# 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.

# Understanding Pythagorean Distance and the Gradient

The Pythagorean Theorem shows how strange our concept of distance is. Using the rule a^{2} + b^{2} = c^{2}, we can trade some “a” to get more “b”.

Starting with

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

# Site updates for BetterExplained.com

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.

# 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!

# Share your insights: aha.betterexplained.com (beta)

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

**Why?**

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

# 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?

# 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 …

Huh?

# 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 Peirce:

It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth.

# Math Webinar Sat 6/19/2010 @ 2pm EST

Math seems to get magical in groups. Maria Droujkova organizes the awesome Math 2.0 interest group, which I found out about recently — I’m bummed I didn’t hear about it till now! There’s a mailing list, webinars with prominent math aficionados (hosts of MathOverflow, Cut the Knot, Art of Problem Solving), and other activities geared towards improving the state of math education.