it is known that length is fixed for which we conclude from Pythagoras theorem that irrational number is exist. But see, during the calculation process result varies with the increase or decrease of the digits after decimal point. Again to my understanding if it exists, the digits after decimal number will be fixed. Again see one unit length can be divided by three geometrically but mathematically there is no end of digits after decimal number. Does it demand any other number system to over come this problem? Explain please
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I believe in infinity number , But we are following decimal number system , is it possible to introduce a new number system where we will be able to find the exact value of this type of numbers.
The square root of 2 was the first number proved irrational, and that article contains a number of proofs. The square roots of all natural numbers which are not perfect squares are irrational and a proof may be found in quadratic irrationals.
An irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate nor are repeating. For example, the number π starts with 3.14159265359..., but no finite number of digits can represent it exactly and it does not end in a segment that repeats itself infinitely often. The same can be said for any irrational number.
In mathematics, an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate nor are repeating. Moreover we may consider it as complement of rational number. Also , geometrically, it may be considered as a diameter of a circle.
By definition, every real number is the limit of a series of rational numbers. If this series is not itself rational then it is irrational. The existence of irrational numbers means that the set of rational numbers is not closed under addition. Closure is thus attained when the extension to real numbers is made. From this perspective and also from the perspective of geometry, irrational numbers are more of the rule than exceptions to the rule. In fact, many irrational numbers can be constructed geometrically, famous examples involving Pythagoras theorem and/or the circle.
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