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<\/a><\/p>\nConstructions such as these are called \u201ccontinued fractions\u201d.\u00a0Though they go on forever, there is<\/em> a way to evaluate them.\u00a0 Before we do that, let\u2019s get a feel for the answer we seek by figuring out an upper and lower limit on the value.<\/p>\n 1.\u00a0 x<\/em> is less than 2\/3.<\/strong><\/p>\n We know this because the numerator is 2 and the denominator is 3 + something positive.\u00a0 If the denominator is MORE than 3, then the fraction is LESS than 2\/3.\u00a0 But how much less?\u00a0 Well, there is a limit on that as well\u2026<\/p>\n 2. x<\/em> is more than 6\/11.<\/strong><\/p>\n As we just said, the denominator is more than 3, but how much more?\u00a0 Look again at the \u201csomething positive\u201d which is added to the denominator:\u00a0 it is in fact the same expression x<\/em> that we are hoping to find.\u00a0 We don\u2019t know its value, but we know it is less than 2\/3.\u00a0 So now we have:<\/p>\n The fraction will have its smallest value when the denominator is the biggest it can be.<\/p>\n That gives us a lower limit on the value of x:<\/p>\n But what if we want an exact value?\u00a0 It turns out that we\u2019ve already noticed the key to this puzzle.<\/p>\n Look again at the continued fraction.<\/p>\n The denominator is more than 3\u2026but notice just how much more:\u00a0 the expression added to 3 is the same as the original expression, x<\/em>.<\/p>\n That means we can re-write the continued fraction this way:<\/p>\n We multiply both sides by 3+x<\/i> and distribute on the left hand side to get:<\/p>\n Were you expecting that? \u00a0To evaluate the continued fraction, we have to solve a quadratic equation.\u00a0 This means that we are going to have to choose a method.\u00a0 I\u2019m going to vote for completing the square (which I will return to in my next post).\u00a0 But if you prefer, you can use the quadratic formula.\u00a0 Either way, you\u2019ll get:<\/p>\n It\u2019s \u00a0the positive answer that we want here. \u00a0And its nice to see that it falls in the range we expected.<\/p>\n I still like completing the square better.\u00a0 Before we get to that, here\u2019s one more fun example that will end with a twist.<\/p>\n THE GOLDEN CONTINUED FRACTION:<\/strong><\/p>\n Let’s call it an exercise (or two):<\/p>\n Show that x must be less than 2 and more than 1.5<\/i>.<\/strong><\/p>\n (You can post your reasoning in a comment if you like.)<\/p>\n But again, what if we want the exact value?\u00a0 We have to notice that once again, the expression we seek is hidden inside the continued fraction.\u00a0Here it is again:<\/p>\n Look closely \u2013 that \u201csomething\u201d is the same expression we started with.\u00a0 It\u2019s x<\/em>!<\/p>\n So we can re-write the original equation:<\/p>\n And if we multiply both sides by x:<\/p>\n Or:<\/p>\n We still have to solve this quadratic equation.\u00a0 But those (both?) of you who have been following along with me might remember that we have seen this equation before.\u00a0 This was exactly the quadratic equation we landed on when looking for the Golden Ratio. \u00a0The one equation answers two seemingly<\/i> unrelated questions:<\/p>\n 1. What is the value of this expression with a continued fraction?<\/p>\n 2. What is the aspect ratio of a rectangle from which you can remove a square, leaving behind a smaller rectangle with the same aspect ratio?<\/p>\n We\u2019ll know both of these when we solve that quadratic equation.<\/p>\n <\/p>\n (That was my idea of a cliff-hanger ending.)<\/p>\n","protected":false},"excerpt":{"rendered":" Here is an interesting fraction to ponder: Constructions such as these are called \u201ccontinued fractions\u201d.\u00a0Though they go on forever, there is a way to evaluate them.\u00a0 Before we do that, let\u2019s get a feel for the answer we seek by … Continue reading <\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/a><\/p>\n