let

and let [x] be the greatest integer less than or equal to x. define

to be the fractional part of x. On

we define addition modulo 1 by a + b mod 1 = (a + b) = fractional part of a + b.

my book then says that

defined by

is a homomorphism between the additive group of the real numbers and the additive group of the reals modulo 1 since

.

what i don't understand is how (x + y) = (x) + (y). after experimenting with numbers such as 0.5 and 0.8, i find that (0.5 + 0.8) = (1.3) = 1.3 - [1.3] = 0.3 while (0.5) + (0.8) = 0.5 - [0.5] + 0.8 - [0.8] = 1.3. so is my book wrong? is this map is not a homomorphism? help clearing this up will be greatly appreciated.