In previous classes we have more explored of the world of real numbers and encountered irrational numbers.
Here we'll continue more learn about the real numbers.
Euclid's division algorithm, as the name suggests, has to do with divisibility of integers. Stated simply, it says any positive integer acan be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b.
Theorem 1.1 (Euclid's Division Lemma) : Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.
An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.
The word algorithm comes from the name of the 9th century Persian mathematician al-Khwarizmi. In fact, even the word ‘algebra’ is derived from a book, he wrote, called Hisab al-jabr w'al-muqabala.
A lemma is a proven statement used for proving another statement.
Euclid's division algorithm clearly. To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:
Step 1 : Apply Euclid's division lemma, to c and d. So, we find whole numbers, q and r such that c = dq + r, 0 ≤ r < d.
Step 2 : If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and r.
Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.
We you have seen that any natural number can be written as a product of its prime factors. For instance, 2 = 2, 4 = 2 x 2, 253 = 11 x 23, and so on.
Theorem 1.2 (Fundamental Theorem of Arithmetic) : Every composite number can be expressed ( factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
The Fundamental Theorem of Arithmetic says that every composite number can be factorised as a product of primes. Actually it says more. It says that given any composite number it can be factorised as a product of prime numbers in a ‘unique’ way, except for the order in which the primes occur.
So, for example, we regard 2 x 3 x 5 x 7 as the same as 3 x 5 x 7 x 2, or any other possible order in which these primes are written.
The prime factorisation of a natural number is unique, except for the order of its factors.
We have learnt about their existence and how the rationals and the irrationals together made up the real numbers.
A number ‘s’ is called irrational if it cannot be written in the form, p/q where p and q are integers and q ≠ 0.
Theorem 1.3 : Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer.
Theorem 1.4 : Root √2 is irrational.
When rational numbers have either a terminating decimal expansion or a non-terminating repeating decimal expansion. In this section, we are going to consider a rational number, say (p/q) (q ≠ 0, and explore exactly when the decimal expansion of p/q is terminating and when it is non-terminating repeating (or recurring).
Theorem 1.5 : Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form , p/q where p and q are coprime, and the prime factorisation of q is of the form 2n5m, where n, m are non-negative integers.
Theorem 1.6 : Let x =p/q be a rational number, such that the prime factorization of q is of the form 2n5m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.
Theorem 1.7 : Let x =p/q, where p and q are coprimes, be a rational number, such that the prime factorisation of q is not of the form 2n5m, where n, m are non-negative integers. Then, x has a decimal expansion which is non-terminating repeating (recurring).