Thursday, February 11, 2010

Maths part II: Arithmetic and geometry

So let get into another post about the beauty and power of math, this time I will talk about arithmetic and geometry. The first one because it is the next logical step after numbers, it’s the branch of mathematic everyone is the most famillar with and it is the oldest part of math, while the second countributed so much to our understanding of the universe early in our history that including it later would do it much disrespect.

Arithmetic in short the manipulation of number, allow us to work with many quantities and predict an outcome. For a simple example, what happen when someone has 4 stick and you remove 2 and 2/3 of one (your left with 1 and 1/3 of another). Arithmetic includes a lot of way to manipulate number, at the beginning we have addition and substraction (which are basically the same thing, one with positive integers and the another with negative). Next in the “simple” operation we have multiplication, division and exponentionation. These operations are mostly the same operation, exponentionation is repeted multiplicaiton of the same number, division is multiplication by a fraction. Now, multiplication are basically addition, so we could technically say that we have only one operation on number, the addition. However keep in mind that writing everything as addition is very time and space consumming. After the sample operation of addition we have more complex operation such as finding the roots of a number (square, cube or even higher power) the root of an equation is in short the exponentiation by a fractional number of another number, and also logarithm of number. The logrithmic operation is much more complex, it consist in finding the number that elevate another number to give the number we are taking the logarithm from. To be clearer, say we have log(10 000)=4, that means that 10 to the 4th power is 10 000 (the 10 is implied in the log without a base denominator). Now, this might seem to old very little power, but being able to predict relationship between quantities and answer question those relationship imply is a corner stone to our thinking.

Now to talk about another ancient field of math, geometry. Geometry is the study of shapes, sizes, positions and proprieties of space. At first, geometry was described for a plane following a set of axiom that forms a basic for logically dedicing a lot of theorems and from those theorems information about the nature of shapes, sizes, posititions and all. This form of geometry called Euclidian geometry (for the greek that formulated it as a set of axiom) is what most of you learned in high school and use most of the time to infere information about the spatial relation and distribution of things. If you combine this with cartesian coordinate you get a very powerful tool to move about and locate thing on the Earth (as long as they are not too far appart). But geometry doest not stop there, we have found through experiment and observation we have discovered that we can formulate other axiom that describ other type of space. Such as for exemple, if our universe was on a sphere, the proprieties of a sphere lead to some interesting phenomeon such as having square triangles with more then 1 90 degree angle, or that the shortest distance between two point is not found as a straight line but as the arc of a circle. Many other geometry are possible for space, in fact curved space is an observed thing in astrophysic notably around very dense and massive object. Oh, one last thing about the power of geometry, understanding shapes, size and position might seems like something that not really spectacular, however one should recall that the greek measured the size of the Moon, the diameter of the Earth, the distance between the Earth and the Moon and many other quantities with amazing precision using only Euclydian geometry principles!

Now, adding arithmetic to number gives humanity the power to predict relation between quantities and manipulate, trade and use number as a model of what is happening or will happen in simple terms. If you have knowledge of geometry too you get an in dept knowledge of the shape of the world you live in, the shape of the object around you, ways to relate to them and predict their positions. In short with just math that was availlable to the ancient greek you can understand and model quite a lot of the world. Your models are time consuming to use but still you have a phenomenal understanding. My next post will show you how math power can make those model faster to use, become more abstract to represent even unknown quantity and even study very small and very dynamic things while I talk about, algebra and calculus.

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