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Topic: Floor function quotient (Read 836 times) |
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NickH
Senior Riddler
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Floor function quotient
« on: Mar 15th, 2003, 10:29am » |
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Let x be a real number and n a positive integer. Show that
[[nx]/n] = [x],
where [x] is the greatest integer less than or equal to x.
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towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
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Posts: 13730
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Re: Floor function quotient
« Reply #1 on: Mar 16th, 2003, 7:56am » |
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...
[[n x ]/n]
=
[[n * ([x]+ x-[x]) ]/n]
=
[[n * [x] + n * (x-[x]) ]/n]
= {n * [x] is integer}
[[x] + [n * (x-[x]) ]/n]
= {n * (x-[x]) < n}
[[x]]
= [x]
...
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