## Latest Blog Posts (see all)

## Popular Articles

## How To Learn Trigonometry Intuitively

## List of Insights

Imaginary numbers (rotation)+ e^x (smooth growth) → Circular motion (Euler's Formula)

Euler's Formula+ Integration (step-by-step analysis)→ Convert to circles, step-by-step (Fourier Transform)

Combinations+ Combinations+ Combinations→ Birthday Paradaox

## Intuitive Learning

## ADEPT method

Learning starts with an analogy.## Why Learn Math?

Vocabulary for thinking## Math Intuition

Start with the cat, not its DNA## Bow & Arrow Intuition

First, have a good shot.## Pencil, Then Ink

Don't hide the process## Analogies

Useful, even if "wrong"## Cartoons

Simplify ideas to a caricature## Intuition Isn't Optional

The recipe for knowledge that lasts.## Arithmetic & Numbers

## Mental Math Shortcuts

A few memorable conversions## Sum 1 to 100

Rearrange to simplify## Visual Arithmetic

Every operation is a transformation## Sense of Scale

Bring large numbers down to everyday terms## Fencepost Problem

Counting spans or points?## Number Systems

When the odometer ticks over## Everyday Logarithms

Counting digits## Modular Arithmetic

Variations on clocks## Algebra & Counting

## Combinations

Multiply possibilities, divide redundancies## Primes

Atoms within numbers## The Equals Sign

Set, check, or define?## Factoring Equations

Break down the tepee## Parametric Equations

Extract the true cause## Types of Graphs

Map (distance matters) or non-map## Square Numbers

Use geometry, algebra, calculus## Navigate a Grid

Convert decisions to letters & permute## Combinations & Multiplication

Choices multiply.## Ratios

How much oomph, how often?## Geometry

## Pythagorean Theorem

How every shape changes## Pythagorean Distance

Compare any quantities## Similarity

Step back. Smaller shape, same ratios.## Pythagorean Rescale

Work from a unit triangle## Precalculus & Trigonometry

## Radians

Angles from the mover’s perspective## Finding Pi

Surround that critter## Sine Waves

The natural sway found in circles, springs## The Number e

Perfectly smooth growth## Natural Log

Time needed to grow## Imaginary Numbers

Numbers can rotate## Complex Numbers

Arithmetic goes 2d## Complex Multiplication

Scale & rotate## Exponents

Grow numbers in the expand-o-tron## Think With Exponents

Logs are causes, exponents are effects## Trigonometry

Visualize a dome, wall, and ceiling## Law of Sines

Every angle has an equal perspective.## Law of Cosines

Keep track of interacting parts.## Trig Identities

Draw the circles yourself.## Statistics

## Statistics

See footprints, guess the animal## Averages

"Typical" depends on the relationship## Birthday Paradox

23 people, many possibilities## Bayes Theorem

Extra info? Adjust the odds.## Short Bayes’ Theorem

Track new info in a table## Monty Hall

Original door vs. best of the other two## Calculus

## 1 Minute Calculus

Analyze patterns with x-ray and time-lapse vision## 0.999… = 1?

What's your number system?## Limits

Guess what happened when you blinked## Limits / Infinitesimals

Make a model perfect## Integration

Piece-by-piece multiplication## Derivative Intro

Measurements depend on the instrument## Derivatives II

Imagine linked machines## Derivatives III

Quotient, exponents, logs## Calculus Bank Account

Raises change income, changing the balance## Vector Calculus

## Circulation & Curl

Total twist, twist at a point## The Gradient

Microwave the doughboy, quick## Gradient Details

Follow the best tradeoff## Flux

Amount of bananas crossing a surface## Divergence

Flux per unit volume## Dot Product

Mario Kart speed boost## Cross Product

All the uneven parts## Engineering / University Math

## Linear Algebra

Math with mini-spreadsheets## Euler’s Formula

Imaginary exponents make circles## Fourier Transform

Patterns have circular ingredients## Programming Tools

## Version Control

Track files over time## Distributed VCS

Pass around changes## Git Insights

#1: There's a staging area## GDB Debugging

A quickstart guide## Web Development

## Faster Javascript

Minify, compress, move to end## HTTP Caching

Don't redownload until expired## Gzip Compression

Zip files before sending## Starting Ruby on Rails

Get past the gotchas## Understanding MVC

Fat models, thin controller, thin view## Javascript Reference

1-page summary## Load XML/JSON

Treat a script as data## XSRF Attacks

Attacker spoofs a legit request## Debugging with Firefox

Use firebug## Making Bookmarklets

A quick javascript snippet## Low-level Programming

## GUIDs

You'll never run out. Ever.## Binary Files

Space efficiency vs. marshalling## Byte Order

Computers speak different languages## Unicode

Not every char is 2 bytes## XOR Swap

Cleverness, not for real life## Fast 1/sqrt(x)

Newton's method, magic initializer## Symbol Tables

Deobfuscate object code## Networking

Telnet is an IM conversation## Business

## Rule of 72

time = 72/interest## Pareto Principle

Sometimes 80% is enough## Stock Market

A transparent store## Accounting Basics

Everything is owned by someone## Interest Rates

A formula for every type of interest## Debt & Leverage

Invested debt multiplies return## Assorted Musings

# Why Is BetterExplained Different?

Most classes walk through a bunch of details, top-down. BetterExplained takes a blurry-to-sharp approach. Here’s the difference:

Learning facts in order seems logical, but doesn’t work for me. I need the big picture up front, then we can sharpen the details:

Learning is about experiencing a true *Aha!* moment, not memorizing facts that are quickly forgotten. If this approach resonates with you, welcome!

# About Kalid

I enjoyed math until a poorly-taught class nearly destroyed that passion. A last-minute Aha! moment showed me math could make sense, even be enjoyable, when presented with:

- A friendly, curious attitude
- A mix of intuitive and technical understanding
- A focus on lasting insight

I share explanations that helped, hoping they help you too. I’m thrilled that BetterExplained now reaches millions every year, and has appeared in blogs for the New York Times and Scientific American. Read more…