Number Theory is the branch of mathematics that deals with the study of positive numbers and arithmetic operations based on them. Numbers are the mathematical entities that are used for counting. Since the development of human civilization, numbers have always been a source of fascination for various mathematicians across the globe.
In this article on Number Theory, we will learn about what is number theory, the history of number theory, the number system, types of numbers, operations on numbers, arithmetic operations, and many more in detail.
What is Number Theory?
Number theory is a branch of mathematics that deals with the properties and relationships of numbers, particularly integers. It explores patterns, structures, and properties of numbers and seeks to understand their fundamental properties and behaviors.
Key topics in number theory include prime numbers, divisibility, modular arithmetic, Diophantine equations, and the distribution of prime numbers. Number theory has applications in various fields such as cryptography, computer science, and physics, and it plays a foundational role in many areas of mathematics.
History of Number Theory
Number theory, originating in ancient Mesopotamia circa 1800 BC with Plimpton 322’s discovery, encompasses Pythagorean triplets and Babylonian astronomy.
Pythagoras furthered this with his triplets, influencing Euclid’s postulates in geometry and arithmetic. Indian mathematicians like Aryabhatta and Brahmagupta made key contributions, while the European Renaissance saw Pierre De Fermat’s perfect numbers and prime theory, and Leonhard Euler’s analytical number theory.
Lagrange and Legendre refined prime number theory, and Carl Friedrich Gauss advanced quadratic reciprocity and computational methods. These efforts shaped modern number theory, impacting diverse mathematical fields and practical applications.
Branches of Number Theory
Number theory is vast and diverse, thus to understand it better it is classified into various sub branches given in the following table:
| Branch |
Description |
| Elementary Number Theory |
Studies basic properties of integers like divisibility, primes, factorization, and congruences. |
| Analytic Number Theory |
Applies tools from analysis to study the distribution of prime numbers and other number-theoretic functions. |
| Algebraic Number Theory |
Investigates algebraic structures involving number fields, rings of integers, and algebraic integers. |
| Diophantine Equations |
Focuses on finding integer solutions to polynomial equations, such as Fermat’s Last Theorem. |
| Additive Number Theory |
Studies properties of additive structures within integers, including partitions and arithmetic progressions. |
| Multiplicative Number Theory |
Focuses on multiplicative properties of integers, such as multiplicative functions and the distribution of primes. |
Number System
The number system is a system for representing numbers on the Number Line in Number Theory using a collection of symbols and rules. These symbols, which run from 0 to 9, are referred to as digits. The Number System is used to conduct mathematical computations ranging from complex scientific calculations to simple counting of Toys for a Kid or the number of chocolates left in the box.
Types of Number System
Numbers can be written in different bases and various forms. The different types of number systems are mentioned below:
| Number System |
Description |
| Decimal Number System |
Base 10 representation using digits 0 to 9. |
| Binary Number System |
Base 2 representation using digits 0 and 1. |
| Octal Number System |
Base 8 representation using digits 0 to 7. |
| Hexadecimal Number System |
Base 16 representation using digits 0 to 9 and letters A to F. |
| Roman Number System |
Ancient system using letters: I (1), V (5), X (10), L (50), C (100), D (500), M (1000). Other numbers are represented using combinations of these letters. |
Note: We can also interconvert these numbers from one form to other form of number system.
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Types of Numbers
Numbers are classified into various categories based on their properties. Let’s learn the various types of numbers.
| Type |
Description |
Examples |
| Natural Numbers |
Counting numbers starting from 1, denoted by N, including positive integers. |
1, 2, 3, 4, 5, . . . |
| Whole Numbers |
Natural numbers including zero, denoted by W. |
0, 1, 2, 3, 4, . . . |
| Integers |
Whole numbers along with their negatives and zero, denoted by Z. |
. . ., -3, -2, -1, 0, 1, 2, 3, . . . |
| Rational Numbers |
Numbers expressible as a ratio of two integers, denoted by Q. |
1/2​, 3/4. 7/3 . . . |
| Irrational Numbers |
Numbers that cannot be expressed as a ratio of two integers, often with non-repeating, non-terminating decimals. |
√2​, π, e, . . . |
| Real Numbers |
Union of rational and irrational numbers, representing all points on the number line, denoted by R. |
-3.14, 0, 2.718, √3​, . . . |
| Imaginary Numbers |
Numbers of the form bi, where b is a non-zero real number and i is the imaginary unit. |
2i, −3i, 1/2​i, . . . |
| Complex Numbers |
Numbers of the form a+bi, where a and b are real numbers, and i is the imaginary unit, denoted by C. |
3+4i, 2−i, −1+2i, . . . |
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Numbers based on Divisibility
A number may be divisible by a certain number or may not be divisible a certain number. Hence based on divisibility they can be classified as follows:
| Type |
Description |
Examples |
| Prime Numbers |
Natural numbers greater than 1 with no positive divisors other than 1 and itself. |
2, 3, 5, 7, 11, 13, 17, … |
| Composite Numbers |
Natural numbers greater than 1 that are not prime, having divisors other than 1 and itself. |
4, 6, 8, 9, 10, 12, 14, … |
| Odd Numbers |
Integers not divisible evenly by 2. |
-3, -1, 1, 3, 5, 7, … |
| Even Numbers |
Integers divisible evenly by 2. |
-4, -2, 0, 2, 4, 6, … |
What is Arithmetic?
Arithmetic is a field of mathematics that studies the characteristics of classical operations on numbers, such as addition, subtraction, multiplication, division, exponentiation, and root extraction.
Arithmetic is a fundamental aspect of number theory, which, along with algebra, geometry, and analysis, is regarded as one of the premieres of modern mathematics. The phrases arithmetic and higher arithmetic were used as synonyms for number theory until the early twentieth century, and are now occasionally used to refer to a wider section of number theory.
Read More: Arithmetic
Arithmetic Operations
There are four basic arithmetic operations addition, subtraction, multiplication and division. These operations are helpful in daily life calculations as well as complex calculations. The four basic operations are discussed below:
| Operation |
Description |
Symbol |
Example |
| Addition |
Combining two or more numbers to find their sum. |
+ |
5 + 3=8 |
| Subtraction |
Finding the difference between two numbers. |
– |
9 − 4 = 5 |
| Multiplication |
Repeated addition; finding the product of two numbers. |
×, * |
2 × 6 = 12 |
| Division |
Sharing or partitioning a quantity into equal parts. |
÷, / |
12 ÷ 3 = 4 |
| Exponentiation |
Repeated multiplication; raising a number to a power. |
ab |
23 = 8 |
| Square Root |
Finding the number which, when multiplied by itself, gives the original number. |
√ |
√(16) =4 |
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Applications of Arithmetic
Arithmetic has got numerous applications in our daily lives. The above-discussed arithmetic operations are used to solve various types of arithmetic operations. Following are the related articles that define the applications of arithmetic.
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Properties of Numbers
The numbers follow various properties. These properties are closure property, commutative property, associative property, distributive property, identity property, and inverse property. These properties are defined for different numbers and different operations.
| Property |
Description |
Example |
| Closure Property |
The result of an operation on two numbers from a set is also a member of that set. |
Addition: 2+3=5 is a member of the set of integers. |
| Commutative Property |
Changing the order of operands in an operation does not change the result. |
Addition: 2+3=3+2 |
| Associative Property |
Changing the grouping of operands in an operation does not change the result. |
Addition: (2+3)+4=2+(3+4)
Multiplication: (2×3)×4=2×(3×4) |
| Identity Property |
The result of an operation with a specific identity element and any number remains the same. |
Addition: 5 + 0 = 0 + 5=5 |
| Inverse Property |
For every element, there exists an inverse element such that their sum is the identity element. |
Addition: 5+(−5)=0 |
| Distributive Property |
The operation distributes over addition or subtraction. |
Multiplication over addition: 2×(3+4)=2×3+2×4 |
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Applications of Number Theory
Arithemtic has got wide range of applications in mathematics from finding the factors and multiples of a number to calculate the compounding value of money. Let’s learn the different applications of Arithmetic in brief.
Place Value and Face Value
Place Value and Face Value are important component a digit in a Number. Face Value is the the value by which a digit is recognised in mathematics irrespective of its position. For Example, in 5, 58, 568, the Face value of 5 is always five. However, the place value is the value of a digits due to its position. Place value is obtained by multiplying face value of number with its position. For example, in 5 the place value of 5 is 5 × 1 = 5 as 5 is in ones place. In 58, the place value of 5 is 5 × 10 = 50 as 5 is in tens place.
The place value also varies from indian system to international system. In Indian system the place value in increasing order is given as Ones, Tens, Hundred, Thousand, Ten Thousand, Lakh, Ten Lakh, Crore, Ten Crore etc.
Factors and Multiples
A number is said to be a factor of other number if it divides other number exactly. The other number which gets divided is called Multiple. If the all the factors are prime, then the method is called Prime Factorization. Factors and Multiples has got application in Number Theory to find HCF and LCM of two or more numbers which are used in arithmetic Problems.
Divisibility Rules
Divisibility Rules are important to find with what numbers a number is divisible. Since, it is very difficult to divide and check if a given number is divisible by particular number or not, here divisibility rules comes as a saviour to make the calculation easy.
Exponents
In Theory of Numbers we come across numbers which are raised to some powers, for example, 23 where 2 is called the base and 3 is called the exponent. Exponents are used in Number Theory to represent very large numbers such as distance between two celestial bodies and also very small numbers such as mass of an atom. The problems involving exponents are solved using laws of exponents.
Roots
We know that exponents are the used to represent large numbers. But in case we have a large number which is a result of exponent raised to some base. This base is called Root of the Number. We often come across problems involving square root and cube root. To solve these questions we need to learn the concept of roots.
Logarithms
In Number Theory, Logarithm is used to find the exponent for which a number is raised to give a certain certain result. For Example, 23 = 8, in this case, the base 2 can be find out using the concept of roots but the power to which 2 is raised to give 8 as result can be find out using the concept of logarithms. The reverse process of logarithm is called Antilog. Logarithm is valid only for positive numbers.
Number Game
A number game is any of a variety of puzzles and games that include mathematical concepts. Mathematical games and puzzles range from simple amusements to complex problems, some of which have never been solved. Arithmetic, algebra, number theory, group theory, combinatory (problems of configurations or designs), symbolic logic, can all be included. The origins of number games and mathematical recreations, as well as the values on which they are based, are covered in details, such as explanations of the puzzles, games, and recreations are discussed in the articles below:
Number Theory Examples
Example 1: Find the value of
i. 23 × 25
ii. (3/4)-2
iii. (43)2
Solution:
i. 23 × 25 = 2(3 + 5) = 28
ii. (3/4)-2 = (4/3)2 = 42/32 = 16/9
iii. (43)2 = 4(3 × 2) = 46
Example 2: Convert 25 into Binary
Solution:
To convert 25 into binary we divide 25 successively by 2
25/2 gives Q = 12 and R = 1
12/2 Gives Q = 6 and R = 0
6/2 gives Q = 3 and R = 1
3/2 gives Q = 1 and R = 1
Now Q = 1 is not divisible by 2
Hence, the binary form of 25 is 11101
Example 3: Find the square root of 576
Solution:
We will find the prime factors of 576
576 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3
Therefore, square root of 576 = √576 = √(2 × 2 × 2 × 2 × 2 × 2 × 3 × 3)
Hence, √576 = √(2 × 2 × 2 × 2 × 2 × 2 × 3 × 3) = 2 × 2 × 2 × 3 = 24
Example 4: If the HCF of two numbers is 4 and their LCM is 120 and one of the numbers is 8, find the other number
Solution:
We know that Product of Two Numbers = Product of HCF and LCM
⇒ First Number × Second Number = HCF × LCM
⇒ 8 × Second Number = 4 × 120
Hence, Second Number = (4 × 120)/8 = 60
Example 5: Solve logb5 + logb6
Solution:
From Log Rules we know that logpx + logpy = logp(xy)
Hence, logb5 + logb6 = logb(5 × 6) = logb30
Example 6: A boy scored 480 out of 500 marks in his board exams. Find the percentage of marks obtained by him.
Solution:
Obtained Marks = 480
Full Marks = 500
Percentage of Marks = (480/500)×100 = 96%
Number Thoery Books
- Elementary Number Theory by David M. Burton
- An Introduction to the Theory of Numbers by G.H. Hardy and E.M. Wright
- A Classical Introduction to Modern Number Theory by Kenneth Ireland and Michael Rosen
- Algebraic Number Theory by Jürgen Neukirch
Practice Questions on Number Theory
Q1. Solve the following
- 34 × 35 × 36
- 46/45
- 23 × 33
Q2. Find the Prime Factors of Following Numbers
Q3. A Man Purchased a Scooter for 45000 and then Sold it at a profit of 10%. Find the selling price of the scooter.
Q4. A man borrowed 2000 rupees for 3 years compound manually at the rate of 5% annually. Find the amount paid by him after 3 years.
Q5. Convert the following decimal numbers into binary equivalent
Q6. Solve the following
Q7. Find the HCF and LCM of following Numbers
- 16 and 40
- 25 and 300
- 15 and 180
FAQs on Number Theory
What is Number Theory?
Number Theory or Theory of Numbers is a branch of mathematics that deals with positive numbers and its applications.
Who are Credited to have First Developed the Number Theory?
People of Babyloian Civilization are credited to have first developed the concept of Number Theory.
Who are some popular Number Theorists?
Some of the popular Number Theorists include Pythagoras, Euclid, Fermat, Gauss, Aryanhatta, Brahmagupt and Ramanujan.
What are the Branches of Number Theory?
The different Branches of Number Theory are Elementary Number Theory, Algebraic Number Theory, AnalyticaL Number Theory and Dipohantine Number Theory.
What is Pythagorean Triplet?
Pythagorean Triplet is a collection of three numbers such that the square of the largest number is equal to the sum of the squares of the rest two numbers.
What are the different Numbers System in which Numbers can be written?
The different forms in which numbers can be written are Binary System, Decimal System, Octal System and Hexadecimal System.
What is Hardy Ramanujan Number?
1729 is known as Hardy Ramanujan Number as it can be represented as sum of cubes of two different pairs of number. 1729 = 123 + 13 and also 1729 = 103 + 93
Which is the only Even Prime Number?
2 is the only even prime number.
What is BODMAS?
BODMAS stands for Bracket, Of, Division, Multiplication, Addition and Subtraction. It is the order of priority to solving the operations where Bracket operations to be done first and subtraction to be done at last.
Who is the Father of Number Theory?
Pierre de Fermat is regarded as the father of Number Theory.
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