Do Irrational Numbers Provide Reliable Seeded Random Number Streams?

[Chris Miller]

Are irrationals effective (seeded) random number input streams? E.g., here's 17^1/2 to 2496 (8K bits) decimal places:

4.123105625617660549821409855974077025147199225373620434398633573094954346337621593587863650810684296684544040939214141615301420840415868363079548145746906977677023266436240863087790567572385708225521380732563083860309142749804671913529322147978718167815796475906080565496973900766721383689212106708921029005520264976997227788461399259924591373145657819254743622377232515783073400662476891460894993314102436279443386280552637475060905080869257482675403757576927464631666351033096817122919874195864431971054705958485725931943603620656058152613585046428067872150064104914222367522243486737258047037771274998566571218570432100303602606506487154690698281546846459564503441849930597639509078619959043334207783036732466105002383305603648597891517738125149725101393295630516977396156134483704021469549517283774775128332086775432479301964503858945967736521957022356481292823232373091650044755709460165721749143175547451122718361635317492475624065195560022755934398822460451518623945769412122844523427764255912670433259808358492948699826803953313743117174259527446589487487995234588945341405362382216244897199383951723677736924815737740851390093874931215463256061131831025557439033296612065618997712372326637094637265748287231103821317707944772530780064797150678855818398908582322952162045692531561131920561778167348673892987498555458433124657440601285674187836856748287375368481499912380912509688262497561558561564648512481411215339812008301990104085930861062338067167258928822842126024770742381463219000159897259624750259093195821477134731917698441102319595010755969064406071469393642506505486427451376148387657994040157094448093988829687785599616740434885132269481081155527404578852192449351277914392107618272900599754897954312002170599256849466159387615532763790999551021418146407027743000792920490048243451252056874608711185472125959950450945481146458616792396812655520824304951486086333901135640630166910365347510269977305391756449457931367201466774922328035237350342552906155151563822733055685642499595284430373695550870606523686673143876906178092044034814292694187010803880108829021799880687696158697329126877696593420355145932219374892438065025680057784508087680163993195284779117186081306592021305623343958242917631701153763240709357137139143819622153188015764287100179012418273573753050688530502962200477947919154553842693338844717836001761469185048394692345476131766736808878375874811980697221956669693993181703367244994221297365315344095040553005626348185982780376611654361701359383702910820779

It seems to satisfy randomness: e.g., .5 bit on/off probability, (provably) non-repeating, values evenly distributed; but will this continue to be the case as it's taken to infinite decimal places?

 

[mathman]

Probably. Almost all numbers have this property.

 

[mfb]

We don't know. As one of the requirements to serve as random number generator you would want a number to be normal. While most real numbers are normal, it is very hard to prove this property for specific numbers. The numbers where we have this proof are not suitable for a pseudo-RNG.

 

[Chris Miller]

Thank you, mathman. "Probably"? Interesting choice of words. All irrationals probably have this property? But no others. Both sets (i.e., reals that have this property and reals that don't) are infinite. So I have some trouble with "almost all."

Thanks, mfb. I love a good "We don't know." One of those things that can neither be proved nor disproved? Would you agree that irrationals tend to be slow to compute, but work as pseudo-RNGs as far as we've calculated any of them? What blows my mind a little (doesn't take a lot) is that, if they are "normal" (thanks for the link and term), then as their bits are calculated to infinity, the number of consecutive on (or off, or any other arrangement of) bits we would eventually (and infinitely) encounter also approaches infinity with a probability that approaches 1.

 

[jbriggs444]

All irrationals probably have this property?

Not all irrationals are normal. Only "most". A classic example of a non-normal irrational is 0.101001000100001...

In order to make a statement about probability, one would need to come up with a distribution first.

 

[Chris Miller]

Right, one could construct infinite abnormal irrational's with non-repeating patterns.

Any distribution (bit stream) for which the probability of bit change is 0.5 (as in a coin toss). Like if one were (somehow able) to examine infinite coin flips, there should be infinite junctures at which the number of consecutive heads approached infinity with a probability approaching 1.

 

[mathman]

My understanding: unit interval with uniform distribution - almost all (i.e. probability = 1) numbers are normal.