what do curly brackets mean in this number theory equation?

Question

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

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. https://math.stackexchange.com/a/1819322/346651 So what does $\{x\}$ mean?

Its defined in Brian Scott's comment on that answer. – Tim kinsella Jun 10 '16 at 20:24 — Tim kinsella, Jun 10 '16 at 20:24

MSE · Accepted Answer · 2016-06-10T20:59:16.340

3

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 $x\in \mathbb R$.

edited Jun 10 '16 at 20:59

answered Jun 10 '16 at 20:23

MSE

Did you mean $[0, 1)$? I have not seen $[0, 1[$ used before. – Jossie Calderon Jun 10 '16 at 20:47
Yes, I corrected it. – MSE Jun 10 '16 at 20:59
@JossieCalderon $[0, 1[$ is French notation. – Jack M Jun 10 '16 at 21:03

score 2 · Answer 2 · answered Jun 10 '16 at 20:23

2

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

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

answered Jun 10 '16 at 20:23

Edward Evans

2 Answers2