Number Theory

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. Numbers are fundamental to mathematics. They are analogous to the alphabet of languages. The high importance of numbers created curiosity among Number Theorists across the globe to propose the Theory of Numbers.

The branch of Number Theory especially deals with the relationship between positive numbers and how they are related to each other via different operations. The relation between them is also studied under ‘Higher Arithmetic’. In this article on Number Theory, we will learn about what is Number Theory, the history of number theory, number system, types of numbers, operations on numbers, arithmetic operations, and many more in detail.

What is Number Theory?

Number Theory is the branch of pure mathematics which studies the set of positive whole numbers and intergers and the relationship between them. Number Theorists are interested in studying the properties of various numbers and performing elementary operations and calculations using them. These elementary operations are called Arithmetic Operations. The operations further used to solve complex problems and real life situations. The application of number theory is studied in ‘Higher Arithmetic’.

History of Number Theory

We will study the history of number theory briefly by learning its origin and then learn about the Early Modern Number Theory given by the Number Theorist of ancient civilization.

Origin of Number Theory

The earliest evidence of Number Theory was determined after the finding of fragmented tables in Mesopotamia that dates back to 1800 BC. The table obtained is called Plimpton 322. This table contains Pythagorean Triples in Babylonian Sexagesimal System. It is assumed that these numbers were used for Babylonian Astronomy.

Pythagoras used to study mathematics in Italy among his devotees. It is believed that Pythagoras learnt mathematics from Babylonians. The most important work of Pythagoras was development of Pythagorean Triplets which states that sum of square of two smaller numbers is equal to the square of the largest number.

After Pythagoras, the era of Euclid came. Euclid was a Greek Mathematician and logiacian. He gave postulates for geometry and also for arithmetic. Euclid Postulates is studied in class 9 and Euclid Division Lemma is studied in class 10. Euclid contributed to the theory of perfect numbers and discovery of prime number theory.

Indian Mathematician also contributed to the number theory. Aryabhatta proved simultaneous congruences can be solved using kuttaka or the pulveriser method. Brahamagupta contributed to the study of the indefinite quadratic equations.

Early Modern Number Theory

Early Modern Number Theory was given by Number Theorist during the period of European ressaince. The contribution of the various mathematician in the Early Modern Number Theory is discussed below:

The first contribution was Pierre De Fermat. Fermat worked for the perfect numbers and also gave theory on prime numbers. His most important contribution was Fermat Little Theorem.

Leonhard Euler was also a Number Theory enthusiast who worked on the proof of Fermat Little Theorem. He initiated the work on analytical number theory which involves the work on sum of four squares, distribution of prime numbers etc.

Several other mathematician such as Langrange who gave proof of Fermat’s and Euler’s work, Legendgre gave Prime Number Theorem and Carl Friedrich Gauss who gave the law of quadratic reciprocity and also developed a section of computational which includes primality tests.

Branches of Number Theory

With the passage of time the field of Number Theory got matured, several mathematicians started taking interest into it and gave new theories which led to the splitting of Number Theory into several branches. Let’s have a glance on the branches of number theory.

Elementary Number Theory

The branch of elementary number theory does not involve the use of complex analysis. It uses basic arithmetic, geometry and algebra to derive a solution.

Algebraic Number Theory

Algebraic Number Theory mainly deals with Algebraic Number. An algebraic number is defined as a complex number which can be a possible solution to a polynomial equation.

Analytical Number Theory

Analytical Number Theory deals with the study of integers from the perspective of real and complex analysis. It is mainly concerned with the number theory on estimates of density and sizes rather than identities.

Dipohantine Number Theory

Dipohantine Number Theory deals with the finding solution of Dipohantine Equations. We know that any curve can be mathematically represented by an equation. Dipohantine Geometry deals with finding of the rational and integral points on these curves and also tries to know how they are distributed over the curve.

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:

Decimal Number System: In this the number is represented with base 10. In this system the numbers are represented using 10 digits from 0 to 9.

Binary Number System: In Binary Number System, numbers are represented with base 2. In this system numbers are represented using only two digits 0 and 1.

Octal Number System: In Octal Number System, numbers are represented with base 2. In this system, numbers are represented using 8 digits from 0 to 7.

Hexadecimal Number System: In Hexadecimal Number System, the numbers are represented with base 16. This system uses combination of digits from 0 to 9 and alphabets from A to F to represent numbers.

We can also interconvert these numbers from one form to other form of number system.

The standard form that we use today for representing numbers were not used across the globe uniformly. Different civilizations have their own way of representing numbers. Among all those Roman Number System caught attention of everyone.

Roman Number System is the form of representing numbers using letters by people of Roman Empire. In this system I stands for 1, V means 5, X means 10, L means 50, C means 100, D means 500 and M means 1000. The other numbers are represented using these letters only by writing them before or ahead of the standard numbers depending upon how much they are larger or smaller.

Types of Numbers

Numbers are classified into various categories based on their properties. Let’s learn the various types of numbers.

Natural Numbers: Counting numbers are called Natural Numbers. These numbers starts from 1 and go till infinity.

Whole Numbers: All the natural numbers along with zero are collectively called Whole Numbers.

Integers: All the natural numbers, zero and negative of natural numbers are called Integers.

Fractions: A number represented in the form of a/b where a and b are whole numbers and b is not equal to zero is called Fraction.

Rational Numbers: A number represented in the form of p/q where p and q are integers and q is not equal to zero is called a Rational Number.

Irrational Numbers: A number which can’t represented in the form of p/q where p and q are integers and q is not equal to zero and whose decimal expansion is non terminating and non repeating is called an Irrational Number.

Real Numbers: All the natural numbers, whole numbers, integers, fractions, rational numbers and irrational numbers are collectively called Real Numbers.

Complex Numbers: A number represented in the form of a + ib where a is the real part and ib is the imaginary part is called a Complex Number.

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:

Even Numbers: All the numbers which are divisible by 2 are called Even Numbers. These numbers has 2 as common factor. Example, 2, 4, 6, 8 etc.

Odd Numbers: The numbers which are not divisible by 2 are called Odd Numbers. Example, 1, 3, 5, 7 etc.

Prime Numbers: The numbers which are divisible by 1 and itself are called Prime Numbers. Prime Numbers has only two factors, 1 and the number itself. Example, 2, 3, 5, 7, etc.

Composite Numbers: The numbers which has more than two factors are called Composite Numbers. Example, 4, 6, 8, 9 etc.

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.

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:

Addition: Addition basically means totalling up. It is represented by plus(+) symbol. Addition helps in counting up total number of articles or total bill. Suppose you have 10 chocolates and your friend gave you 5 more chocolates. Hence you have 10 + 5 = 15 chocolates.

Subtraction: Subtraction basically means finding the difference between two numbers. It is represented by minus(-) symbol. It is helpful in comparing quantities like which which one is how much greater or lesser than other or in finding the remaining balance.

Let’s say John has 10 apples and Adam has 6 apples. Then we can say that John has more apples. But the question is how much more then we need to find difference. For this we subtract 10 and 6 i.e. 10 – 6 = 4. Hence, John has four more apples than Adam.

Multilplication: Multiplication is another way of addition. Multiplication is represented by cross(⨯) symbol. It is helpful when you have to add same numbers for larger number of times.

For Example, if we have to find how much chocolate to be purchased to distribute 2 chocolates to each student in a class of 20 students. For this we have to add 2 + 2 +….. 20 times. This can be hectic and complex for larger numbers. To simplify this we perform multiplication, hence we will multiply 2 and 20 which is equal to 2 ⨯ 20 = 40. Hence 40 chocolates to be purchased in total. One must learn tables to perform multiplications easily.

Division: Division is basically used to find the share or to distribute equally. It is reverse process of multiplication. It is represented by (÷) symbol.

Suppose we have 30 sweets and we have to find how much sweets we should distribute to each student among 15 students so that each get equal number of sweets. Hence we need to the share of each student for this we will divide 30 by 15. Hence, 30 ÷ 15 = 2. Hence we will give 2 sweets to each student.

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.

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.

Closure Property

Closure Property states that arithmetic operation between two same types of numbers result in the same number. The validity of closure property for different numbers over different operations are different. In case of Integers and Rational Numbers closure property is valid over addition, subtraction and multiplication but not for division. It can be represented as (a * b) = c where a, b, c all are integers or all are rational numbers and * is addition, subtraction or multiplication.

Commutative Property

Commutative Property states that the operation between two numbers yields the same result even if their positions are interchanged. It is only valid for addition and multiplication. For Example, a + b = b + a is Commutative Law of Addition and a ⨯ b = b ⨯ a is Commutative Law of Multiplication. Commutative Law is not valid for subtraction and division.

Associative Property

Associative Property states that the operation between three numbers always gives the same result irrespective of the order of operation between the numbers. Associative Law is only valid for Addition and Multiplication and not for Subtraction and Division. Associative law for addition is given as (a + b) + c = a + (b + c) and Associative Law of Multiplication is given as (a ⨯ b) ⨯ c = a ⨯ (b ⨯ c).

Distributive Property

Distributive Property as the name suggests that if a number is commonly multiplied to the sum or difference of two numbers then the number commonly multiplied gets distributed to both the numbers and then sum or difference is calculated between the product of the numbers. Distributive Law of multiplication is defined over addition and subtraction. Distributive Law over addition is given as a(b + c) = ab + ac and Distributive Law of multiplication over subtraction is given as a(b – c) = ab – ac

Identity Property

Identity Property states that if operation is performed on a number then the result is the number itself. Identity Property is given generally for addition and multiplication. In case of addition, it is called Additive Identity and in case of multiplication it is called Multiplicative Identity. Additive Identity states that if a number is added to zero then the result is the number itself. Additive Identity is represented as (a + 0 = a). Multiplicative Identity states that if a number is multiplied with 1 then the product is the number itself. Multiplicative Identity is represented as (a ⨯ 1 = a).

Inverse Property

Inverse Property is also defined for addition and subtraction. In case of addition it is called Additive Inverse and in case of multiplication it is called Multiplicative Inverse. Additive Inverse states that if a number is added to the negative of itself then the result is zero. Additive Inverse is represented as a + (-a) = 0. Multiplicative Inverse states that if a number is multiplied with its inverse or reciprocal then the product is 1. Multiplicative Inverse is represented as a ⨯ 1/a = 1.

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.

• HCF and LCM
• Fundamental Theorem of Arithmetic

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.

• Laws of Exponents
• Negative 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.

• nth roots of a number
• Squares and Square Roots
• Cubes and Cube 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.

• Log Formulas
• Natural Log Formula
• Laws of Logarithm

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:

• Playing with Numbers
• Game of Numbers
• Letters for Digits

Solved Examples on Number Theory

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 log_b5 + log_b6

Solution:

From Log Rules we know that log_px + log_py = log_p(xy)

Hence, log_b5 + log_b6 = log_b(5 × 6) = log_b30

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%

Practice Questions on Number Theory

Q1. Solve the following

i. 3⁴ × 3⁵ × 3⁶

ii. 4⁶/4⁵

iii. 2³ × 3³

Q2. Find the Prime Factors of Following Numbers

i. 236

ii. 729

iii. 150

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

i. 112

ii. 56

iii. 65

Q6. Solve the following

i. log_c24 – log_c12

ii. log₁₀23

Q7. Find the HCF and LCM of following Numbers

i. 16 and 40

ii. 25 and 300

iii. 15 and 180

Number Theory – FAQs

1. What is Number Theory?

Number Theory or Theory of Numbers is a branch of mathematics that deals with positive numbers and its applications.

2. 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.

3. Who are some popular Number Theorists?

Some of the popular Number Theorists include Pythagoras, Euclid, Fermat, Gauss, Aryanhatta, Brahmagupt and Ramanujan.

4. 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.

5. 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.

6. 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.

7. 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

8. What is Co-Prime Number?

Co-Prime Numbers are those numbers whose HCF is 1.

9. Which is the only Even Prime Number?

2 is the only even prime number.

10. 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.