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.

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.

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…

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.

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.

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