Saturday, December 15, 2018

Real Numbers:

Real Numbers:
Real numbers can be both positive or negative, and they are denoted by the symbol R. Numbers like a natural number, decimals, and fraction comes under the real number.

Euclids division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers.

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.

Let us see how the algorithm works, through an example first. Suppose we need to find the HCF of the integers 120 and 50. We start with the larger integer, that is, 120. 

Then we use Euclids lemma to get
120 = 50 x 2 + 20
Now consider the divisor 50 and the remainder 20, and apply the division lemma to get
50 = 20 x 2 + 10
Now consider the divisor 20 and the remainder 10, and apply the division lemma to get
20 = 10 x 2 + 0
Notice that the remainder has become zero, and we cannot proceed any further.
We claim that the HCF of 120 and 50 is the divisor at this stage, i.e., 10. You can easily verify this by listing all the factors of 120 and 50. 

Euclids division lemma/algorithm has several applications

A fruitstall has 120 apple and 50 mangoes. He wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the number of mangoes and apple that can be placed in each stack for this purpose?
Solution : This can be done by trial and error. But to do it systematically, we find HCF (120, 50). Then this number will give the maximum number of fruits in each stack and the number of stacks will then be the least. The area of the tray that is used up will be the least.
Now, let us use Euclids algorithm to find their HCF. We have :
120 = 50 x 2 + 20
50 = 20 x 2 + 10
20 = 10 x 2 + 0
So, the HCF of 120 and 50 is 10.
∴ The fruitseller can make stacks of 10 mangoes and 10 apples

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.

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

Rational Numbers and Their Decimal Expansions
Rational numbers have either a terminating decimal expansion or a non-terminating repeating decimal expansion.

Real number is a rational number of the form p/q where the prime factorisation of q is of the form 2n5m and n, m are some non-negative integers.

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