TAOCP 1.2.4 Exercise 10
By equation (1), x \bmod y = x - y\left\lfloor \frac{x}{y}\right\rfloor.
Section 1.2.4: Integer Functions and Elementary Number Theory
Exercise 10. ▶ [10] What are $1.1 \bmod 1$, $0.11 \bmod 0.1$, $0.11 \bmod (-0.1)$?
Verified: yes
Solve time: 1m10s
By equation (1),
$$ x \bmod y = x - y\left\lfloor \frac{x}{y}\right\rfloor. $$
For $1.1 \bmod 1$,
$$ 1.1 \bmod 1 =1.1-\left\lfloor 1.1\right\rfloor =1.1-1 =0.1. $$
For $0.11 \bmod 0.1$,
$$ 0.11 \bmod 0.1 =0.11-0.1\left\lfloor \frac{0.11}{0.1}\right\rfloor =0.11-0.1\lfloor 1.1\rfloor =0.11-0.1 =0.01. $$
For $0.11 \bmod (-0.1)$,
$$ 0.11 \bmod (-0.1) =0.11-(-0.1)\left\lfloor \frac{0.11}{-0.1}\right\rfloor =0.11+0.1\lfloor -1.1\rfloor =0.11+0.1(-2) =-0.09. $$
Hence
$$ \boxed{1.1 \bmod 1=0.1,\qquad 0.11 \bmod 0.1=0.01,\qquad 0.11 \bmod (-0.1)=-0.09.} $$