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A BetterExplained Guide To Calculus

Kalid — Mon, 09 Nov 2009 05:37:30 +0000

I’ve struggled with how to write about calculus. The standard techniques seem to be:

The “bag of formulas”: memorize ‘em and move on
The anal-retentive, rigorous treatment: written by math robots, for math robots!
The happy smiles tour: oversimplifications without examples (Calculus helps scientists solve problems!)

No, nyet, nein! I know what I need: intuition (What does it really mean?) followed by examples to back it up. I want a calculus series that lets calculus be calculus — wild, interesting, and fun.

The Explanatory Approach

I started writing in a vacuum, but realized I don’t remember calculus. I need a refresher — in fact, I need the insights I want to share! These articles are for us both (it’s what I’d want to relearn the subject), and here’s my approach:

I’m reading Elementary Calculus: An Infinitesimal Approach [free pdf]. It teaches calculus using its original approach (infinitesimals), not the modern limit-based curriculum. My goal is intuition, so this works well.

As I study the chapters, I’ll share the insights I find and the concepts I struggled with.

I’ll sprinkle examples along the way. They’re a gut check, not the focus (if you want practice problems, the book has plenty).

It’s a lack of insights, not information, that makes calculus hard. We don’t need another course repeating the definitions that confused us the first time (Here’s the definition of a limit, again!).

We shouldn’t be struggling with the true meaning of a subject centuries after its invention. This is my intuition-laced hat in the ring.

The Calculus Articles

The goal is to be concise, informal, and fun. Dabble, skim and ignore the examples if needed — focus on the insights. The elegance of calculus can be appreciated progressively: we don’t need astrophysics to enjoy a starry night.

Learning Math

Developing Your Intuition For Math

Calculus Overview

Small numbers: Limits and Infinitesimals

Learning Calculus: Overcoming Our Artificial Need for Precision
Understanding the need for small numbers (in progress)

Measuring Changes: Derivatives

Accumulating Changes: Integrals

A Calculus Analogy: Integrals as Multiplication

This post is the table of contents for the series. Happy math.

A Simple Introduction To Computer Networking

Kalid — Mon, 16 Mar 2009 16:00:30 +0000

Most networking discussions are a jumble of acronyms. Forget the configuration details -- what are the insights?

Networking is about communication
Text is the simplest way to communicate
Protocols are standards for reading and writing text

Beneath the details, networking is an IM conversation. Here's what I wish someone told me when learning how computers communicate.

TCP: The Text Layer

The Transmission Control Protocol (TCP) provides the handy illusion that we can "just" send text between two computers. TCP relies on lower levels and can send binary data, but ignore that for now:

TCP lets us Instant Message between computers

We IM with Telnet, the 'notepad' of networking: telnet sends and receives plain text using TCP. It's a chat client peacefully free of ads and unsolicited buddy requests.

Let's talk to Google using telnet (or putty, a better utility):

telnet google.com 80
[connecting...]
Hello Mr. Google!

We connect to google.com on port 80 (the default for web requests) and send the message "Hello Mr. Google!". We press Enter a few times and await the reply:


...
Bad Request
Your client has issued a malformed or illegal request
...

Malformed? Illegal? The mighty Google is not pleased. It didn't understand us and sent HTML telling the same.

But, we had a conversation: text went in, and text came back. In other words:

Protocols: The Forms To Fill Out

Unstructured chats is too carefree -- how does the server know what we want to do? We need a protocol (standard way of communicating) if we're going to make sense.

We use protocols all the time

Putting to: and from: addresses in certain places on an envelope
Filling out bank forms (special place for account number, deposit amount, etc.)
Saying "Roger" or "10-4" to indicate a radio request was understood

Protocols make communication clear.

Case Study: The HTTP Protocol

We see HTTP in every url: http://google.com/. What does it mean?

Connect to server google.com (Using TCP, port 80 by default)
Ask for the resource "/" (the default resource)
Format the request using the Hypertext Transport Protocol

HTTP is the "form to fill out" when asking for the resource. Using the HTTP format, the above request looks like this:

GET / HTTP/1.0

Remember, it's just text! We're asking for a file, through an IM session, using the format: [Command] [Resource] [Protocol Name/Version].

This command is "IM'd" to the server (your browser adds extra info, a detail for another time). Google's server returns this response:

HTTP/1.0 200 OK
Cache-Control: private, max-age=0
Date: Sun, 15 Mar 2009 03:13:39 GMT
Expires: -1
Content-Type: text/html; charset=ISO-8859-1
Set-Cookie: PREF=ID=5cc6...
Server: gws
Connection: Close


(Google web page, search box, and cute logo)

Yowza. The bottom part is HTML for the browser to display. But why the junk up top?

Well, suppose we just got the raw HTML to display. But what about errors: if the server crashed, the file wasn't there, or google just didn't like us?

Some metadata (data about data) is useful. When we order a book from Amazon we expect a packing slip describing the order: the intended recipient, price, return information, etc. You don't want a naked book just thrown on your doorstep.

Protocols are similar: the recipient wants to know if everything was OK. Here we see infamous status codes like 404 (resource not found) or 200 (everything OK). These headers aren't the real data -- they're the packing slip from the server.

Insights From Protocols

Studying existing, popular systems is a great way to understand engineering decisions. Here are a few:

Binary vs Plain Text

Binary data is more efficient than text, but more difficult to debug and generate (how many hex editors do you know to use?). Lower-level protocols, the backbone of the internet, use binary data to maintain performance. Application-level protocols (HTTP and above) use text data for ease of interoperability. You don't have religious wars about endian issues with HTTP.

Stateful vs. Stateless

Some protocols are stateful, which means the server remembers the chat with the client. With SMTP, for example, the client opens a connection and issues commands one at a time (such as adding recipients to an email), and closes the connection. Stateful communication is useful in transactions that have many steps or conditions.

Stateless communication is simpler: you send the entire transaction as one request. Each "instant message" stands on its own and doesn't need the others. HTTP is stateless: you can request a webpage without introducing yourself to the server.

Extensibility

We can't think of everything beforehand. How do we extend old protocols for new users?

HTTP has a simple and effective "header" structure: a metadata preamble that looks like "Header:Value".

If you don't recognize the header sent (new client, old server) just ignore it. If you were expecting a header but don't see it (old client, new server), just use a default. It's like having an "Anything else to tell us?" section in a survey.

Error Correction & Reliability

It's the job of lower-level protocols like TCP to make sure data is transmitted reliably. But higher-level protocols (like HTTP) need to make sure it's the right data. How are errors handled and communicated? Can the client just retry or does the server need to reset state?

HTTP comes with its own set of error codes to handle a variety of situations.

Availability

The neat thing about networking is that works on one computer. Memcached is a great service to cache data. And guess what? It uses plain-old text commands (over TCP) to save and retrieve data.

You don't need complex COM objects or DLLs - you start a Memcached server, send text in, and get text out. It's language-neutral and easy to access because any decent OS supports networking. You can even telnet into Memcached to debug it.

Wireless routers are similar: they have a control panel available through HTTP. There's no "router configuration program" -- you just connect to it with your browser. The router serves up webpages, and when you submit data it makes the necessary configuration changes.

Protocols like HTTP are so popular you can assume the user has a client.

Layering Protocols

Protocols can be layered. We might write a resume, which is part of a larger application, which is stuffed into an envelope. Each segment has its own format, blissfully unaware of the others. Your envelope doesn't care about the resume -- it just wants the to: and from: addresses written correctly.

Many protocols rely on HTTP because it's so widely used (rather than starting from scratch, like Memcached, which needs efficiency). HTTP has well-understood methods to define resources (URLs) and commands (GET and POST), so why not use them?

Web services do just that. The SOAP protocol crams XML inside of HTTP commands. The REST protocol embraces HTTP and uses the existing verbs as much as possible.

Remember: It's All Made Up

Networking involves human conventions. Because plain text is ubiquitous and easy to use, it is the basis for most protocols. And TCP is the simplest, most-supported way to exchange text.

Remembering that everything is a plain text IM conversation helps me wrap my head around the inevitable networking issues. And sometimes you need to jump into HTTP to understand compression and caching.

Don't just memorize the details; see protocols as strategies to solve communication problems. Happy networking.

Developing Your Intuition For Math

Kalid — Thu, 08 Jan 2009 19:44:28 +0000

Our initial exposure to an idea shapes our intuition. And our intuition impacts how much we enjoy a subject. What do I mean?

Suppose we want to define a “cat”:

Caveman definition: A furry animal with claws, teeth, a tail, 4 legs, that purrs when happy and hisses when angry…
Evolutionary definition: Mammalian descendants of a certain species (F. catus), sharing certain characteristics…
Modern definition: You call those definitions? Cats are animals sharing the following DNA: ACATACATACATACAT…

The modern definition is precise, sure. But is it the best? Is it what you’d teach a child learning the word? Does it give better insight into the “catness” of the animal? Not really. The modern definition is useful, but after getting an understanding of what a cat is. It shouldn’t be our starting point.

Unfortunately, math understanding seems to follow the DNA pattern. We’re taught the modern, rigorous definition and not the insights that led up to it. We’re left with arcane formulas (DNA) but little understanding of what the idea is.

Let’s approach ideas from a different angle. I imagine a circle: the center is the idea you’re studying, and along the outside are the facts describing it. We start in one corner, with one fact or insight, and work our way around to develop our understanding. Cats have common physical traits leads to Cats have a common ancestor leads to A species can be identified by certain portions of DNA. Aha! I can see how the modern definition evolved from the caveman one.

But not all starting points are equal. The right perspective makes math click — and the mathematical “cavemen” who first found an idea often had an enlightening viewpoint. Let’s learn how to build our intuition.

What is a Circle?

Time for a math example: How do you define a circle?

There are seemingly countless definitions. Here’s a few:

The most symmetric 2-d shape possible
The shape that gets the most area for the least perimeter (see the isoperimeter property)
All points in a plane the same distance from a given point (drawn with a compass, or a pencil on a string)
The points (x,y) in the equation x² + y² = r² (analytic version of the geometric definition above)
The points in the equation r* sin(t), r* cos(t), for all t (really analytic version)
The shape whose tangent line is always perpendicular to the position vector (physical interpretation)

The list goes on, but here’s the key: the facts all describe the same idea! It’s like saying 1, one, uno, eins, “the solution to 2x + 3 = 5″ or “the number of noses on your face” — just different names for the idea of “unity”.

But these initial descriptions are important — they shape our intuition. Because we see circles in the real world before the classroom, we understand their “roundness”. No matter what fancy equation we see (x² + y² = r²), we know deep inside that a circle is “round”. If we graphed that equation and it appeared square, or lopsided, we’d know there was a mistake.

As children, we learn the “caveman” definition of a circle (a really round thing), which gives us a comfortable intuition. We can see that every point on our “round thing” is the same distance from the center. x² + y² = r² is the analytic way of expressing that fact (using the Pythagorean theorem for distance). We started in one corner, with our intuition, and worked our way around to the formal definition.

Other ideas aren’t so lucky. Do we instinctively see the growth of e, or is it an abstract definition? Do we realize the rotation of i, or is it an artificial, useless idea?

A Strategy For Developing Insight

I still have to remind myself about the deeper meaning of e and i — which seems as absurd as “remembering” that a circle is round or what a cat looks like! It should be the natural insight we start with.

Missing the big picture drives me crazy: math is about ideas — formulas are just a way to express them. Once the central concept is clear, the equations snap into place. Here’s a strategy that has helped me:

Step 1: Find the central theme of a math concept. This can be difficult, but try starting with its history. Where was the idea first used? What was the discoverer doing? This use may be different from our modern interpretation and application.
Step 2: Explain a property/fact using the theme. Use the theme to make an analogy to the formal definition. If you’re lucky, you can translate the math equation (x^{2 + y}2 = r^2) into a plain-english statement (”All points the same distance from the center”).
Step 3: Explore related properties using the same theme. Once you have an analogy or interpretation that works, see if it applies to other properties. Sometimes it will, sometimes it won’t (and you’ll need a new insight), but you’d be surprised what you can discover.

Let’s try it out.

A Real Example: Understanding e

Understanding the number e has been a major battle. e appears all of science, and has numerous definitions, yet rarely clicks in a natural way. Let’s build some insight around this idea. The following section will have several equations, which are simply ways to describe ideas. Even if the equation is gibberish, there’s a plain-english idea behind it.

Here’s a few popular definitions of e:

The first step is to find a theme. Looking at e’s history, it seems it has something to do with growth or interest rates. e was discovered when performing business calculations (not abstract mathematical conjectures) so “interest” (growth) is a possible theme.

Let’s look at the first definition, in the upper left. The key jump, for me, was to realize how much this looked like the formula for compound interest. In fact, it is the interest formula when you compound 100% interest for 1 unit of time, compounding as fast as possible.

Definition 1: Define e as 100% compound growth at the smallest increment possible.

The article on e describes this interpretation.

Let’s look at the second definition: an infinite series of terms, getting smaller and smaller. What could this be?

After noodling this over using the theme of “interest” we see this definitions shows the components of compound interest. Now, insights don’t come instantly — this insight might strike after brainstorming “What could 1 + 1 + 1/2 + 1/6 + …” represent when talking about growth?”

Well, the first term (1 = 1/0!, remembering that 0! is 1) is your principal, the original amount. The next term (1 = 1/1!) is the “direct” interest you earned — 100% of 1. The next term (0.5 = 1/2!) is the amount of money your interest made (”2nd level interest”). The following term (.1666 = 1/3!) is your “3rd-level interest” — how much money your interest’s interest earned!

Money earns money, which earns money, which earns money, and so on — the sequence separates out these contributions (read the article on e to see how Mr. Blue, Mr. Green & Mr. Red grow independently). There’s much more to say, but that’s the “growth-focused” understanding of that idea.

Definition 2: Define e by the contributions each piece of interest makes

Neato.

Now to the 3rd, and shortest definition. What does it mean? Instead of thinking “derivative” (which turns your brain into equation-crunching mode), think about what it means. The feeling of the equation. Make it your friend.

It’s the calculus way of saying “Your rate of growth is equal to your current amount”. Well, growing at your current amount would be a 100% interest rate, right? And by always growing it means you are always calculating interest — it’s another way of describing continuously compound interest!

Definition 3: Define e by as a function always growing by 100% of your current value

Nice — e is the number where you’re always growing by exactly your current amount (100%), not 1% or 200%.

Time for the last definition — it’s a tricky one. Here’s my interpretation: Instead of describing how much you grew, why not say how long it took?

If you’re at 1 and growing at 100%, it takes 1 unit of time to get from 1 to 2. But once you’re at 2, and growing 100%, it means you’re growing at 2 units per unit time! So it only takes 1/2 unit of time to go from 2 to 3. Going from 3 to 4 only takes 1/3 unit of time, and so on.

The time needed to grom from 1 to A is the time from 1 to 2, 2 to 3, 3 to 4… and so on, until you get to A. The first definition defines the natural log (ln) as shorthand for this “time to grow” computation.

ln(a) is simply the time to grow from 1 to a. We then say that “e” is the number that takes exactly 1 unit of time to grow to. Said another way, e is is the amount of growth after waiting exactly 1 unit of time!

Definition 4: Define the time needed to grow continuously from 1 to as ln(a). e is the amount of growth you have after 1 unit of time.

Whablamo! These are four different ways to describe the mysterious e. Once we have the core idea (”e is about 100% continuous growth”), the crazy equations snap into place — it’s “possible” to translate calculus into English. Math is about ideas!

What’s the Moral?

In math class, we often start with the last, most complex idea. It’s no wonder we’re confused — we’re showing DNA and expecting students to see the cat.

I’ve learned a few lessons from this approach, and it underlies how I understand and explain math:

Search for insights and apply them. That first intuitive insight can help everything else snap into place. Start with a definition that makes sense and “walk around the circle” to find others.
Develop mental toughness. Banging your head against an idea is no fun. If it doesn’t click, come at it from different angles. There’s another book, another article, another person who explains it in a way that makes sense to you.
It’s ok to be visual. We think of math as rigid and analytic — but visual interpretations are ok! Do what develops your understanding. Imaginary numbers were puzzling until their geometric interpretation came to light, decades after their initial discovery. Looking at equations all day didn’t help mathematicians “get” what they were about.

Math becomes difficult and discouraging when we focus on definitions over understanding. Remember that the modern definition is the most advanced step of thought, not necessarily the starting point. Don’t be afraid to approach a concept from a funny angle — figure out the plain-English sentence behind the equation. Happy math.