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Enjoy clear, intuitive lessons about imaginary numbers, exponents, and more.

Latest Articles

Why Is Better Explained Different?

Most lessons offer low-level details in a linear, seemingly logical sequence. Better Explained focuses on the big picture — the Aha! moment — and then the specifics. Here’s the difference:
I know which approach keeps my curiosity and enthusiasm. The learning strategy is the ADEPT Method:

Learning isn’t about memorizing facts to pass a test. It’s about unlocking the joy of discovery when an idea finally makes sense. If this approach resonates with you, welcome aboard.

About Kalid Azad

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 Better Explained now reaches millions every year, and has appeared in blogs for the New York Times and Scientific American. Read more…

Popular Articles

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


Useful, even if "wrong"


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


Multiply possibilities, divide redundancies


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.


How much oomph, how often?

Quadratic Formula

Rearrange a square

Polynomial Intuition

Decmials, interactions, function DNA


Pythagorean Theorem

How every shape changes

Pythagorean Distance

Compare any quantities


Step back. Smaller shape, same ratios.

Pythagorean Rescale

Work from a unit triangle
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Precalculus & Trigonometry


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


Grow numbers in the expand-o-tron

Think With Exponents

Logs are causes, exponents are effects


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.

Hyperbolic Trig

Mini exponentials and logarithms



See footprints, guess the animal


"Typical" depends on the relationship

Birthday Paradox

23 people, many possibilities

Bayes Theorem

Extra info? Adjust the odds.

Short Bayes’ Theorem

Adjust for false positives

Monty Hall

Original door vs. best of the other two

Calculus Course

Calculus Guide

Learn the basics, fast.

Lesson 1

1-Minute Summary

Lesson 2

X-Ray Vision

Lesson 3

3D intuition

Lesson 4

Integrals, Derivatives

Lesson 5

Computer Notation

Lesson 6

Improved Algebra

Lesson 7

Linear Changes

Lesson 8

Squared Changes

Lesson 9


Lesson 10


Lesson 11

Fundamental Theorem of Calculus

Lesson 12

Rules: Add, Multiply, Invert

Lesson 13

Patterns In the Rules

Lesson 14

Rules: Powers, Division

Lesson 15

Archimedes' Formulas

Calculus Summary

The big insights. All together now.

Calculus (Articles)

1 Minute Calculus

Analyze patterns with x-ray and time-lapse vision

0.999… = 1?

What's your number system?


Guess what happened when you blinked

Limits / Infinitesimals

Make a model perfect


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

Integral of sin

Whoa: the horizontal change

Taylor Series

DNA of a function

Vector Calculus

Circulation & Curl

Total twist, twist at a point

The Gradient

Microwave the doughboy, quick

Gradient Details

Follow the best tradeoff


Amount of bananas crossing a surface


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


Fancy, programmable multiplication

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


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


You'll never run out. Ever.

Binary Files

Space efficiency vs. marshalling

Byte Order

Computers speak different languages


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


Telnet is an IM conversation


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

Know Thyself

If you're Hulk, be Hulk.

Simplicity vs. Complexity

Complexity, not power, is the enemy.

Notes on Happiness

Passion, foolishness, dancing

Brevity is Beautiful

Zero guilt.