In my understanding of modular arithmetic, any number $x \bmod{1}$ should be zero since any number is divisible by 1. I'd also like to understand what is meant by "almost all real numbers $\theta > 1$". If you can explain either one, I'll accept the answer.
You can see both expressions in context in the image below.
This passage is taken from Donald E. Knuth's The Art of Computer Programming, volume 2, seminumerical algorithms, chapter 3, page 152.
For non-integer values, $x$, we can define the function $x\text{ mod } m$ (the Modulo operation) which is equivalent to $$f(x,m)=x\text{ mod } m=x-m\left\lfloor\frac{x}m\right\rfloor$$ In the case $m=1$ we have $$f(x,1)=x\text{ mod } 1=x-\lfloor x\rfloor=\{x\}$$ where $\{x\}$ denotes the fractional part of $x$.