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Question Number 6449 by Temp last updated on 27/Jun/16

x=0.112123123412345123456... decimal pattern: 1, 12, 123, ... Can this be represented as a fraction? or is this number trancendental?

$${x}=\mathrm{0}.\mathrm{112123123412345123456}... \\ $$$$\mathrm{decimal}\:\mathrm{pattern}:\:\mathrm{1},\:\mathrm{12},\:\mathrm{123},\:... \\ $$$$\mathrm{Can}\:\mathrm{this}\:\mathrm{be}\:\mathrm{represented}\:\mathrm{as}\:\mathrm{a}\:\mathrm{fraction}? \\ $$$$\mathrm{or}\:\mathrm{is}\:\mathrm{this}\:\mathrm{number}\:\mathrm{trancendental}? \\ $$

Commented by prakash jain last updated on 28/Jun/16

123456 may have some inputs on this sequence.

$$\mathrm{123456}\:\mathrm{may}\:\mathrm{have}\:\mathrm{some}\:\mathrm{inputs}\:\mathrm{on}\:\mathrm{this} \\ $$$$\mathrm{sequence}. \\ $$

Commented by prakash jain last updated on 27/Jun/16

What comes after 12345689 12345678910? If the pattern is non−repeating then it will be irrational. It is generally very hard to prove if a number is trancendental.

$$\mathrm{What}\:\mathrm{comes}\:\mathrm{after} \\ $$$$\mathrm{12345689} \\ $$$$\mathrm{12345678910}? \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{pattern}\:\mathrm{is}\:\mathrm{non}−\mathrm{repeating}\:\mathrm{then}\:\mathrm{it} \\ $$$$\mathrm{will}\:\mathrm{be}\:\mathrm{irrational}. \\ $$$$\mathrm{It}\:\mathrm{is}\:\mathrm{generally}\:\mathrm{very}\:\mathrm{hard}\:\mathrm{to}\:\mathrm{prove}\:\mathrm{if}\:\mathrm{a} \\ $$$$\mathrm{number}\:\mathrm{is}\:\mathrm{trancendental}. \\ $$

Commented by Temp last updated on 28/Jun/16

12345678910, 1234567891011, etc. and I see

$$\mathrm{12345678910},\:\mathrm{1234567891011},\:{etc}. \\ $$$$\mathrm{and}\:\mathrm{I}\:\mathrm{see} \\ $$

Commented by nburiburu last updated on 28/Jun/16

this number is generalized as x=Σ_(n=0) ^∞ (((n−((⌊(((√(8n+1))−1)/2)⌋(⌊(((√(8n+1))−1)/2)⌋+1))/2)+1)/(10^(n(n+1)/2) )) ) (Smadaranche crecendo sequence / 10 to the sum from 1 to n) It is not a rational number.

$${this}\:{number}\:{is}\:{generalized}\:{as} \\ $$$${x}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{{n}−\frac{\lfloor\frac{\sqrt{\mathrm{8}{n}+\mathrm{1}}−\mathrm{1}}{\mathrm{2}}\rfloor\left(\lfloor\frac{\sqrt{\mathrm{8}{n}+\mathrm{1}}−\mathrm{1}}{\mathrm{2}}\rfloor+\mathrm{1}\right)}{\mathrm{2}}+\mathrm{1}}{\mathrm{10}^{{n}\left({n}+\mathrm{1}\right)/\mathrm{2}} }\:\right) \\ $$$$\left({Smadaranche}\:{crecendo}\:{sequence}\:/\right. \\ $$$$\left.\mathrm{10}\:{to}\:{the}\:{sum}\:{from}\:\mathrm{1}\:{to}\:{n}\right) \\ $$$$ \\ $$$${It}\:{is}\:{not}\:{a}\:{rational}\:{number}. \\ $$$$ \\ $$

Commented by prakash jain last updated on 28/Jun/16

This is good information. Thanks. I will look up derivation of the formula.

$$\mathrm{This}\:\mathrm{is}\:\mathrm{good}\:\mathrm{information}.\:\mathrm{Thanks}. \\ $$$$\mathrm{I}\:\mathrm{will}\:\mathrm{look}\:\mathrm{up}\:\mathrm{derivation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{formula}. \\ $$

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