# Number Theory

Numbers have the ability to be grouped together in many different ways to form arithmetic. Arithmetic uses all types of numbers from natural numbers, integers, rational numbers, and irrational numbers to form different types of equations. These equations and the numbers being used in them make up the number theory. The number theory goes back to the first discoveries of ancient number systems, and the beginnings of early mathematics. The number theory also deals with the number’s own properties such as primes and congruencies. In today’s world the number theory has been broken down into different groups containing the elementary number theory, analytic number theory, algebraic number theory, geometric number theory, and computational number theory. All of this information gives number theorists a large quantity of information to research, and the ability to come up with new ideas.

The number theory starts off with the understanding of numbers. Numbers can have positive value, negative value and no value at all. They also can be put into fractions or decimals which vary the amount of the numbers actual worth. Some numbers will go on forever never having one exact amount of value. These different types of numbers all are included in the number theory. Numbers also are given a certain base depending on what number system is being used. The base value of a number is what gives you the ability to add, subtract, multiply, and divide to give you the correct answer. There are different answers depending on what base is being used. The number theory deals with all of these different types of calculations and number principles. A few examples of the different variations of numbers used in the number theory are: natural numbers 1, 2, 3, integers -1, -2, -3, rational numbers ¼, ½, ¾, irrational numbers 2/3, √2. These are all different forms of values used in the number theory. (Wikipedia Number Theory)

A key part of the number theory is arithmetic. In early forms of arithmetic Giuseppe Peano formed three basic concepts: zero, a number, and a successor. He used these steps to come up with the first basic rules of arithmetic, The Peano Axioms of Arithmetic. The rules are: “1. Zero is a number.

2. If n is a number, then the successor of n is a number.

3. Zero is not the successor of a number.

4. If the successors of two numbers are equal, then the numbers themselves are equal. 5. If a set S of numbers contains zero and the successor of every number in S, then every number is in S.”(Mollin p.5) Through these rules basic operations of arithmetic began to be achieved. The first two basic operations that can be achieved through this are addition and subtraction. Addition is the process of taking a number and giving more to it to make it larger, and subtraction is the ability to take a number and take a certain amount away from it to make it smaller. These basic operations gave the pathway of many new laws to be formed to add the number theory. Examples of these laws are Commutative Law, and Associative Law. These added grouping elements to simple addition and subtraction arithmetic. An example of these Laws goes as follows: Commutative: 2 + 3 = 5, which is the same as 3 + 2 = 5

Associative: (2 + 3) + 5 = 10, which is the same as (2 + 5) + 3 = 10 These simple new rules helped add to the number theory by giving new equations using the same numbers will still give the same answer. The number theory is part of all the different laws used in arithmetic to help find easier and quicker ways to solve simple mathematics. As arithmetic got more advanced mathematicians started multiplication and division. These forms of arithmetic worked off of each other to form a larger number in multiplication or a smaller number in division. This depending highly on which integers were being used, and also the way the problem was set up. For example: Multiplication: 3 * 9 = 27

Division: 27 / 9 = 3

The answer to the multiplication...

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