Recall Square Take-away lake a rectangular piece of paper and remove from it the
ID: 3120685 • Letter: R
Question
Recall Square Take-away lake a rectangular piece of paper and remove from it the largest possible square. Repeat the process with the left-over rectangle. What different things can happen? Can you predict when they will happen? Prove that if the original piece of paper has sides with lengths whose ratio is rational, there exists a unit length such that with respect to this unit length the measures of both the length and width are integers. Prove that if the original piece of paper has sides whose ratio is rational, then the process terminates. Prove that if the original piece of paper has sides whose ratio is irrational, the process does not terminate. Start with a page which is Squareroot 2 times 1 units. Apply the process twice and use your observations to give a proof that Squareroot 2 is irrational.Explanation / Answer
2. The process terminates because after the removal of the squares for some number of times we'll eventually end up with a square.
If the sides of the rectangle are say p units and q units where p and q are positive integers then that square to be removed would have side equal to the greatest common divisor of p and q.
Let p > q .
If p = q then the process will terminate as we'll get a square.
If q < p after we remove a square qxq form the rectangle then we'll have a rectangle of size (p-q)xq.
=> the new rectangle would have a side smaller in dimension then the original rectangle.It's smaller by atleast 1 unit and the other dimension remains the same.
THis might not go on forever as in that case one length would have to be replaced infinite number of times.
Eventually we'll end up with a square of 1x1 units^2 and the process terminates as p=q.
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