Understand The Importance Of Transcendental Numbers

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Numbers are generally categorized into sets that share similar, distinct characteristics. These may be that they are all even or all odd, or even simply just real. The transcendental numbers are in essence another way to categorize a particular set of numbers. To understand what numbers the transcendentals encompass it must first be understood what they do not. To start there are the natural numbers, which can be found through counting, none of these are transcendental. A short side note, if a set of numbers is said to be closed under an operation then that means if any two numbers within said set are operated on with that operation the result will end with a number in that set. Then there are the integers, gotten by adding closure under subtraction. Next come the rationals, found through further closure under …show more content…

Add in all the non-complex n roots and now all the algebraic real numbers are accounted for, which none of are transcendental. So what are the transcendentals? All non-algebraic numbers; so virtually the majority of the real number set. This can be said confidently with the support from the following logic: if there are infinitely many real numbers and a countable set of algebraic numbers, then the remainder of numbers in the real number set is infinite, with these said numbers being the transcendental set, and an infinite of any degree is greater in size than any countable number. With this understanding of the transcendentals on a conceptual level though it is still extremely difficult to prove any given number to be transcendental. This is because it is easy to set up any non-zero polynomial equation with rational coefficients and find its roots, but it is not as straightforward to set one up and not find any. Why is

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