the greatest integer function - New

[logistic_guy]

Find the following values of the greatest integer function.

\(\displaystyle \bold{a)}\) \(\displaystyle \left[-\frac{1}{4}\right]\)

\(\displaystyle \bold{b)}\) \(\displaystyle \left[-\frac{22}{7}\right]\)

\(\displaystyle \bold{c)}\) \(\displaystyle \left[\frac{5}{4}\right]\)

\(\displaystyle \bold{d)}\) \(\displaystyle \left[\left[\frac{1}{2}\right]\right]\)

\(\displaystyle \bold{e)}\) \(\displaystyle \left[\left[\frac{3}{2}\right] + \left[-\frac{3}{2}\right]\right]\)

\(\displaystyle \bold{f)}\) \(\displaystyle \left[3 - \left[\frac{1}{2}\right]\right]\)

 

[puctw]

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[logistic_guy]

\(\displaystyle \bold{a)}\) \(\displaystyle \left[-\frac{1}{4}\right] = \left\lfloor -\frac{1}{4} \right\rfloor = \left\lfloor -0.25 \right\rfloor = -1\)

 

[logistic_guy]

\(\displaystyle \bold{b)}\) \(\displaystyle \left[-\frac{22}{7}\right] = \left\lfloor-\frac{22}{7}\right\rfloor = \left\lfloor-3-\frac{1}{7}\right\rfloor = -4\)

 

[pka]

\(\displaystyle \bold{b)}\) \(\displaystyle \left[-\frac{22}{7}\right] = \left\lfloor-\frac{22}{7}\right\rfloor = \left\lfloor-3-\frac{1}{7}\right\rfloor = -4\)

If [imath]x \in \Re[/imath] then the the floor function, [imath]\lfloor{x}\rfloor[/imath], returns the largest integer not greater than [imath]\bold{x}[/imath]
You, as one at your level, should know at once that [imath](-\frac{22}{7})=(-3\frac{1}{7})[/imath].
Thus [imath]\lfloor{-\frac{22}{7}}\rfloor=-4[/imath]

 

[logistic_guy]

\(\displaystyle \bold{c)}\) \(\displaystyle \left[\frac{5}{4}\right] = \left\lfloor\frac{5}{4}\right\rfloor = \left\lfloor 1 + \frac{1}{4}\right\rfloor = 1\)

If [imath]x \in \Re[/imath] then the the floor function, [imath]\lfloor{x}\rfloor[/imath], returns the largest integer not greater than [imath]\bold{x}[/imath]
You, as one at your level, should know at once that [imath](-\frac{22}{7})=(-3\frac{1}{7})[/imath].
Thus [imath]\lfloor{-\frac{22}{7}}\rfloor=-4[/imath]

Thanks a lot professor pka. Your golden advices have been noted down and studied deeply by me.