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[mathjax]The topic in my geometry class today involved finding solutions to quadratic equations. The actual topic was on the ratio of lengths of secants in a circle, but that’s for another post. For a specific example, consider this problem. Two secants intersect outside a circle. The first secant line has segments of 15 (between the…
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There are things I wish I could say where you could hear them but the smile I painted on to the rhythm of the metronome cannot be so easily belied as that Somewhere at night my faith capsized, rammed against a frozen memory floating on the surface And for a moment, I reached out and…
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Pascal’s Triangle [mathjax]Pascal’s Triangle represents the coefficients of a binomial such as \(x + 1\) raised to a power. Row n of the triangle lists the coefficients of \((x + 1)^{n-1}\). Here are the first few rows of Pascal’s Triangle: \[\newcommand\cn[3]{\llap{#1}#2\rlap{#3}} \begin{array}{c} &&&&&&\cn{}{1}{}\\ &&&&&\cn{}{1}{}&&\cn{}{1}{}\\ &&&&\cn{}{1}{}&&\cn{}{2}{}&&\cn{}{1}{}\\ &&&\cn{}{1}{}&&\cn{}{3}{}&&\cn{}{3}{}&&\cn{}{1}{}\\ &&\cn{}{1}{}&&\cn{}{4}{}&&\cn{}{6}{}&&\cn{}{4}{}&&\cn{}{1}{}\\ &\cn{}{1}{}&&\cn{}{5}{}&&\cn{1}{}{0}&&\cn{1}{}{0}&&\cn{}{5}{}&&\cn{}{1}{}\\ \cn{}{1}{}&&\cn{}{6}{}&&\cn{1}{}{5}&&\cn{2}{}{0}&&\cn{1}{}{5}&&\cn{}{6}{}&&\cn{}‌​{1}{} \end{array}\] For instance, row 4 is…
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Introduction and Terms [mathjax]Recently, a post on the G+ Mathematics community involved how to determine \(x\) and \(y\) when: \[3x + 5y = 12 \\ x + y = 2\] This is generally referred to as simultaneous equations or a system of equations. As a general rule, for such a problem to be solvable, you…
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Terms [mathjax]For every polygon, there is a largest circle that fits entirely within that polygon. If that circle touches all sides of the polygon, then it is said to be inscribed; it is called the incircle, and its center is called the incenter (which is then also called the polygon’s incenter). Every triangle has an incenter…
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This one strikes me, and apparently others, as highly counter-intuitive, but it’s true because of mathematics! Take any two places in the world; call these points A and B. Take any two paths between A and B that are the same distance; call these paths C and D. Let C(x) be as far down path…
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“There’s a shadow in the mirror with a glimmer of the one that was.” My creative juices are like a gas-powered lawn mower that’s been kept a few too many seasons. I pull the cord and the engine kicks over a few times and I think that it’s the time that things will engage, but…
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Just for fun… Here’s the pseudocode for the method of building a binary number from a decimal number, based on the Russian peasant method of multiplication: function mybin(mydec) { mybin = “”; do while mydec > 0 { if mydec is odd: { mydec = mydec – 1; mybin = “1” + mybin;Â } else: mybin…
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Russian peasants do too much work [mathjax]There is a method of multiplication called the Russian peasant method. I’ve seen it mentioned here and there, but I was not explicitly educated in the process; it struck me as being more trouble than it was worth, and I didn’t previously bother to dig farther into it. I…
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[mathjax]This problem was brought to my attention on G+, but I wasn’t satisfied with the solution presented. There are actually two versions, the one that was originally presented on G+ and the corrected one that matches the standard version. I’ll discuss the standard version first. Standard version: All seats on an airplane are assigned. However,…