Elections are democracy in action. People go to polls and express their preferences, and somehow we must aggregate the preferences...
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Products of Sums of Two Squares
Here’s a nice theorem due to Fibonacci, in 1202. Theorem. If integers N and M can each be written as the...
Continue reading...Kakeya Needle Problem
What is the smallest-area convex set in the plane inside which a needle (unit straight line segment) can be reversed...
Continue reading...Eccentricity of Conics
To each conic section (ellipse, parabola, hyperbola) there is a number called the eccentricity that uniquely characterizes the shape of the curve. A circle...
Continue reading...Descartes’ Rule of Signs
Given a polynomial such as: x4 + 7×3 – 4×2 – x – 7 it is possible to say anything about how many positive real...
Continue reading...Repunit Fun
A repunit is a whole number consisting of only 1’s, such as 1, 111, or 111111111. (It’s like a repeating unit.) These numbers have...
Continue reading...Sine of (1/55555….)
Take your calculator, and enter a number consisting of several 5’s, such as 5555555. Now take the reciprocal of this number. Now take the...
Continue reading...Divisibility by Eleven
It is easy to tell that the following are multiples of 11: 22, 33, 44, 55, etc. But how about:...
Continue reading...Behold! the Pythagorean Theorem
Figure 1 shows one of the simplest proofs of the Pythagorean Theorem. It is also perhaps the earliest recorded proof, known...
Continue reading...Why Does 0.999… = 1?
Consider the real number that is represented by a zero and a decimal point, followed by a never-ending string of...
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