#### Welcome to the world of mathematics. The beauty of math is incredible. It demonstrates the power and creativity of the Lord.

Mathematics has many branches. All of these branches are interconnected in various interesting ways. We find that Trigonometry and Calculus are closely related to Geometry. As we study Diophantine equations we might suddenly find ourselves deep in number theory. As we are working with complex numbers we might suddenly find that Quaternions or Octonions provide a more effective answer to our problem. Some times we just have to stop and admire the designs and intriguing patterns that appear. When we finally find the answer to our mathematics problem we usually realize that not only does it all work together in an incredible way, but it also illustrates the power and the glory of God. Here are a few of the many interrelated branches of mathematics. Each could be subdivided many times.
• ## Number Theory:

Mathematics starts with numbers. The most basic of which are the counting numbers. 1,2,3,4,5,6,7,8,9,10... Number theory is the science which was created to study this very basic building block of math. Number theory is all about the positive real whole numbers - the integers (The words positive real and whole have special meanings in mathematics) and their relationships with each other and what laws can be applied to certain classes of integers. One of the most thoroughly investigated areas of Number Theory is Prime Numbers. The study of numbers soon branched to negative numbers, fractional numbers, irrational numbers, intransitive numbers, imaginary numbers, complex numbers, quaternions, octonions, and many more....
• ## Arithmetic

Arithmetic is used to add, subtract, multiply and divide most of these numbers. Arithmetic starts with addition and subtraction. Subtraction can be thought of as undoing addition. When repeated additions of the same number are required, multiplication is used, with division to undo it. Division is where the fractional numbers come in. Some whole numbers when divided do not produce a whole number so those problems are often left incomplete as fractions. If we decide to complete them we create decimals. Either way we can not produce a whole number. When repeated multiplications of the same number are required we use exponents or powers. Because the power and the base are not interchangeable, the operation of exponents can be undone in one of two ways: roots or logarithms. If we want the base we find the root and if we want the exponent we take the logarithm.
• ## Algebra

Sometimes not all the numbers in an equation are known so we use letters or variables to stand for the unknown numbers. For example: x+y=z instead of 1+2=3; x stands for 1, y stands for 2, z stands for 3. The power of algebra, however, is that this allows us to generalize this equation for any number. Algebra gives us the tools to find these unknown numbers.
• ## Diophantine equations:

This class of equations is very closely related to Number Theory in that it is concerned with only integer values for the given variables. One of the most famous Diophantine equations: Fermat's last theorem is xn + yn = zn has no integer solutions for n>2. According to my personal research it may be possible to generalize this equation: xn + yn = zn has no integer solutions for |n|>2 permitting negative values for n. Although I have not found a proof of this, a precursory search of the first 10,000 integers given for x and y yields no results.
• ## Set Theory:

This is the study of groups of numbers and other objects and the relationships between them. This field of study was used in an unsuccessful attempt to try to reduce all of mathematics to a set of axioms. Strange infinite sets were discovered such as the set of all sets, or the set of all sets except itself. These led to interesting logic problems such as "This statement is false" or "This statement can not be proven" and to paradoxical story problems such as the barber paradox proposed by Bertrand Russel. The counterintuitive result of these forays was Kurt Godel's First and Second Incompleteness Theorems.
• ## Geometry:

Geometry is the study of shapes and their mathematical properties. Although there are many types of geometry the most basic one and the one typically referred to is classical or Euclidean geometry. First given as axioms by Euclid, it generally describes our perception of reality most accurately. Other branches of geometry have appeared, however, as soon as some of the basic assumptions were challenged. These branches include spherical geometry and hyperbolic geometry. In some instances these provide us with valuable additional tools to understand or describe our world.
• ## Topology:

This extension of Geometry is the study of shapes in space. More specifically, it is the study of how shapes can be transformed through stretching and squeezing to reach another shape. Size and angles become irrelevant. All that is important is whether a face must be broken or joined to reach another transformation. Consequently Topologists are often accused of being unable to tell the difference between a doughnut and a coffee cup. At first glance, this area of study does not even seem to be related to mathematics at all. The underlying concepts are quite mathematical, however.
• ## Chaos Theory:

This is another extension of Geometry with an emphasis on the results of non-linear non-periodic equations. It's name - Chaos Theory stems from the fact that it's primary use is to describe and understand things that appear totally chaotic and random at first glance. Of all the mathematical fields this one probably has the closest connection to art through the area of fractals. It also does the best job of describing natural phenomena that Euclidean geometry is unable to approach.
• ## Calculus:

This field of mathematics wraps up and builds on the basic concepts of mathematics. It explores and expands upon the idea of a function, and introduces limits, differentiation, integration, transcendental functions, sums, series, sequences, vectors, and multivariate functions. The three main branches of Calculus are Differential Calculus, Integral Calculus, and Multivariate Calculus.
• ## Programming:

This practical application of mathematics appeared with the advent of microchips and electronics. It uses many ideas from other areas of mathematics, particularly binary arithmetic. Logic and boolean comparisons also dominate this field. It originated with linear programming: doing one command after another until the end of the program or a terminating statement was reached. As computers became faster a new type of programming appeared - object oriented programming. This allowed programs to be much more flexible and reusable as well as supporting parallel processing. As programming languages have also become more user oriented and abstracted from the original machine code, higher level languages have become possible.
• ## Statistics:

This field of mathematics is concerned with probability and percentages, and the likelihood that certain events will occur. One of the main areas concerns probability distributions or the probability that a given event will occur. Although statistics can be very powerful, it has often been used to distort and skew the facts, and has consequently acquired a bad name. The saying Figures never lie, but liars figure. is frequently used in reference to Statisticians.
• ## Catastrophe Theory:

This branch of mathematics is closely related to Chaos Theory. It concentrates on finding the point where a dynamical system suddenly changes course - the catastrophe. Catastrophes can occur in many ways interrupting the normal flow of actions. For example: a dam breaking, a pot starting to boil, a nuclear reactor core reaching a critical mass and initiating a meltdown, or even a more mundane occurrence such as a writer getting interrupted and not finishing his
• ## Game Theory:

This theory studies the interrelationships of multiple players each attempting to get the best result for themselves and observes how the rules will affect the results and how to figure the best moves. When each player has decided on the best course of action the game has reached a Nash Equilibrium. Game theory even studies what happens when someone decides not to follow the rules. It is not only used in games, however, it is also used in business, politics, and war.