Number Theory

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 Ð'ј, Ð'Ð..., Ð'Ñ*, irrational numbers 2/3, в?Ñ™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 equation will then become the number being divided into. These new mathematics help form larger numbers and contribute highly to the number theory.

The history of the number theory goes all the way back to the beginnings of numbers. In early Babylonian era we first began seeing humans understand mathematics and begin to use math to solve problems. In these times the number theory was at its lowest level with simple mathematics. Math then was completely different than how it is looked at today considering their base value was a different number than the one we use today. With a different base value all of the answers they had researched came up to be different then how we understand it today. Math continued in other parts of the world as time went on being used in Egypt and early Greece. The Greeks were the first to study the properties of numbers. This study all started with the mathematician Euclid in his book titled “Elements.” Euclid wrote many sections about the properties and fundamental beginnings of numbers. In this book is where many number theorists got their start. They began by translating many Arabic writings to try and discover the different types of math being done in early Egypt. Through these early studies the first algebra and number theory were being formed. These works in Greece were slowly passed through Europe and started Western interest in mathematics. (Grinstein and Lipsey p. 510)

The number theory also has strong dealing with the properties of numbers. One property of a number is whether or not the number is prime. Prime numbers deal with the divisibility of a number. A number can be considered prime if it is only divisible by itself and 1. For example:

Prime Numbers: 5, because 5 is only divisible by 5 and 1. 23, because 23 is only divisible by 1 and 23. 37, because 37 is only divisible by 1 and 37.

These numbers were first proven by Carl Friedrich Gauss a mathematician who specialized in the number theory. Primes are an essential part of the number theory, but mathematicians are still very unsure about. One thing confusing about primes is that they have no direct pattern and come up randomly between numbers. (Mollin p. 42, Grinstein and Lipsey p. 510)

Gauss also had another major impact on the properties of numbers in the number theory. His other contribution came with congruencies. The statement of congruencies that he showed through his

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