Real numbers include all rational and irrational numbers. There are several key properties and theorems regarding real numbers: 1) Euclid's division lemma states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. This lemma is used to calculate the highest common factor of two integers. 2) The fundamental theorem of arithmetic states that every integer greater than 1 can be written as a unique product of prime factors, regardless of their order. 3) Rational numbers are numbers that can be written as a ratio of two integers. Irrational numbers are all other real numbers that are not rational and have non-repe

1. Real Numbers
By Atharav Porwal
X-A

2. Index
• Introduction
• Euclid’s Division Lemma
• The Fundamental Theorem of Arithmetic
• Rational and Irrational numbers
• Their Decimal expansions

3. Introduction
• Real number is a value that represents a quantity along
a line.
• It consist of all rational and irrational numbers
• It has further subsets:-
1)Natural
2)Whole
3)Integers

4. • Natural Numbers:- The positive integers
(whole numbers) 1, 2, 3, etc., and
sometimes zero as well.
• Whole numbers:- A number without
fractions; an integer.

5. Euclid’s Division Lemma
• If we have two positive integers a and b,
then there exists unique
integers q and r which satisfies the
condition a = bq + r where 0 ≤ r ≤ b.
• To calculate the Highest Common Factor
(HCF) of two positive integers a and b we
use Euclid’s division algorithm.

6. Example of Euclid’s Division
Lemma

7. The Fundamental Theorem Of
Arithmetic
• Also called as Unique Factorization
Theorem, sates that every integer greater
than 1, then that product is unique, and
that the order of the factors does not
matter.
• For example:- 1200 = 24
× 31
× 52
= 3 × 2 ×
2 × 2 × 2 × 5 × 5 = 5 × 2 × 3 × 2 × 5 × 2 ×
2 = etc.

8. Rational Numbers
• Are the numbers that can be written as a
ratio. That means it can be written as a
fraction.

9. Irrational Numbers
• All numbers that are not rational are
considered irrational. An irrational no. is
endless non-repeating digits to the right of
the decimal point.