Study Guide

Logic and Proof

Logic and Proof Introduction


Most people think that mathematics is all about manipulating numbers and formulas to compute something. While numbers play a starring role (like Brad Pitt or Angelina Jolie) in math, it's also important to understand why things work the way they do. That means at its core, math is more about logic than it is about numbers (or Hollywood movie stars, for that matter).

For example, the area of a rectangle can be found by multiplying its length times its width. Big whoop. But why does area work like that? Once we know the answer to that question, we can approach related questions. Quietly, though, because we don't want to scare them away.

To explain this "why" (em cee ay?), mathematicians have developed the idea of "proof." Proof is a touching drama starring Gwyneth Paltrow, Jake Gyllenhaal, and Anthony Hopkins. Or, you know, an explanation of why something is true.

First, you list your assumptions, and then you argue your way to a desired conclusion, explaining each step along the way. By the end, you should have convinced yourself and anybody who may be reading your proof, like Anthony Hopkins (it could happen!) or your math teacher (far more likely) that you're right. It feels wonderful to bask in the glow of mathematical truth, doesn't it?

But don't break out your suntan lotion just yet. Before we can prove anything, we need to clarify the language we'll use. After all, it's hard to argue with somebody who speaks a different language, right? Don't worry if it seems abstract and mysterious at first. Just like any language, you have to use it in order to really learn it, so it'll become much more natural as we start working out some proofs later.

Logic and Proof Resources

Websites

Euclid's Definitions and Postulates
We're studying Euclidean geometry and this website gives a list a pretty thorough (and well-organized) list of all of Euclid's geometric original definitions and postulates. The writing is a bit flowery, but not difficult to follow. If you're achin' for some Euclid, this website will be heaven on earth (or the Internet).

eHow: How To Do Formal Geometry Proofs
If proofs have got your knickers in a twist, this website will lay it all out for you. Complete with helpful tips and a full set of instructions, it'll get you started chipping away at proofs sooner than you can say, "formal geometry proof." Thanks, eHow!

Theorems and Postulates
If you understand the idea of proofs, but can't quite keep the properties and postulates and theorems straight, this might help you out a bit. Not only does it clearly list out a bunch of handy dandy theorems and postulates, but it also categorizes them so you can find whatever you're looking for without a giant magnifying glass.

A Background of Constructivism
The whole basis for our logic and reasoning is that these shapes, lines, and stuff can be measured. This is based in the idea of constructivism. Want more information from the wonderful (and very scholarly) world of academia? You got it.

Purplemath: Induction Proofs
Induction got you in a dither? Let the fine folks at Purplemath help bulk up your induction chops.

Dr. Math: Proof by Induction
Proving stuff by induction can be known to cause headaches, stomach cramps, and spontaneous yelling. Let the good Doctor prescribe you some solid practice, and you'll be feeling tip-top in no time.

Art of Problem Solving: Proof by Contradiction
A short-and-sweet breakdown of how proofs by contradiction work, including Euclid's famous proof that there's an infinite number of prime numbers.

Videos

Yay Math: Intro to Proofs
Watch Mr. Robert Ahdoot, a high school math teacher (and multi-gazillionaire) as he explains the properties of equality and introduces proofs. As he says, proofs are probably one of the most difficult concepts in geometry, so hopefully his flashy bling isn't too distracting.

MathPlanet: If-Then Statements
If the fact that math can make statements seems fundamentally wrong to you, we understand. Fortunately, there's a whole planet (or a website masquerading as a planet, anyway) dedicated to math and helping us understand these statements. If you watch the videos and explore the various topics, then it'll all come together. Did you catch that if-then statement?

Proof by Mathematical Induction
Listen to Paul's dulcet tones as he takes you step-by-step through a slightly more complicated proof by induction. Paul is good people.

Spoonful of Maths: Proof by Contradiction
In this video, Dr. Herke runs us through a proof that there's no smallest rational number greater than zero. Say what? It's a seriously weird idea, but contradiction is your friend here.

Games and Tools

Two-Column Proofs Practice Tool
If you feel comfortable enough with definitions and angles and lines and stuff, you may want to give this tool a try. It's just like Build-A-Bear, except in this case, you're building a proof. It won't protect you from monsters while you sleep, but it will protect you from failing geometry. Fill in the statements and the reasons from the options provided and conquer those proof-monsters in no time.

ClassZone: Geometry Quizzes
Practice makes perfect (unless you're doing it wrong). Testing yourself and getting instant results? There's no way to do that wrong. Choose any subject you feel confident or helpless about and complete the online quiz. It'll tell you if you're a master in that area, or if you might need a little more practice before you're ready to tackle a full-on test.

Math Goodies: Conditional Statements
If you're struggling with conditional statements and how to represent them, especially, take a gander at this. It'll give you thorough explanations with charts and everything, plus a mini-quiz at the end. Feel free to explore other topics in the unit if you feel a little uneasy with the material. No need for a permission slip.

Scatter
Need more help minding your p's and q's? This quick little game will enable you to tell the difference between a converse and a contrapositive without breaking a sweat. Drag the definition to match the symbols as fast as you can, and your p's and q's will have minds of their own. Not really though, because that'd be weird.