# 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...

# 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...

# 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...

# 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...

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...

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...

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...

# 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...