Using Logarithms in the Real World

Logarithms are everywhere. Ever use any of the following phrases?

6 figures
Double digits
Order of magnitude

You’re describing numbers in terms of their powers of 10 — a logarithm. Ever mention an interest rate or rate of… Read article

Understanding Pythagorean Distance and the Gradient

The Pythagorean Theorem shows how strange our concept of distance is. Using the rule a² + b² = c², we can trade some “a” to get more “b”.

Starting with

means “A 13-inch pizza equals a 13-inch pizza”. Sure.… Read article

Understanding Why Complex Multiplication Works

Seeing imaginary numbers as rotations was one of my favorite aha moments:

i, the square root of -1, is a number in a different dimension! Once that clicks, we can use multiplication to “combine” rotations of two complex numbers:… Read article

Intuitive Understanding of Sine Waves

Sine waves confused me. Yes, I can mumble "SOH CAH TOA" and draw lines within triangles. But what does it mean?

I was stuck thinking sine had to be extracted from other shapes. A quick analogy:

You: Geometry is

… Read article

Understanding Calculus With A Bank Account Metaphor

Calculus examples are boring. "Hey kids! Ever wonder about the distance, velocity, and acceleration of a moving particle? No? Well you're locked in here for 50 minutes!"

I love physics, but it's not the best lead-in. It makes us wait… Read article

Understanding Why Similarity "Works"

Similarity has bothered me for a long time. Why do all circles have the same formula for area — how do we know nothing sneaky happens when we make them larger? In physics, don’t weird things happen when you scale… Read article

Surprising Patterns in the Square Numbers (1, 4, 9, 16…)

A quick puzzle for you — look at the first few square numbers:

1, 4, 9, 16, 25, 36, 49…

And now find the difference between consecutive squares:

1 to 4 = 3
4 to 9 =

… Read article

Intuitive Understanding Of Euler's Formula

Euler's identity seems baffling:

It emerges from a more general formula:

Yowza -- we're relating an imaginary exponent to sine and cosine! And somehow plugging in pi gives -1? Could this ever be intuitive?

Not according to 1800s mathematician Benjamin… Read article

Fun With Modular Arithmetic

A reader recently suggested I write about modular arithmetic (aka “taking the remainder”). I hadn’t given it much thought, but realized the modulo is extremely powerful: it should be in our mental toolbox next to addition and multiplication. Instead of… Read article