floor function of $5.999\dots$

Question

What is $\left\lfloor5.\overline9\right\rfloor$ where $\lfloor.\rfloor$ denotes the floor function?

I believe it to be $6$ since converting $5.\overline9$ to fractional form yields $6$. Meanwhile my friend insists that $\left\lfloor5.\overline9\right\rfloor = 5$ since actually $5.\overline9 \ne 6$ but rather is closer to $6$ than any other real number.

Is $\left\lfloor5.\overline9\right\rfloor$ $6$ or $5$?

Welcome to Mathematics Stack Exchange. You are correct. $5.\overline9=6$ — J. W. Tanner, Oct 12 '20 at 14:33
Actually $5.\bar 9$ is exactly $6$ , not just close to $6$ , so the floor-function of $5.\bar 9$ is $6$ as well. — Peter, Oct 12 '20 at 14:51

score 3 · Accepted Answer · answered Oct 12 '20 at 15:03

3

You are correct.

Actually $5.\overline9=6$ (cf. this post),

so $\lfloor5.\overline9\rfloor=\lfloor6\rfloor=6$.

answered Oct 12 '20 at 15:03

J. W. Tanner

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floor function of $5.999\dots$

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