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# Real Numbers Definition

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 Sub Topics Let us consider a number like $\frac{\sqrt{-5}}{2}$ , $\sqrt{-7}$. These kind of numbers can be expressed in terms of i like $\sqrt{7}$ i, 2.5 i etc. where i represents the imaginary numbers. To differentiate the imaginary numbers from the numbers existing in real there came a concept of Real Numbers. Real numbers are defined as those numbers which do not have i (imaginary numbers) as a part of it. It is a Set of both Rational and Irrational Numbers. It is represented by R. Example: R = {-3, -2, -1, 0, 0.5, 2, 5......}. In short it represents particular or fixed amount of quantity. It includes positive or negative numbers, natural number, whole numbers, fractional numbers, integers etc.

## History of Real Numbers

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The history of Real Numbers starts with two major periods - first one is classical Greek mathematics and the second one is the rigorization and the formalization of mathematics which developed in the 19th century. But in that time, there was lack of knowledge and weakness to understand the real number, and because of this reason the mathematician were unable to expand their ideas.

The Natural Numbers were developed or understood by Greek mathematicians. They also developed an idea of how we call the Whole Number ratios, rational and real numbers. But now a day’s these systems are not treated as they were known in that time.

When any number system has been developed to modern standards, it is necessary to develop that number system in the Method of Exhaustion. The real numbers and the integral Calculus demonstrate the Method of Exhaustion. But this was not subsequently matched with formalization of mathematics in the 19th century. Mathematicians in the 19th century further analyzed these ideas and then improved the ideas on which this field is based.

The Indian and Chinese mathematicians developed the science of the acceptance of zero, negative, real, integral and fractional numbers. Due to this development and the development of Algebra, the Irrational Numbers are treated as algebraic objects by Arabic mathematicians in the middle ages. Abu Kamil Shuja ibn Aslam an Egyptian mathematician first considered irrational numbers as solution of quadratic equations or coefficients to square roots and Cube roots in an equation and so on. Around 500 BC, Greek mathematicians realized the need for irrational numbers particularly the Square root of two. In this way the real number system was developed.