x.denominator() % y
d = d.inverse_mod(y)
return (n * d) % y
```

Note that all the modulo in the above code are between integers. Hence the rationale is: in x % y with x a rational and y an integer should be considered as a computation in ZZ / y ZZ.

This is indeed inconsistent with the behavior of % when x is a floating point number... where the definition is of x % y is the unique real number in [-y/2, y/2) of the form x + n y with n integer.

```
sage: 3.2 % 1
0.200000000000000
sage: 3.7 % 1
-0.300000000000000
```

PS: you can look at the complete source code of the modulo operator with

```
sage: Rational.__mod__??
```

3 | No.3 Revision |

Each Sage object might behave differently with respect to the modulo operator %. ~~Namely in ~~In your situation this is ~~because of ~~due to the implementation ~~within ~~of this operator for rational numbers. Namely x % y if x is rational and y integer Sage will do the following

```
def modulo_for_rational(x,y):
x = ZZ(x)
n = x.numerator() % y
d = x.denominator() % y
d = d.inverse_mod(y)
return (n * d) % y
```

Note that all the modulo in the above code are between integers. Hence the rationale is: in x % y with x a rational and y an integer should be considered as a computation in ZZ / y ZZ.

This is indeed inconsistent with the behavior of % when x is a floating point number... where the definition is of x % y is the unique real number in [-y/2, y/2) of the form x + n y with n integer.

```
sage: 3.2 % 1
0.200000000000000
sage: 3.7 % 1
-0.300000000000000
```

PS: you can look at the complete source code of the modulo operator with

```
sage: Rational.__mod__??
```

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