what do curly brackets mean in this number theory equation?

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$$(\lfloor y\rfloor-1)(\lfloor x\rfloor-1)=\{x\}+\{y\}+1$$

From looking at the LaTeX, I can see the left-hand side symbols mean the floor of the variable, but the right-hand side doesn't give much clue.

I thought it had no special meaning until I pointed out this answer, which got two downvotes. So what does $\{x\}$ mean?

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2 Answers

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It's the fractional part, that mean $$\{x\}=x-\lfloor x\rfloor .$$ For example, if $x=3.456$, then $$\{x\}=x-\lfloor x\rfloor= 3.456-3=0.456.$$

Notice that $\{x\}\in [0,1)$ for all $x\in \mathbb R$.

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$\lbrace x \rbrace$ is called the fractional part of $x$. That is, $$\lbrace x \rbrace = x - \lfloor x\rfloor.$$

E.g. If $x = \frac{3}{2}$ then $\lbrace \frac{3}{2}\rbrace = \frac{3}{2} - 1 = \frac{1}{2}.$

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