Fractional Part Function | Brilliant Math & Science Wiki

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The floor function \( \lfloor x \rfloor\) is defined to be the greatest integer less than or equal to the real number \( x \). The fractional part function \( \{ x \}\) is defined to be the difference between these two:

Let \( x\) be a real number. Then the fractional part of \(x\) is

\[\{x\}= x -\lfloor x \rfloor.\]

This is the graph of the function \( y=\{x\}.\) This is the graph of the function \( y=\{x\}.\)

For nonnegative real numbers, the fractional part is just the "part of the number after the decimal," e.g.

\[ \{3.64 \} = 3.64 - \lfloor 3.64 \rfloor = 3.64 - 3 = 0.64. \]

But for negative real numbers, this is no longer the case:

\[ \{-3.64 \} = -3.64 - \lfloor -3.64 \rfloor = -3.64 - (-4) = 0.36. \]

Note that in both cases \( \{ x \} \) is nonnegative.

The following are some examples of how fractional part functions work:

  • \( \{ 1 \} = 1 - 1 = 0. \)
  • \( \left\{ \sqrt{2} \right\} = \sqrt{2}-1 = 0.4142\ldots. \)
  • \( \{ \pi \} = \pi - 3 = 0.14159\ldots. \)
  • \( \left\{ -\frac{17}5 \right\} = -\frac{17}5 - (-4) = \frac35. \)

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