Wednesday, February 10, 2010

Math part 1: Numbers.

As I said in my last post this should be the first post about maths. Let me first give you a little history about why I’m doing these posts. Well it all started last time I met with Yofed on her visit to Quebec city. I made the claim that math could potentially explain and represent everything, so Yofed challenged me with the old explain love with math (quickly answered by simplistic 1+1=2 answer) and blue+yelow (answered with a not so simple answer of adding to wavelenght of blue and yellow to get green, which I quickly admited was wrong but that there was a way to add blue and yelow mathmatically to show that it would give green for our eyes). Now the conversation continued with a “you should make a post about the power of math.” So instead I choose to make many post about it!

So before I start I must say that I am not an historian of science, nor a mathematician. The ideas expressed below might not be completely acurate. Now that it’s out of the way, I’ll first start by writing about the numbers, the part of mathematics most people are arguably famillar with. Now the power of by themselve is only descriptive but we are deceived by their ubiquity into forgetting just how powerfull they are. Lets start at the beginning of number, the positive integers, you know the first number you know (1, 2, 3, …). Those number can represent almost all physical quantity, from the number of pencil, to the number of stars in the univers as long as it is something physical the positive integer can represent it.Now, you can not, with those nomber, easilly quantify lack of something. For that we extend to the integer set, adding the minus sign before a nombre give us a representation and quantification for the lack of something. Your missing 15$ so you have -15$, simple. Then we have the zero, strangely a relatively recent invention, that recent the absence of quantity, emptiness, it’s the very usefull quantification of nothing.

Not with the natural number set (integer +0) you can arguably represent any physical quantity, however you can’t easilly represent part of a whole. So we have another set (that include the natural numbers), that we call the rational set. The set includes the rational fraction, i.e fraction that can represent “physical” fraction of a thing. For exemple, 1/2, 1/4 and 1/10 would be these some of these fraction, they are constructed from any number that can be constructed from the ratio of two non-zero natural number. We now have a lot of descriptive power, however there are still quantity that cannot be represente, for exemple, the square root of 2. To represent these quantities we have another set greater then the rational that is call the irrational set. It includes the rational number as well as all those quantity that cannot be expressed as the ratio of two non-zero natural number. They describ slightly more esoteric quantity so as the lengh of the hypothenus of some triangle and some logarithm (more on those in a later post) they might also include very fondamental constant in natural such as pi and the euler number e (it’s still not clear if pi or e are member of the rational or irrational number).

Now with the irrational set we can pretty much describ every quantity you find in natural (in fact I’m pretty sure you can describ everything physical thing with them), however math descriptive power is not limited to nature, or our 4-dimentional univers. Lets imagine for exemple you want to find the square root of a negative number. Now, you know that two negative numbers squared give you a positive number while two positive numbers squared also give you a positive number. While it might seem unlikely to encounter such a thing in nature they are essential to solving many equation. So to represent those quantity that have no “real” manifestation we expend our knowledge toward another set of number called the complex (or imaginary) number. These numbers are very special because you need 2 real numbers to represent them, for exemple one such number is 3+2i. Now, what do we need them for? Well they are usefull in engineering, quantum physics, chaos theory, and I could not easilly (or shortly) explain how they are used and why. You’ll probably never have to use them if you’re not working in advanced scientific field, but you probably encounter device who work on principle needing such number to be understood.

So just from the number in math we can gather a lot of information about the world, quantitative information. In fact, number do not just give information about the world but they can give us information about world event imaginary or world without relation to our own. In a way if you try to consider math a language for describing models of reality and predicting reality from those model you could think of number as part of this language alphabet. It’s only a part of this alphabet because number are not the building block of the communication of mathematic, equation are. And while number are usefull to solve equation and to obtaint physical representaiton of that solution they are by themselve meaning less. In fact, I think you could say that numbers are what we would use to translate between the models and the “real world”. In the next post about math, I will talk about two of the oldest branch of mathematique, arithmetics and geometry.

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