3320prelimSol

# 3320prelimSol - Math 3320 Prelim — Solutions Fall 2009 1...

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Unformatted text preview: Math 3320 Prelim — Solutions Fall 2009 1. Determine how a 37 × 8 rectangle can be decomposed into 9 non-overlapping squares (of various sizes), and draw a figure showing how the 9 squares are arranged in the rectangle. Solution : The continued fraction for 37 / 8 is 4 + 1 upslope ր 1 + 1 upslope ր 1 + 1 upslope ր 1 + 1 upslope ր 2 so the numbers of triangles of each size are 4 , 1 , 1 , 1 , 2, and the decomposition is 2. Find two different linear fractional transformations that take the edge ( 1 / 2 , 1 / 1 ) in the Farey diagram to the edge ( 5 / 3 , 3 / 2 ) (with 1 / 2 going to 5 / 3 and 1 / 1 going to 3 / 2). Solution : One such transformation has matrix ± parenleftBigg 5 3 3 2 parenrightBiggparenleftBigg 1 1 2 1 parenrightBigg − 1 = ± parenleftBigg 5 3 3 2 parenrightBiggparenleftBigg 1 − 1 − 2 1 parenrightBigg = ± parenleftBigg − 1 − 2 − 1 − 1 parenrightBigg = ± parenleftBigg 1 2 1 1 parenrightBigg This matrix has determinant − 1 so it is orientation-reversing. To get another trans- formation with determinant + 1 we can change the sign of one column in either of the original two matrices, for example ± parenleftBigg 5 3 3 2 parenrightBiggparenleftBigg − 1 1 − 2 1 parenrightBigg − 1 = ± parenleftBigg 5 3 3 2 parenrightBiggparenleftBigg 1 − 1 2 − 1 parenrightBigg = ± parenleftBigg 11 − 8 7 − 5 parenrightBigg 3. Determine the value of the continued fraction 1 upslope ր 1 + 1 upslope ր 1 + 1 upslope ր 1 + 1 upslope ր 4 . Solution : We learned two ways to do this. The first was to set x = 1 upslope ր 1 + 1 upslope ր 1 + 1 upslope ր 4 and do some algebraic manipulations to get a quadratic equation for x , then solve this equation, then do a little more algebra involving some square roots to get the final...
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3320prelimSol - Math 3320 Prelim — Solutions Fall 2009 1...

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