 Hello and welcome to the session. In this session, we will learn some more important theorems of real numbers. First let's discuss fundamental theorem of arithmetic. According to this theorem we have that every composite number can be expressed as a product of primes and this factorization is unique the order in which the prime factors occur. The theorem basically establishes the importance of prime numbers and prime numbers are the basic building blocks of the positive integers. Also, if the prime factorization of a number is known to us, then we get the complete knowledge about all factors of that number. Let's consider a composite number 60. Now we will show that the prime factorization of this composite number is unique. This is the prime factorization of the composite number 60. As you can see this is unique. This cannot be factorized further. So generally fundamental theorem of arithmetic says that given any composite number there is one and only one way to write it as a product of primes. The next theorem that we shall discuss says let p be a prime number if this prime number p divides a square then p divides a where a is the positive integer. Next we prove root 5 is irrational. For this our first step is we assume root 5 to be any rational number with its denominator not equal to 0. Suppose the numerator and the denominator have a common factor other than 1. So we divide by the common factor and we get this rational number of the form a upon b where the numerator and denominator are co-prime. So we get this of this form which shows that 5 divides a square. So from the above theorem that we have stated we get that 5 divides a. Thus this a can be written in the form this where we take c to be some integer. We substitute this value of a in this to get which further shows that 5 divides b square from the theorem stated above we can say that 5 divides b. This shows that a and b have at least 5 as a common factor but we already know that a and b are co-prime. So this is a contradictory statement hence our assumption that root 5 is a rational number is not and thus we get that root 5 is irrational. Now we shall discuss rational numbers and their decimal expansions. First of all we have that if x is a rational number of the form p upon q where the prime factorization of q is of the form 2 raised to the power n into 5 raised to the power m where n and m are non-negative integers then we say that x has a decimal expansion which terminates. So for any rational number we will have a look at the denominator and the prime factorization of its denominator if it's of the form 2 raised to the power n into 5 raised to the power m then it has a decimal expansion which terminates. This is the basic idea. Let's consider a rational number now the prime factorization of its denominator is given by this or this can also be written in this form which is same as the form 2 raised to the power n into 5 raised to the power m hence we say the given rational number has a decimal expansion which terminates. Next we have if x is a rational number of the form p upon q and the prime factorization of its denominator that is q is not of the form 2 raised to the power n into 5 raised to the power m where n and m are non-negative integers then we say that the rational number x has a decimal expansion which is non-terminating repeating. Consider a rational number now if you look at the prime factorization of its denominator we see that it is not of the form 2 raised to the power n into 5 raised to the power m. So we say that this rational number has a decimal expansion which is non-terminating repeating. So this completes the session for today hope you have understood the concept of the fundamental theorem of arithmetic and the rational numbers and their decimal expansions. Have a good day.