# Introduction of Number Theory in Mathematics

*I am a PhD student of mathematics. I have complete MS in math from the University of Pakistan and have been writing online since 2020.*

## Introduction of Number Theory in Mathematics

## Introduction of Number Theory in Mathematics

**Introduction:**

The study of integers (such as whole numbers) and related objects is known as number theory. The distribution of prime numbers among integers is a subject that number theorists' study, as is the shape and quantity of solutions in systems of polynomial equations with integer coefficients. Number Theory as a branch has its roots in the B. Cs, specifically during the time of one Euclid.

The "Father of Geometry," Euclid of Alexandria, was a brilliant mathematician who developed one of the earliest known "algorithms"(a collection of sequential operations). The main objective of number theory is to identify interesting and unexpected correlations between various types of numbers and to establish the reality of these relationships.

**What are Numbers in Mathematics?**

An arithmetic value used to indicate amount is called a number. A number is a mathematical concept that is used for counting, measuring, and labelling. Thus, mathematics is built on numbers.

**Types of Numbers:**

There are 7 types of numbers like,

- Natural Numbers.
- Whole Numbers.
- Integers.
- Rational Numbers.
- Irrational Numbers.
- Real numbers.
- Complex numbers.
**Natural Numbers (N):**

A natural number is one that naturally and frequently appears. It is a whole, non-negative number as a result:

N = {0, 1, 2, 3.}.

**Remark:**

Zero is neither positive nor negative. Zero is still regarded as a whole number even though it is not a positive number. Some mathematicians refer to zero as a natural number because it is a whole number and does not have a negative value.

**Whole Numbers (W).**

Any positive number that does not contain a fractional or decimal component is known as a whole number.

**Example**:

The numbers 0, 1, 2, 3, 4, 5, 6, and 7 are all whole numbers.

**Remark:**

0 is the smallest whole number.

**Integers (Z).**

An integer is a whole number that can be positive and negative. The positive integers are called natural numbers and negative integers are called inverse of positive natural numbers.

**Example**:

Z = {-1, 3, -9…...}

**Important Result:**

The product of three consecutive integers is divisible by 6.

**Rational Numbers (Q):**

The numbers which can be written in the form of fraction like u / k where u and k is integers and k is not equal to zero are called the rational numbers.

Example:

1 / 9, 3 /4, 7 /9…...

**Irrational Numbers (Q`)**:

The numbers which cannot be expressed in the form of fraction are called irrational numbers.

**Examples:**

√5, √7, √3 …

**Important Result:**

- There is no rational p such that P ² = 2.
**Real numbers (R):**

The numbers which can be written in the union of both rational and irrational numbers are called real numbers.

R = Rational numbers + Irrational numbers

**Examples:**

1, 7, √5, 0.9……

**Complex numbers (C):**

The numbers which can be written in the form of a+ib, were a and b are the integers and I is iota. The value of iota is √-1.

**Examples:**

3 + i5, 7 + i8,.....

**Definition of Number Theory:**

It is mathematical theory that study the property and relations of integers and their extension both algebraic and analytic.

Number theory has a few key concepts that are essential to understand, like.

**Divisibility:**

Let a, b ∈ z, we say that “a” divides “b” if there exist an integer c ∈ Z, such that, b = ac, then “a” is called divisor or factor of b and b is called multiple of a.

**Mathematical Representation:**

Symbolically it can be represented as, a / b read “a” divides “b”.

**Important Results:**

- a / 0 for all a ∈ Z
- a / a for all a ∈ Z.
- If a / b and c ∈ Z then a / bc.
- If a / b and b / a the a = b
- -1 / a and 1 / a for all a ∈ Z.
- If a / b and b / c then a / c.
- If a / b and a / c then a / bx+cy for all x and y ∈ z.
- If a / c +d then a / c and a / d.
- a / a^n – b^n for all n ≥0 and n ∈ z.
- a +b / a^n + b^n if n is odd.
- a +b / a^n – b^n if n is even.
- If n is odd then 8 / n ²-1

**Applications of Number ****Theory:**

**Applications of Number Theory in Real Life:**

Numerous divisibility tests utilize number theory to determine whether a given integer, "m," is divisible with the integer, "n." This theory is employed not just in mathematics but also in coding, security systems, e-commerce websites, device authentication, and many more fields.

Number theory is used to calculate the cost, journey duration, and distance. Understanding loans for homes, vehicles, education, and other objectives. knowledge of sports.

**Applications of Number Theory in Branch of Physics:**

Recently, it has been rather unexpected to observe how number theory is used by physicists to solve physical issues, and perhaps even more unexpectedly, how physicists' methods are applied to number theory issues.

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

**© 2022 Kinza Javaid**