# Learning math? Think like a cartoonist.

What’s the essential skill of a cartoonist? Drawing ability? Humor? A deep well of childhood trauma?

I’d say it’s an eye for simplification, capturing the essence of an idea.

For example, let’s say we want to understand Ed O’Neill:

A literal-minded artist might portray him like this:

While the technical skill is impressive, does it really capture the essence of the man?... Read article

# How To Think With Exponents And Logarithms

Here’s a trick for thinking through problems involving exponents and logs. Just ask two questions:

Are we talking about inputs (cause of the change) or outputs (the actual change that happened?)

• Logarithms reveal the inputs that caused the growth
• Exponents find the final result of growth

Are we talking about the grower’s perspective, or an observer’s?... Read article

# Understand Ratios with “Oomph” and “Often”

Ratios summarize a scenario with a number, such as “income per day”. Unfortunately, this hides the explanation for how the result came about.

For example, look at two businesses:

• Annie’s Art Gallery sells a single, $1000 piece every day • Frank’s Fish Emporium sells 250 trout at$4/each every day

By the numbers, they’re identical \$1000/day operations, right?... Read article

# How To Learn Trigonometry Intuitively

Trig mnemonics like SOH-CAH-TOA focus on computations, not concepts:

TOA explains the tangent about as well as x2 + y2 = r2 describes a circle. Sure, if you’re a math robot, an equation is enough. The rest of us, with organic brains half-dedicated to vision processing, seem to enjoy imagery.... Read article

# Site Update: New Design + Intuition Cheatsheet

After months of work with the help of Neil, a great designer, and my Excel-blogging friend Andrew, I’m happy to launch a brand-new design.

My goals were to be friendly, readable, and easy-to-navigate. Here’s a quick before-and-after:

## New Logo

Neil did a fantastic job here — I’d been looking for a way to convey a welcoming, conversational tone.... Read article

# A Quick Intuition For Parametric Equations

Algebra is really about relationships. How are things connected? Do they move together, or apart, or maybe they’re completely independent?

Normal equations assume an “input to output” connection. That is, we take an input (x=3), plug it into the relationship (y=x2), and observe the result (y=9).... Read article

# It’s Time For An Intuition-First Calculus Course

Summary: I’m building a calculus course from the ground-up focused on permanent intuition, not the cram-test-forget cycle we’ve come to expect.

Update: The course is now live at http://betterexplained.com/calculus

## The Problem: We Never Internalized Calculus

First off: what’s wrong with how calculus is taught today?... Read article

# Print Edition of “Math, Better Explained” Now Available

I’m thrilled to announce the print edition of Math, Better Explained is available on Amazon:

With the magic of print-on-demand, you can order the book with overnight shipping (Amazon Prime!), and be reading full-color insights tomorrow. Yowza.

I’ve often been asked if a print version can be made, and I’m beaming to say it’s now a reality:

• 12 chapters (~100 pages) of full-color explanations
• Professional-quality typesetting & layout
• Gorgeous, high-resolution text and diagrams
• Compact, easy-to-carry size with comfortable margins (7″ x 10″)

# An Intuitive Introduction To Limits

Limits, the Foundations Of Calculus, seem so artificial and weasely: “Let x approach 0, but not get there, yet we’ll act like it’s there… ” Ugh. Here’s how I learned to enjoy them:

• What is a limit? Our best prediction of a point we didn’t observe.

# Understanding Bayes Theorem With Ratios

My first intuition about Bayes Theorem was “take evidence and account for false positives”. Does a lab result mean you’re sick? Well, how rare is the disease, and how often do healthy people test positive? Misleading signals must be considered.

This helped me muddle through practice problems, but I couldn’t think with Bayes.... Read article

# An Interactive Guide To The Fourier Transform

The Fourier Transform is one of deepest insights ever made. Unfortunately, the meaning is buried within dense equations:

$\displaystyle{X_k = \sum_{n=0}^{N-1} x_n \cdot e^{-i 2 \pi k n / N}}$

$\displaystyle{x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \cdot e^{i 2 \pi k n / N}}$

Yikes. Rather than jumping into the symbols, let's experience the key idea firsthand. Here's a plain-English metaphor:

• What does the Fourier Transform do?