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Floor

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Floor is a function that gives the largest integer below a given real number \scriptstyle x \,, i.e. rounding towards minus infinity. It is generally notated \scriptstyle \lfloor x \rfloor \,, or in computer programming languages, ​floor(x)​. For example, \scriptstyle \lfloor \sqrt{50} \rfloor \,=\, 7 \,. For \scriptstyle x \,\geq\, 0 \,, then \scriptstyle \lfloor x \rfloor \, is the same as the integer part (or truncation, i.e. rounding towards zero) \scriptstyle [x] \, of \scriptstyle x \,; but if \scriptstyle x \,<\, 0 \,, then \scriptstyle \lfloor x \rfloor \, is one less than the integer part (which is now the same as the ceiling \scriptstyle \lceil x \rceil \,). For example, \scriptstyle \lfloor (\pi i)^2 \rfloor \,=\, -10 \,, not –9 (the latter number is the integer part of \scriptstyle (\pi i)^2 \,).

In the OEIS, some sequences of nonintegers are entered by flooring the terms. For example, A013629 gives the floor of the imaginary parts of the nontrivial zeros of the Riemann zeta function (these are given in order of magnitude if the Riemann hypothesis is true). The first zero is \scriptstyle \frac{1}{2} + 14.134725\ldots i \,, and thus the first term of that sequence is 14.

See also

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