User:Er meer/sandbox

Real Numbers

Introduction:

Real numbers are simply the combination of rational and irrational numbers, in the number system. In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line, also.

The set of real numbers consist of different categories, such as natural and whole numbers, rational and irrational numbers and integers. In the table given here, all these numbers are defined with examples.

Natural Numbers

All counting numbers which start from 1. N = {1,2,3,4,……}

All numbers such as 1, 2, 3, 4,5,6,…..…

Whole Numbers:

Collection of zero and natural number. W = {0,1,2,3,…..} All numbers including 0 such as 0, 1, 2, 3, 4,5,6,…..…

Integers:

The collective result of whole numbers and negative of all natural numbers. Includes: -infinity (-∞),……..-4, -3, -2, -1, 0, 1, 2, 3, 4, ……+infinity (+∞)

Rational Numbers:

Numbers that can be written in the form of p/q, where q≠0.

Examples of rational numbers are ½, 5/4 and 12/6 etc.

The decimal expansions of Rational Numbers are:

I. TERMINATING

II. TERMINATING REPEARTING OR RECURRING

III. NON-TERMINATING RELEATING /RECURRING

Irrational Numbers:

All the numbers which are not rational and cannot be written in the form of p/q.

Irrational numbers have non-terminating and non-repeating Decimal expansion like √2,√3 etc.

FUNDAMENTAL THEOREAM OF ARITHMETIC:

Fundamental Theorem of Arithmetic states that every integer greater than 1 is either a prime number or can be expressed in the form of primes. In other words, all the natural numbers can be expressed in the form of the product of its prime factors. To recall, prime factors are the numbers which are divisible by 1 and itself only. For example, the number 35 can be written in the form of its prime factors as:

6 = 2 × 3

Here, 2 and 3 are the prime factors of 6

Euclids Division Leema/ Algorithm:

According to Euclid’s Division Lemma if we have two positive integers a and b, then there exist unique integers q and r which satisfies the condition a = bq + r where 0 ≤ r ≤ b.

The basis of the Euclidean division algorithm is Euclid’s division lemma. To calculate the Highest Common Factor (HCF) of two positive integers a and b we use Euclid’s division algorithm. HCF is the largest number which exactly divides two or more positive integers. By exactly we mean that on dividing both the integers a and b the remainder is zero.

The main disadvantage of EDL is that It cannot be used for calculating the HCF of More than Two Numbers.

Steps used in EDL:

1. Compare the given numbers.

2. Make the larger number dividend and the smaller one divisor.

3. Multiply the divisor by a scalar to get dividend or smaller than that.

4. If the remainder is not equal to zero. Then make Divisor as dividend and remainder as Divisor.

5. Repeat step 4 until remainder will be zero.

6. If the remainder is zero check the divisor at that step. The divisor at that step will be the HCF of given Numbers.