Greatest Integer Functions

Date: 09/27/98 at 18:10:45
From: Laura
Subject: Greatest integer function

I am in 11th grade pre-calculus. We have been asked to solve for the 
graph of [y]=[x]. I am familiar with the greatest integer function of 
y=[x], but I do not know how to go about solving [y]=[x]. Please help.

Thank you,
Laura

Date: 09/27/98 at 18:30:56
From: Doctor Pat
Subject: Re: Greatest integer function

Laura,

This is an unusual question, but interesting. Let's see if we can 
figure out what it means. We want to mark all the points on the 
coordinate plane that would make this equation true.  

First it should be obvious that if y = x then [y] = [x] is true so the 
line y = x must be part of the solution. Okay? 

Now, think about numbers that give the same result when using [x]. 
Any number x so that 0 <= x < 1 has a value of 0. That would be true 
for 0 <= y < 1 also. So any ordered pair (x,y) where both are in the 
interval [0,1), would be included in our solution. Following this idea 
for values between [1,2) etc. it seems that we should expect to have a 
series of squares moving along the line y = x. Shade the square where 
x and y are both between zero and one. The x-axis and y-axis, including 
the origin, are part of the solution out to x = 1 and y = 1, which are 
not in the solution.

Another square starts at (1,1) and goes to (1,2), (2,2), (2,1). Again, 
the bottom and left edge of the square are in the solution and the top 
and right edge of each square are not in the solution.  

Continue to infinity in both directions.  

If you could see all the arm waving I'm doing it might help, but alas, 
that is not yet possible.  

Good luck,

- Doctor Pat, The Math Forum
  http://mathforum.org/dr.math/