Mathinfinity

Here’s a great link to another math blog. This one follows more advanced math topics, and includes tutorials and many great links. Check it out: http://catchydomain.wordpress.com/2008/02/12/math-55s-website-2005/

Time for my favorite topic: pi.

Pi is an irrational number. Its decimal trail extends to infinity, meaning that the complete number will never be calculated (See the infinity post). Pi is generally seen as equaling 3.1415926353…, but is actually far more complex than that. Other than simply using it in area and volume equations, it seems useless. However, pi is extremely valuable to math, and has a rather simple origin.

Pi is derived from the ratio of a circle’s area to its radius squared. For example, imagine a circle with a radius of two units. The area would thus be πr^2, or 4π. This divided by 4, or r^2, equals π. Another method to finding π is through inscribing a polygon with many sides into a circle, taking its area, and dividing it by the radius. This produces a close value to π, but not the actual number.

π cannot be found through any series of integer operations, meaning it is trancendental. The letter π is Greek and origin, and is likely derived from the Greek word for perimeter, περίμετρος. The term was popularized by William Jones and Leonhard Euler, and also has origins in Archimedes work. Pi has been said to be equal to 22/7, but this has been proven untrue. Actually, 22/7 is equal to 3.142857, not π. The attempt to discover and analyze π has lasted since very early times, in Arabic and Greek societies. Its mystery extends to present day, where computers are being used to calulate the most expansive versions of π ever seen.

Pi has a strong presence in modern culture. Pi day, a celebration of the number and its origins, falls on March 14th, or 3/14. Contests to recite the most digits of π have spawned many new records and stories, where some are rumored to know 100,000 digits.

Pi is used in most branches of math. In trigonometry, it is used on the unit circle to find sine, cosine, and other values. It is also used in the radian system, where π represents 180 degrees. In calculus, π is used to find the volume of revolution of a function. In physics, π is often utilized whenever the spherical coordinate systems are present.

Pi is extremely interesting, and is the theme of much mathematical work. For pi to 4 million digits, see http://zenwerx.com/pi.php .

Welcome to my blog. My name is Gavin, I love math, and I strive to understand every application it has. Math can be used to entertain, explain, or even teach. I hope to reflect my passion for the “universal law” through explanations, solutions, and interesting facts and stories. I plan on blending basic arithmetic with calculus, statistics, algebra, and physics, to create a cross-comparison of everything I you see.

Accordingly, I thought I would start off with infinity.

Now to business. Infinity: It is daunting, unbounded, and practically everywhere. It is used in nearly every type of mathematics, from kindergarten numbers (the unbounded number line) to calculus and beyond. However, this does not define what infinity is, or how it is actually assumed. This is a basic introduction to infinity, without complex mathematics or theories.

Infinity dictates an unbounded limit, similar to limits in basic calculus. Imagine a Cartesian plain, two number lines extending horizontally and vertically, while stretching forever. A graph on this plain seems to go on forever. Thus, the set of values “x” limit to infinity, in that |x| -> . Because the number plains stretch to never ending bounds, the set of values continue to infinity as well. This can be thought of another way. In this number line:

|-1-2-3-4-5-6-7-8-9-10-11-12-…>

the values will continue to be represented forever. Infinitely. Without a bound, or a range, set will never stop extending. The end can never be reached, and the pattern will continue until a bound is placed. This is also reflected in algebraic ranges, where a certain set of numbers is given a lower and upper bound. However, on any given linear function, where “x” can equal all real numbers, any number you can imagine will work. The bound formed is thus (- , ), where any negative or positive “x” value will create a viable solution. In these consistently present but often ignored examples, infinity clearly presents itself for understanding and use.

Now what is infinity? At face value, the symbol for infinity is . The commonly accepted view of this symbol’s origin is that the ribbon, , has no start or end point. It simply continues forever. Walking along the path it shows will leave you walking the same course infinitely! Infinity itself is defined by a number that grows beyond a set range, in a mathematical sense. Simply put, no matter how high the number, the solution still exists.

How about another solution to this question, as discovered by countless restless insomniacs? Imagine that you cannot sleep. You have tried everything. Your final solution is to count sheep. You start counting, and make it up to say, 100 sheep. At this point, you realize that no matter how many sheep you count, countless other sheep wait for their call. Infinity has been uncovered!

Infinity cannot be logically computed. It is simply greater than the human brain can muster. Numbers will grow beyond what can even be given a name. Like pi, with its constantly continuing series of decimal numbers, it has no end. Infinity can serve as a means to acknowledge all possible data, or acknowledge the lack of knowledge of that data. However, any way you look at it, it is unbounded, common, and extremely important in life. As said before, this is a basic explanation, and one should seek out more complex answers at their pleasure!