Geometry is a subject that asks you to think logically. This is not an easy thing to do, of course, and we've all known people- including lots of adults- who are unable to do this. If you understand the jumping off point of the class, though, you'll at least be off to a solid start.

About The Number Man

Ralph Schatzki has taught high school algebra, and geometry, as well as trigonometry and statistics.

Referred to as "Mr.Ralph" at Ruamruedee Internatioal School Bangkok, a pretigious school where he taught for many years, he prides himself in being able to identify areas on which students need improvement and in giving them the tools they need to succeed at math.

Geometry is a subject that asks you to think logically. This is not an easy thing to do, of course, and we've all known people- including lots of adults- who are unable to do this. If you understand the jumping off point of the class, though, you'll at least be off to a solid start.


Do you remember when you were a little kid and you asked the seemingly innocuous question, "why?" Perhaps your mother said, "Billy, don't eat food off the floor," and you asked, "why?" Well, depending on how patient your mother was, the conversation might have gone something like this:

"Because the floor is dirty."

"Why?"

"Well, dirt falls down."

"Why?"

"Everything falls down, dear. It's called 'gravity'."

"Why?"


At this point, of course, even a sainted mother would probably give up, since the initial lesson has been completely obscured.

"It just is, alright. Don't ask why."

"Why?"


Let's leave the remainder of this encounter to our own imaginations, shall we?


The point is, one can always ask "why." At some point, we don't have the answers anymore since our knowledge is finite. But the question itself can be asked indefinitely.


A child learns at some point, of course, that his parents don't know everything, and begins to learn based upon observation, trial-and-error and all sorts of other methods. In fact, some might say that experience is the best teacher. (This is not always true. There is always some give-and-take here. A child should know the consequences of his actions, and in many cases a little bit of hurt can go a long way toward teaching a child never to do something again. I'm not, however, going to apply this philosophy to teaching my child to cross a street.)


Geometry is not like that, however. Everything in geometry is based upon a simpler, more basic rule before it.


Ah-ha! you say: where do we start? We can always ask why and get to something more and more basic, so what are the beginning rules based upon? (since at some point we just can't explain further)


Well, they're not based upon anything at all. It's like your mother finally saying to you, "It just is, ok," or, "Because I told you so." It's a little bit disconcerting, then, that this entire branch of mathematics is based upon nothing more than that.


These are the postulates: the things we simply accept without proof. Without them, we have nothing to build upon. Hey, we have to start somewhere!


Really, though, all knowledge is like that. It's just that geometry makes this big to-do about how everything in it is so logical and based upon something else and that it's infallible or something.


I always like to say that at the far end of the universe is a race of alien mathematical idiots, whose entire geometry textbook is devoted to proving the postulates we simply accept as given on page 1. At the other end of the universe is a race of alien mathematical geniuses whose postulates are the most difficult problems in our book. In the end, this is just another way to illustrate the idea that there is a continuum and we simply pick an arbitrary point and start there.


The shortest distance between two points is a straight line.


Two points determine a unique line.


If two distinct lines intersect, then their intersection is a point.


Three noncollinear points determine a unique plane.


If two planes intersect, then their intersection is a line.


Definitions are also like postulates, in that we determine the definition of something. We don't seek to show it- we just accept it. Definitions and postulates are where we start: they're the ground on which we build. Don't question the ground: accept it.


But question everything else!

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Math Tutorial links

• Introduction
• Finding your own way
• Knowing The Words Pre-Algebra, Algebra
• Variables and Equations - Pre-algebra, Algebra
• Postulate , aka Axiom, what is it? - Geometry
• Math Made Easy The Number Man Ralph Schatzki

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