# Chapter 1 – Pure Arithmetic

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

Arithmetic is the oldest, most basic branch of mathematics. The word arithmetic derived from the Greek word arithmos means “number.” It entails analysing numbers, specifically conventional operations like addition, subtraction, multiplication, and division. These are basic operations, although pure arithmetic also involves advanced operations such as percentage, logarithmic functions, exponentiality, and square roots.

#### History of Arithmetic

Carl Friedrich Gauss established the Fundamental Principle of Number Theory in 1801, which states that every integer greater than one can be represented as a product of prime numbers in only one way. The philosophy of numbers is arithmetic. Addition, subtraction, multiplication, and division are the four basic arithmetic operations.

#### Arithmetic operations

While the topic includes several other operations, the fundamental arithmetic operations are addition, subtraction, division, and multiplication.

• Addition (+): Adding is a basic arithmetic operation. It adds two or more values. Example: 4 + 8 = 12, 8 + 4 = 12.
• Subtraction (−): Subtraction is the inverse of addition. It calculates the difference between the two values – the minuend minus the subtrahend. If the minuend is greater than the subtrahend, the difference will always be positive. Example: 7-2=5, 2-7=-5.
• Multiplication (×): Multiplication combines two values, like addition, into a single value. Example: 2 X 3=6
• Division (÷): Division is the opposite of multiplication. It calculates the quotient of two numbers, the dividend divided by the divisor. Example: 20/2=10.

#### What Are Rational And Irrational Numbers?

Rational and irrational numbers are real numbers, but their properties differ. A rational number can be expressed in the form of P/Q, where P and Q are integers and Q is zero. However, an irrational number cannot be expressed in the P/Q form.

An example of a rational number is ⅚, while an irrational number is root 3.

Let us learn more about the difference between rational and irrational numbers with some examples.

Meaning of Rational number

Rational numbers can be expressed as fractions and as positive numbers, negative numbers, and zero. It may be written as p/q, where q is not equal to zero.

The word “rational” is derived from the word ratio, which means comparing two or more values or integer numbers and is known as a fraction. In simple terms, this is the ratio of two integers. However, it is important to remember that every rational number is a whole number, not the case with irrational numbers.

Example: 4/3 is a rational number. This means that integer 4 is divided by another integer 3.

Meaning of Irrational number

Numbers that cannot be classified as rational numbers are called irrational numbers. Let us elaborate. Irrational numbers could be written in decimals but not as fractions, which means that they cannot be written as the ratio of two integers.

Irrational numbers have never-ending non-repeating digits after the decimal point. Given below is an example of an irrational number.

Example: √8=2.828…

#### Rational Vs. Irrational Numbers

The key difference between rational and irrational numbers is that a rational number can be expressed in the form of p/q. In contrast, the irrational number cannot be expressed as p/q (though both are real numbers).

Definitions of rational and irrational numbers

• Rational Numbers: Real numbers that can be represented in the form of a ratio of two integers, say P/Q, where Q is not equal to zero, are called rational numbers.
• Irrational Numbers: Real numbers that cannot be expressed in the form of a ratio of two integers are called irrational numbers.
 Rational Numbers Irrational Numbers Numbers that can be expressed as a ratio of two numbers (p/q form) are called rational numbers. Numbers that cannot be expressed as a ratio of two numbers are called irrational numbers. Rational numbers include finite or recurring numbers. These numbers are non-terminating and non-repeating. Rational numbers include perfect squares, such as 4, 9, 16, 25, etc. Irrational numbers include surds, such as √2, √3, √5, √7. Both the numerator and the denominator are whole numbers, where the denominator is not equal to zero. Irrational numbers cannot be written in the form of a fraction. Example: 3/2 = 1.5 = 3.6767 Example:  √5, √11

#### Conclusion

In this chapter, we learned about arithmetic, which is the foundation of mathematics. We also studied concepts like rational and irrational numbers and their differences.

#### FAQs

1. Define rational and irrational numbers.
Rational numbers can be expressed in the form of a ratio (P/Q when Q is not equal to 0), and irrational numbers cannot be expressed as fractions. But both numbers are real numbers and can be represented on a number line.
2. 3.60551275……is it rational or irrational?
The ellipse (…) after 3.605551275 shows that the number is non-terminating and no number repeats. Hence, it is irrational.
3. What are the main branches of pure mathematics?
The main branch of pure mathematics is:
Algebra
Geometry
Trigonometry
Statistics and Probability
4. What are the fundamentals of mathematics?
The fundamentals of maths include basic arithmetic operations or calculations, such as addition, subtraction, multiplication, and division, taught in primary classes. In higher classes, you will learn concepts, such as algebra, geometry, factors, ratios, etc.
5. Name the basic operations used in arithmetic.
• Addition
• Subtraction
• Multiplication
• Division

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### High School Physics

• Alternating Current
• Atoms
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• Motion in a plane
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• Nuclear Energy
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• Our Environment
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• Physical world
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• Vectors, Scalar Quantities and Elementary Calculus
• Wave Optics
• Waves
• Work, Power and Energy

### High School Chemistry

• Acids, Bases and Salts
• Alcohols, Phenols and Ethers
• Aldehydes, Ketones and Carboxylic Acids
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• Absorption and Movement of Water in Plants
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• Photosynthesis
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• Plant Growth and Development
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• Reproduction and Development in Angiosperms
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• Respiration in Plants
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• Sexual Reproduction in Flowering Plants
• Strategies for Enhancement in Food Production
• Structural Organisation in Animals
• Structural Organisation of the Cell
• The Endocrine System
• The Fundamental Unit Of Life
• The Living World
• The Nervous System and Sense Organs
• Tissues
• Transpiration
• Transport in Plants

### High School Math

• Algebra – Arithmatic Progressions
• Algebra – Complex Numbers and Quadratic Equations
• Algebra – Linear Inequalities
• Algebra – Pair of Linear Equations in Two Variables
• Algebra – Polynomials
• Algebra – Principle of Mathematical Induction
• Algebra – Quadratic Equations
• Binomial Theorem
• Calculus – Applications of Derivatives
• Calculus – Applications of the Integrals
• Calculus – Continuity and Differentiability
• Calculus – Differential Equations
• Calculus – Integrals
• Geometry – Area
• Geometry – Circles
• Geometry – Conic Sections
• Geometry – Constructions
• Geometry – Introduction to Euclid’s Geometry
• Geometry – Three-dimensional Geometry
• Geometry – Lines and Angles
• Geometry – Quadrilaterals
• Geometry – Straight Lines
• Geometry – Triangles
• Linear Programming
• Matrices and Determinants
• Mensuration – Areas
• Mensuration – Surface Areas and Volumes
• Number Systems
• Number Systems – Real Numbers
• Permutations and Combinations
• Probability
• Sequence and Series
• Sets and Functions
• Statistics
• Trignometry – Height and Distance
• Trignometry – Identities
• Trignometry – Introduction

### Middle School Science

• Acids, Bases And Salts
• Air and Its Constituents
• Basic Biology
• Body Movements
• Carbon and Its Compounds
• Cell – Structure And Functions
• Changes Around Us
• Chemical Effects Of Electric Current
• Chemistry in Your Life
• Coal And Petroleum
• Combustion And Flame
• Components Of Food
• Conservation Of Plants And Animals
• Crop Production And Management
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• Electricity And Circuits
• Elements and Compounds
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• Food production and management
• Force And Pressure
• Forests: Our Lifeline
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• Fun With Magnets
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• Getting To Know Plants
• Health and Hygiene
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• Materials: Metals And Non-Metals
• Matter and Its States
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• Micro Organisms: Friend And Foe
• Motion And Measurement Of Distances
• Motion And Time
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• Nutrition In Plants
• Organization in Living Things
• Our Environment
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• Reaching The Age Of Adolescence
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• Reproduction In Plants
• Respiration In Organisms
• Rocks and Minerals
• Separation Of Substances
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• The Living Organisms And Their Surroundings
• Transfer of Heat
• Transformation of Substances
• Transportation In Animals And Plants
• Universe
• Waste-water Story
• Water: A Precious Resource
• Weather, Climate And Adaptations Of Animals To Climate
• Winds, Storms And Cyclones

### Middle School Math

• Addition
• Area and Its Boundary
• Boxes and Sketches
• Data Handling
• Fun With Numbers
• Heavy and Light
• How Many
• Long And Short
• Mapping
• Measurement
• Money
• Multiplication and Factors
• Multiply and Divide
• Numbers
• Parts and Wholes
• Pattern Recognition
• Patterns
• Play With Patterns
• Rupees And Paise
• Shapes And Angles
• Shapes And Designs
• Shapes and Space
• Similarity
• Smart Charts
• Squares
• Subtraction
• Tables And Shares
• Tenths and Hundredths
• Time
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