The Infinite Possibility of a “Normal” Number

You can find anything in the digits of Pi

"Remember me?"

Ever try to memorize the digits of Pi?  The world record is only 67,890 digits – shouldn’t be hard to knock that one down, right?  What makes memorizing Pi so hard, of course, is that the digits seem to be completely random – there’s no apparent pattern to them.  It’s commonly thought that the digits of Pi are completely patternless, so that you can find any possible series of numbers you want in there somewhere.  Out at the 762nd decimal place of Pi, you’ll find six consecutive 9’s (named the Feynman Point in honor of the great physicist and my personal man-crush Richard Feynman).  Now if that mind-bending fact has you on the floor, prepare to be knocked down through the Earth’s crust at the other wonders we’ll be able to find in Pi.

A number is said to be “normal” if its digits appear to be random – more technically speaking, if its digits contain all combinations of numbers equally1.  For example, pick any string of 8 digits, for example “12345678” – then this string of 8 is just as likely to pop up, and pops up just as often, as any other string of 8 digits.  Of course, not every number is like this.  5, for example.  Or more specifically, 5.0, or even more specifically 5.0000000….  Clearly, this number is biased towards the digit “0”.  It is believed (though not yet proved) that Pi is normal, and so its digits contain every possible sequence of numbers you can think of, if you look far enough in the decimal expansion.

What this implies is that you can choose any string of digits you want, and it’ll be in the digits of Pi at some point.  Somewhere in there is your weight, your height, and your IQ.  You’ll also find your cholesterol score, today’s winning lottery number, tomorrow’s winning lottery number, the exact number of hairs on your head, every number stored inside your computer right now … you get the idea.

What’s more, anything that can be encoded into numbers can be found in the digits of Pi, as well.  That means any English text (which can be coded into integers) can be found somewhere.  Your name is in there, your parents’ names, your license plate are all in there.  A copy of every essay you wrote in high school is in there.  A copy of this essay is in there.  Are you now thinking what I’m thinking?  If you are, you immature deviant, check out my other article, the Search for Naughty Words in the Digits of Pi.  Wanna try this yourself?  Check out the The Pi search page, which lets you search for any number you want in the digits of Pi.  Or the Search Pi at NERSC page, which lets you search for any text (after encoding it into integers).  Spend some time looking up any words you want — look up your name, your address, whatever you want. It’s pretty fun to find that the slightest whimsy that pops into your mind is already immortalized somewhere in the digits of Pi.  As you play around with your searching in Pi, you may find yourself disappointed with how hard it is to find longer words — in my experience, any word longer than five or six letters is pretty unlikely to be found in the few million digits of Pi they store at these websites.  But rest assured this is only due to the limits of computer storage — if we could search trillions of digits, we’d find longer naughty words (or whatever else strikes your fancy).

One wrinkle is that we don’t actually know if Pi definitely has this property of being “normal” — so far, no one has been able to prove that Pi is truly normal, though it sure looks random as far out as we can measure.  So what if Pi doesn’t turn out to be “normal”?  What if someone proves that Pi doesn’t have this property of containing every possible number sequence?  No biggie – plenty of other numbers are known to be normal, so just take one of them.  Take Champernowne’s Constant — simply create a number where the decimal expansion is the whole list of all integers.  It’s been shown to be normal.  Or the Copeland-Erdos constant, which is the same deal using only prime numbers.  So we can keep on using Pi in our explorations of this crazy idea, safe in the knowledge that if Pi winds up being unusable, we can choose another normal number.  Don’t feel comfortable with this?  Well, I’m sure you can bang out the proof of Pi’s normality over your next lunch break.

Can we take this over the deep end now?

Now let’s follow this chain of reasoning into the Territory of the Weird, as we so often do on this site.  And let’s ask Dr. Cliff Pickover be our tour guide.  Dr. Pickover has taken this idea of information encoded in Pi to astounding (and somewhat spiritual) conclusions.  He points out that since any information which can be encoded numerically must be found in Pi somewhere, there must be an encoding of ourselves in Pi.  A quote from his site:

“Pi almost surely contains the 1993 Edition of the Encyclopedia Britannica. Moreover, it contains the Windows XP operating system. Moreover, it contains all your thoughts, coded in its digit string. You need not fear death or yearn for the woman you once loved but could never have. You have her in pi where you live forever.”


Imagine encoding the location of every atom in your body at this very moment into a numerical code – that code must be in Pi somewhere, a numerical spec sheet describing how to build an exact replica of you.  That same specification of you, encoded in digits, from every instant in your entire life must also be in there somewhere.  Furthermore, an encoding of your thoughts, every thought you’ve ever had, must be in there.  Dig far enough, and all of the information required to specify everything about you, precisely, must be found in the digits of Pi.

Strange, isn’t it?  If Pi is really a normal number, then eventually in its digits somewhere, you’ll find a long string of digits that completely encodes you.  Invent whatever scheme you want for specifying how to build up a person, and it’ll be in there.  In fact, all possible schemes for encoding you are found in there somewhere.  And the same is true for all the other people you know, all the people you don’t know,  all the people that have ever existed, a bunch of people who never existed — they’re all inside Pi somewhere.  So in a sense, that woman of your dreams that you’ve lost forever, is still contained in the digits of Pi somewhere.  Check out the rest of Dr. Pickover’s site for

What a romantic notion!  Somewhere in Pi, the woman of your dreams is telling you how much she loves you, and that your receding hairline really doesn’t matter to her.  True, but Pi also encodes her dumping piping hot coffee in your lap because you called her fat.  Or her turning into a Tyrannosaurus Rex and destroying your apartment while you’re sitting on the couch watching “Norbit”.  (Despite how preposterous this sounds, it must be true — somewhere in Pi, you’re watching “Norbit”.)  Pi may contain the correct answer to every question you could possibly ask, but it also contains the wrong answer.   Lots of wrong answers.  In fact every possible answer, even nonsensical ones.   Mostly nonsensical ones.

Wait just one goddamn minute here…

I hope we're not all encoded in the movie "Pi", because that movie was insufferable.

This leads to one major philosophical problem with this romantic idea of worlds contained in Pi.  These ideas view the digits of Pi as sort of an information storage mechanism – any info you want is in there somewhere, so all you need is its “address” — the digit where the info starts.  So you want to store the entire works of Shakespeare?  Great, all you need is one number!  Of course that number might be long… might be longer than the entire works of Shakesepeare, in fact.  While in principle the information is perfectly preserved among Pi’s digits, in practice it is unimaginably difficult to retrieve.  Over at my other article on Finding Naughty Words in the Digits of Pi, I found that it was immensely difficult to find any words longer than five or six letters in the first fifty million digits of Pi.  The entire works of Shakespeare likely appear at some astronomically huge decimal place within Pi.  So while it’s true that any information you want is found in there somewhere, in practice it might be more difficult to retrieve it from Pi than to simply recreate the information in the first place.  So buck up and ask that woman out for coffee.

And let’s get to another problem with this admittedly beautiful idea — the inconvenient fact that the digits of Pi are discrete integers.  While the number Pi itself is a real number (not an integer), the digits of pi written out in decimal notation are all integers.  So when we go searching for some number in the digits of Pi, we’re by necessity stuck hunting through integers.  And there are probably limits to the kinds of information that integers alone can encode.

One example is the value of Pi itself.  Does Pi contain a copy of itself somewhere?  Nope – because Pi is an infinite, nonrepeating irrational number, you can’t use simple integers to represent it.  Being irrational means that Pi can’t be represented by the ratio of two integers.  So that’s one bit of information that most assuredly isn’t found in the digits of Pi, nor any other normal number.  How embarrassing.  Even if I grant you the freedom to not be stuck with just integers, by allowing you to throw a decimal point wherever you want in the digit string that encodes your information, you’re still out of luck.  Because irrational numbers can’t be perfectly represented by any terminating or non-repeating decimal number, no irrational numbers can be perfectly encoded in the digits of Pi.

No big deal, you think?  What’s the problem if we chuck a few exotic irrational numbers?  Well, there are far more irrational numbers than rational numbers (those that can be represented by the ratio of two integers).  Integers are “countably infinite”, while real numbers are “uncountably infinite” – in a nutshell there are far more irrational numbers than rational numbers.  So there’s the concern that for encoding random things from the real world, you might need to avail yourself of irrational numbers here and there, and therefore these ideas may not be representable in the digits of Pi.  You could live with approximate representations, I suppose – for example you could encode a JPEG compressed image of the Mona Lisa and find it in Pi somewhere.  But if for some reason specifying the real Mona Lisa requires irrational numbers, then you’d be out of luck encoding it perfectly.

Why would we need irrational numbers to represent anything numerically?  Well, it looks to all appearances that in our universe physical quantities can take on any real-valued number – position, velocity, wavelength, etc. are all allowed to have any old value, not necessarily a rational-number value.  If so, that means that most physical measurements take on irrational values from time to time – and since irrationals are much more common than rationals, most of the physical states of the universe might require irrational numbers to encode as numeric information.  (Note that the quantum nature of the microscopic realm doesn’t necessarily mean some quantities can’t take on continuous values, nor that the quantum states of a discrete state have to be rational numbers.)  So if you want to precisely specify the Mona Lisa exactly, you’ll need to record the exact coordinates of every single elementary particle in the entire painting, and to do that you may need irrational numbers.

Sorry to crap on your idea, Dr. Pickover.  But we’ll come back to your fantastic ideas a couple sections from now.  But first, let’s explore a different way to get to the woman of your dreams morphing into a T-Rex…


Next Up:   Our Infinite Universe   >> 


1.  Another example from mathematics of shitty nomenclature.  (That would be a great band name, “Shitty Nomenclature”.)   We don’t mean these numbers are “ordinary” or “not weird” — we mean a very specific technical property when we call them “normal”.  I personally hate it when plain english words that already have other meanings get carjacked to become mathematical jargon, because then you tend to confuse the plain-english meaning with the intended mathematical meaning.  At best, the word simply fails to convey what you want it to about the mathematical property.  At worst, it kicks up clouds of unintended meaning to the layperson — see “imaginary numbers”.  OK, rant over.

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