Hi All, This might be a silly question but can anyone tell me with certainty if it is possible to convert a transcendental number into a terminating decimal by base changing? If that is possible that is insanely awesome. [edit] Sorry that was completely not what I was wondering. I meant this: Does an integer based number system exist, wherein some transcendental number when converted into this number system is a finite number. I think I worded it ok that time.
First, decimal means base 10. You can't take a transcendental number written in base 10, and change to base 10, and expect the expression to be different. (This is not correct for all real numbers. But you only asked for transcendental.) If you are asking "given a transcendental number x, does there exist a base, so that x can be written finitely in that base?" Then the answer is yes: base x. For example the golden ratio is "10" in base phinary. Edit: I just realized that phi is not transcendental. But the idea is the same.
In other words, given any transcendental number, x, we can write x in the number system having base x, as "10".
Hi Folks & thanks for responding. I think I should have been more explicit in my question though. I'm gonna go back and change it. I meant for the base to be an integer. As far as me calling it a decimal... yeah that was pretty stupid.
like how [itex] 2^{\aleph_0}=10^{\aleph_0} [/itex] which equals [itex] \aleph_1 [/itex] is that what you mean. edit: I confused transcedental with transfinite. but I still think you could change the base.
Change of base from one integer to another integer will not change the nature of a real number. Transcendental will stay transcendental, algebraic will stay algebraic, and rational will stay rational. The only thing noticeable is that the decimal expression of rational numbers will terminate in some bases, but not in others. Example: 1/3 = .3333... in decimal, = .1 in base 3.
All numbers are finite. Any repeating pattern of digits no matter what the base will give you a rational number.