Notes

## Real Numbers

- Let's recap the key topics of the lesson briefly
1
Revisiting Irrationals, Rational And Their Decimal Expansions

Irrational numbers:
All those numbers which can not be expressed as fractions i.e a number is an irrational number if it can not be written as .

In the view of decimal expansion:
All those numbers whose decimal expansion is neither terminating nor repeating.

Eg:etc.

Proving irrationality of numbers:
In order to prove the irrationality of numbers, we will use the method of contradiction and following results:
• The sum or difference of a rational and irrational number is an irrational number.
• The product and quotient of a non-zero rational number and an irrational number is an irrational number.
• If be a prime number and be a positive integer. If divides , then divides .
For example: If divides i.e. divides , then also divides .

Remarks:
• For any prime number , is an irrational number.

Rational numbers:
All those numbers which can be expressed as fractions i.e. a number is a rational number if it can be written as .

In the view of decimal expansion:
All those numbers whose decimal expansion is either terminating or non-terminating repeating.
Eg:,0.02,0.3412,0.121212, . . ., 0.010203, etc.

Theorems to determine the nature of decimal expansions of rational numbers

Theorem 1.
Let be a rational number whose decimal expansion terminates. Then, can be expressed in the form ,where and are co-primes,and the prime factorization of is of the form , where are non-negative integers.
Eg:Let

Thus,, which is in the form of, where.

Theorem 2.
Let be a rational number such that the prime factorization of is of the form , where are non-negative integers. Then has a decimal expansion which terminates.
Eg:Let

.
Thus, has a terminating decimal expansion.

Theorem 3.
Let ,where and are co-primes, be a rational number such that is not of the form , where and are non-negative integers. Then the decimal expansion of is non-terminating and repeating.
Eg: Let

Thus, has decimal expansion which is non-terminating and repeating.
2
Euclid's Division Lemma and Fundamental Theorem of Arithmetic

Real numbers
Collection of all those numbers which are either rational or irrational.

Euclid's Division Lemma:
Let and be any two positive integers. Then there exist unique integers and such that
, where

Application of Euclid's Division Lemma to find HCF
In order calculate HCF of any two positive integers, say and with ,we may follow the following steps:
Step 1.Apply Euclid's division lemma to and to obtain and such that .
Step 2.If ,then is the HCF of and .
Step 3.If apply Euclid's division lemma to and and obtain two whole numbers and such that .
Step 4.If ,then is the HCF of and .
Step 5.If ,then applied Euclid's division lemma to and and continue the above process till the remainderis zero.The divisor at this stage i.e. , or the non-zero remainder at the previous stage, is the HCF of and .

Eg:Use Euclid's division lemma to find the HCF of and .
Solution:Given integers are and . Clearly,. Applying Euclid's division lemma to and ,we get
Since, the remainder . So, we apply the division to the divisor and remainder to get

Now, we apply division lemma to the new divisor and new remainder to get

The remainder at this stage is zero. So the divisor at this stage or the remainder at the previous stage. So is the HCF of and .

Fundamental theorem of arithmetic:
Every composite number can be factorized as a product of primes and this factorization is unique expect for the order in which the prime factors occur.
In general, a composite number, say, can be factorized as where are prime numbers and written in ascending order i.e..
Also, if any prime number occurs more than one time in the factorization, then we will combine the same primes and get powers of primes.
Eg:The number can be expressed as .
Also, by fundamental theorem of arithmetic, we can say if there is any other prime factorization of other than mentioned above such as or ,etc. all are equal.
Application of Fundamental Theorem of Arithmetic to obtain HCF and LCM

In order to find HCF and LCM of two numbers, we may follow the following criteria:
Step 1.Factorize each of the given positive integers and express them as a product of powers of primes in ascending order of magnitudes of primes.
Step 2.To calculate the HCF, identify the common prime factors and find the smallest exponent of these common factors. Now raise these common prime factors to their smallest exponents and multiply them to get HCF.
To find LCM, list all the prime factors occurring in the prime factorization of given positive integers but take all the primes only once. For each of these factors, find the greatest exponent and raise each prime to the greatest exponent and multiply them to get LCM.

Eg:Find HCF and LCM of and by prime factorization method.
Sol:Given positive integer are and their prime factorization is given as

And
Therefore,HCF
And LCM.

Remarks:
• We may also use the following result to find any one of HCF or LCM if we are given the other:
If and are the two positive integers, then
HCFLCM