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Equations with the Floor Function


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9 replies to this topic

#1 James33

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Posted 09 June 2012 - 04:24 PM

For those that don't know, the floor function which I will denote by [x] is the largest integer less than or equal to x so for example [6]=6, [8.7]=8 and [-4.3]=5. Solving equations using this function can be very interesting and there are often loads of ways to do it.

An easier one to start off:

Solve -6x^2+[x^4]+19=0

This one is harder;

Solve [x^3]-x^2+6[x]=23
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#2 CaptainEd

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Posted 09 June 2012 - 05:19 PM

I'm more accustomed to [-4.3] = (-5)
Is that what you mean? Or do you really intend to take the absolute value of the floor?
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#3 James33

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Posted 09 June 2012 - 05:20 PM

I'm more accustomed to [-4.3] = (-5)
Is that what you mean? Or do you really intend to take the absolute value of the floor?


Yes that is what I meant.
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#4 WitchOfSecrets

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Posted 10 June 2012 - 12:14 AM

For the first:

Spoiler for solved

Edited by WitchOfSecrets, 10 June 2012 - 12:16 AM.

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#5 superprismatic

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Posted 10 June 2012 - 02:09 AM

Spoiler for for the first

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#6 WitchOfSecrets

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Posted 10 June 2012 - 03:13 AM

Whoops. I did [x]^4, not [x^4]. I think SP is right.
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#7 CaptainEd

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Posted 10 June 2012 - 04:15 AM

Spoiler for an answer for second problem

Edited by CaptainEd, 10 June 2012 - 04:18 AM.

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#8 James33

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Posted 10 June 2012 - 09:39 AM

Superprismatic is correct for the first, CaptainEd is correct for the second but there are much better ways of doing these than looking at the graph, also you have to prove there are no other solutions.
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#9 CaptainEd

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Posted 11 June 2012 - 10:15 PM

Spoiler for Not proud about brute forcing, but there is little more to say


I'd like to add that this whole area is bizarre, and yet it does seem (you've said so, and SP has shown an example) that one could do some sort of analysis for problems in this area. Thank you, James33, for a glimpse into a weird world!

Edited by CaptainEd, 11 June 2012 - 10:24 PM.

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#10 James33

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Posted 11 June 2012 - 10:29 PM

Spoiler for Not proud about brute forcing, but there is little more to say


I'd like to add that this whole area is bizarre, and yet it does seem (you've said so, and SP has shown an example) that one could do some sort of analysis for problems in this area. Thank you, James33, for a glimpse into a weird world!


Thats very similar to the way I was expecting it to be solved, a lot of these problems cannot be solved perfectly analyticly but, as you did, can reduce the possibilities of x to a small, finite number of options which can be eliminated without much trouble.
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