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# Revision Notes on Real Numbers

## Euclid’s Division Lemma It is basically the restatement of the usual division system. The formal statement for this is-

For each pair of given positive integers a and b, there exist unique whole numbers q and r which satisfies the relation

a = bq + r, 0 ≤ r < b, where q and r can also be Zero.

where ‘a’ is a dividend, ‘b' is divisor, ‘q’ is quotient and ‘r’ is remainder.

∴ Dividend = (Divisor x Quotient) + Remainder

### Natural Numbers

Non-negative counting numbers excluding zero are known as natural numbers.

i.e. 5, 6, 7, 8,  ……….

### Whole numbers

All non-negative counting numbers including zero are known as whole numbers.

i.e. 0, 1, 2, 3, 4, 5, …………….

### Integers

All negative and non-negative numbers including zero altogether known as integers.

i.e. ………. – 3, – 2, – 1, 0, 1, 2, 3, 4, ………….. ### Algorithm

An algorithm gives us some definite steps to solve a particular type of problem in a well-defined manner.

### Lemma

A lemma is a statement which is already proved and is used for proving other statements.

## Euclid’s Division Algorithm

This concept is based on Euclid’s division lemma. This is the technique to calculate the HCF (Highest common factor) of given two positive integers m and n,

To calculate the HCF of two positive integers’ m and n with m > n, the following steps are followed:

Step 1: Apply Euclid’s division lemma to find q and r where m = nq + r, 0 ≤ r < n.

Step 2: If the remainder i.e. r = 0, then the HCF will be ‘n’ but if r ≠ 0 then we have to apply Euclid’s division lemma to n and r.

Step 3: Continue with this process until we get the remainder as zero. Now the divisor at this stage will be HCF(m, n). Also, HCF (m, n) = HCF (n, r), where HCF (m, n) means HCF of m and n.

## The Fundamental Theorem of Arithmetic

We can factorize each composite number as a product of some prime numbers and of course, this prime factorization of a natural number is unique as the order of the prime factors doesn’t matter.

• HCF of given numbers is the highest common factor among all which is also known as GCD i.e. greatest common divisor.

• LCM of given numbers is their least common multiple.

• If we have two positive integers  ‘m’ and ‘n’ then the property of their  HCF and LCM will be:

HCF (m, n) × LCM (m, n) = m × n. ### Rational Numbers

The number ‘s’  is known as a rational number if we can write it in the form of m/n where  ‘m' and ‘n’ are integers and n ≠ 0, 2/3, 3/5 etc.

Rational numbers can be written in decimal form also which could be either terminating or non-terminating. E.g. 5/2 = 2.5 (terminating) and (non-terminating).

### Irrational Numbers

The number ‘s’ is called irrational if it cannot be written in the form of m/n, where m and n are integers and n≠0 or in the simplest form, the numbers which are not rational are called irrational numbers. Example - √2, √3 etc.

• If p is a prime number and p divides a2 , then p is one of the prime factors of a2 which divides a, where a is a positive integer.

• If p is a positive number and not a perfect square, then √n is definitely an irrational number.

• If p is a prime number, then √p is also an irrational number.

### Rational Number and their Decimal Expansions

• Let y be a real number whose decimal expansion terminates into a rational number which we can express in the form of a/b, where a and b are coprime, and the prime factorization of the denominator b has the powers of 2 or 5 or both like 2n5m, where n, m are non-negative integers.

• Let y be a rational number in the form of y = a/b, so that the prime factorization of the denominator b is of the form 2n5m, where n, m are non-negative integers then y has a terminating decimal expansion.

• Let y = a/b be a rational number, if the prime factorization of the denominator b is not in the form of 2n2m, where n, m are non-negative integers then y has a non-terminating repeating decimal expansion.

• The decimal expansion of every rational number is either terminating or a non-terminating repeating.

• The decimal form of irrational numbers is non-terminating and non-repeating.