algebra

Multiplication by 111

If you liked the Fun Fact Multiplication by 11, you’ll enjoy seeing how to take that idea one step farther. Here’s a quick way to multiply by 111. To multiply a two-digit number by 111, add the two digits and if the sum is a single...

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Squaring Quickly

You may have seen the Fun Fact on squares ending in 5; Here’s a trick that can help you square ANY number quickly. It’s based on the algebra identity for the difference of squares, but with a twist! Can you figure it out? 542 = 50 * 58...

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Multidimensional Newton’s Method

You’ve probably heard of Newton’s Method from your calculus course. It can be used to locate zeros of real-valued functions. But did you know that it is possible to define a multi-dimensional version of Newton’s Method for functions from Rn to Rn? Here’s how it goes. The derivative...

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Really Complex Matrices

If you know how to multiply 2×2 matrices, and know about complex numbers, then you’ll enjoy this connection. Any complex number (a+bi) can be represented by a real 2×2 matrix in the following way! Let the 2×2 matrix [  a b ] [ -b a ] correspond...

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Taylor-made Pi

After learning about the Taylor series for 1/(1+x) in calculus, you can find an interesting expression for Pi very easily. Start with 1/(1+w) = 1 – w + w2 – w3 + … Now substitute x2 for w: 1/(1+x2) = 1 – x2 + x4 – x6 + … Then integrate both sides...

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Banach-Tarski Paradox

Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using rigid motions only, form TWO solid balls, EACH THE SAME SIZE AND SHAPE as the original? This theorem is known as the Banach-Tarski paradox. So why...

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Sierpinski-Mazurkiewicz Paradox

If you’ve seen the Banach-Tarski paradox, you know that it is possible to cut a solid 3-dimensional ball into 5 pieces and reassemble the pieces using only rigid motions to form two solid balls each the same size as the original. The construction depends on the Axiom...

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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 roots it has, just by looking at it? Here’s a striking theorem due to Descartes in 1637, often known as “Descartes’ rule of signs”: The number...

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Fundamental Group

You are abducted by space aliens and dumped, blindfolded, on a strange asteroid! Removing your blindfold, you decide to play “Columbus” by walking in one direction forever to see if you can determine whether your asteroid is flat or curved. But you leave a trail...

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Perfect Shuffles

We know from the Fun Fact Seven Shuffles that 7 random riffle shuffles are enough to make almost every configuration equally likely in a deck of 52 cards. But what happens if you always use perfect shuffles, in which you cut the cards exactly in half and perfectly...

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