Vector Calculus: Understanding the Dot Product

I see the dot product as directional multiplication. But multiplication goes beyond repeated counting: it’s applying the essence of one item to another.

Normal multiplication combines growth rates: “3 x 4″ can mean “Take your 3x growth and make it 4x larger (i.e., 12x)”.... Read article

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 return? It's the logarithm of your growth.... Read article

Math, Better Explained available on the Kindle Store!

Hi all, I wanted to announce that Math, Better Explained is available on the Kindle store!

If you’re looking for a digital gift this holiday season, I know just the ticket ;).

A lot of work was done to convert the original PDF but keep the image and equation formatting the same, and readable on eInk and color screens.... Read article

Intuition, Details and the Bow/Arrow Metaphor

My favorite analogies explain a thought and help you explore deeper truths. Here’s a metaphor that captures my stance on learning:

• Rote details are arrows, intuition is the bow.

Our goal is to hunt down problems. You can use arrows alone, sure, but intuition is the framework that makes details astoundingly useful.... 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

$\displaystyle{13^2 + 0^2 = 13^2}$

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

After a few years, I thought it was time for a new layout. The goals:

• Be warm & friendly
• Be clean & readable (more whitespace & larger fonts)
• Be easy to skim & search (search results and posts have image previews)

BetterExplained isn’t an authority lecturing you on facts: it’s an excited friend sharing what actually helped when learning.... 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:

Yowza, did that ever blow my mind: add angles without sine or cosine!... Read article

I’ve made aha.betterexplained.com to share aha! moments. In 3 words, “Twitter meets Wikipedia”.

Why?

1. Writing articles hurts: research, collect thoughts, organize, filter the best, and write. This takes 20+ hours, and most articles languish half-done.

2. People don’t share “Ah, I get it!” moments on Wikipedia — that’s not its goal.

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 about shapes, lines, and so on.

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 till science class (9th grade?) and worse, it implies calculus is "math for science class".... 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 things (particles, insects, small children) to gargantuan sizes?... 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   = 5
9 to 16  = 7
16 to 25 = 9
25 to 36 = 11
…