Category: Mathematics
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[mathjax]I was recently wondering why Euclid, the geometer, published a proof that there is an infinite number of primes. I should have known that his proof is geometric. It is: “Let A, B, and C be distinct lengths that cannot be further divided into lengths of whole numbers (other than the unit segment). Let \(\overline{DE}\)…
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The first shoe: Multiplying [mathjax]Adding complex numbers is a straightforward task. Given two numbers, \(a + bi\) and \(c + di\), the sum is the sum of the real portion and the sum of the imaginary portion: \((a + c) + (b + d)i\). When working with the standard form, multiplication (and division) is a…
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[mathjax]I was reminded of the cylindrical wedge that casts shadows of a triangle, a square, and a circle, and it got me wondering: What if I wanted to create such a shape with an equilateral triangle as one of its shadows? The wedge shown either casts an isosceles (but not equilateral) triangular shadow, or it…
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[mathjax]Students often struggle with the concept of multiplying negative numbers, particularly with the notion that multiplying two negative numbers results in a positive. I’ve seen numerous attempts by teachers and teacher educators to explain why, conceptually, it is that the product of two negative numbers is a positive number. However, what if the underlying presupposition…
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[mathjax]The game of Set consists of 81 cards. Each card has one, two, or three identical symbols of one of three shapes (oval, diamond, or squiggle), in one of three colors (red, green, or purple) and one of three textures (solid, hollow, striped). A “set” consists of three cards where each of the four attributes…
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[mathjax]This is an example of a common sort of story problem encountered in standardized tests: “1. A team of five professionals can do a certain job in nineteen days; a team of nine apprentices can do the same job in the same amount of time. Assuming all professionals work at the same rate and all…
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[mathjax]Here’s an extension to the problem in my previous post. Time has run out, and a player is at the free throw line. If he makes the first shot, he gets a second try. If he makes both shots, his team wins; if he misses the first, his team loses. Otherwise, it’s a tie game,…
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[mathjax]At a recent workshop on collaboration, the other participants and I were presented with a version of this problem: Adam hits 60% of his free throws. He gets fouled just before the buzzer, and his team is down by one point. Based on the rules in play, he can shoot up to twice. If he…
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[mathjax]I’ve seen variations of this one a few times, so I thought I’d give it a quick write-up. The simpler version is: Given two circles that are tangent and a line that is cotangent to them, what is the length of the segment between the points of tangency? To make things easier, the radii are…
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[mathjax]Most people are aware of two or three basic tests for divisibility by a prime number: A numberย n is divisible by 2ย iff it ends in an even number (0, 2, 4, 6, or 8) by 5ย iff it ends in a 0 or a 5 by 3ย iffย its digits add up to a multiple of 3 These…