In The Big Bang of Numbers, I take the reader through a thought experiment, where together, we build all the different kinds of numbers, and then, from these, go on to trying to build geometry, algebra, patterns, physics, and everything in the universe. In Chapter 4, we just learned how to construct irrational numbers like pi and the square root of 2, whose decimal expansions go on and on, without repeating. In Chapter 5, we start discussing how this can help us in the universe we are building.
So how do the irrationals you’ve just created enhance your universe?
On an aesthetic level, they provide a sense of completeness. You can rest secure in the knowledge that you’ve created each and every decimal expansion possible, without any omissions. If you felt a touch of mathematical appreciation producing the rationals, that’s now grown to full-fledged intellectual satisfaction. However, this benefit is somewhat abstract. What do irrationals contribute in practical terms to your universe?
Most mathematicians would answer precision. With irrationals at your disposal, you can identify the exact square root of 2, the exact ratio of circumference to diameter. But there’s a gap here, since you can never actually write down the exact value of or pi. Each has an infinite expansion, so in reality, you’d be forced to use an approximation.
And that would be fine. For instance, NASA has been reported to use the value of pi to fifteen decimal places, π=3.141592653589793, for all its calculations. This turns out to be precise enough to track even its most far-flung satellites. As one of NASA’s mission managers has stated, were they to use 40 digits instead, they’d be able to calculate the circumference of the entire visible universe to an accuracy of the diameter of a hydrogen atom! In other words, it would be overkill.
NASA’s experience suggests that while building our universe, you might never need the exact value of an irrational – you could get away with approximations that were close enough. In other words, rationals, with enough decimals, would suffice. Which returns us to the question, what do irrationals bring to the table?
Later, we’ll see how essential irrationals are for motivating the construction of a straight line (without which, we’d be unable to create the 3-D space we live in). But right now, let me introduce another crucial aspect they inspire in the universe. To do so, I need to first take you on a detour, by acquainting you with a very curious irrational many of you are unlikely to have seen before.
A crystal ball of digits
Imagine a decimal point with each of the whole numbers 1,2,3… strung together after it, in order:
0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15….
I’ve written down the first fifteen naturals above, so the next two digits would be 16, the two after that 17, and so on. The complete expansion would contain the sequence of all the naturals. Notice this gives a real number lying between 0 and 1.
Clearly, this infinite expansion doesn’t repeat – so the number can’t be a fraction, it must be irrational. The number’s mathematical name is the Champernowne constant, but let’s call it the “crystal ball” constant. Why? Because it contains the answer to every question about the future posed by you or anyone else!
To see this, think of the simple numerical codes used by kids in their cloak and dagger games. I remember making the substitutions a = 1, b = 2, c = 3, up to z = 26. (Perhaps you tried this as well?) Using this code, the word “cab” becomes the whole number 312. One can convert sentences, articles, even books this way – after adding some extra integers for punctuation, spaces between letters, and so on. For instance, the entire 587,287 word text of War and Peace can be reduced to a single integer using the code! A very long integer it’s true, but still one of finite length.
Now the crystal ball irrational contains every natural number, so it must contain the one corresponding to “cab.” This is easy to verify – just read along the list to the 312th natural listed, and voila! it’s there.
But by the same token, this irrational also contains the much larger integer for War and Peace. Proceed along the list far enough and you’ll eventually have to get to it. In fact, you can find the entire Harry Potter series in this irrational number, along with all the plays by William Shakespeare. Everything ever written – every poem, every treatise, every issue of the New York Times exists in it.
Moreover, it also contains everything that ever will be written in the future. Tomorrow’s newspaper, for instance – or one from next year. Which means that if you’re wondering what stocks to buy or whether we’ll be submerged due to melting icecaps, the answer’s in there.
This book I’m writing is in there as well – wouldn’t it be great if I could just copy it down without having to actually slog through all the work ahead? Unfortunately, it’s inaccessible, since I don’t know at which digit it starts. I also have no way of differentiating my book from all the slightly changed variations contained in the number, or from radically altered versions, or, for that matter, from the many tracts of utter gibberish. The same is true of what stocks to purchase, since mixed in with what’s actually going to happen are reports of every possible future imaginable.
In 1941, the Argentinian author Borges published a short story called The Library of Babel, about a rambling multi-roomed library that holds every book of 410 pages ever written or that ever might be written. However, nobody knows where anything is shelved, so the entire collection is useless. The crystal ball irrational is an expanded version of that library, since it stores every bit of information that might ever exist. A virtual flash drive if you will, which, sadly for us – or perhaps fortunately – is perfectly encrypted.
OK, all this was the detour – or subterfuge, even – before I unveiled the irrationals’ most important contribution. The answer: it’s not the coded knowledge they contain, but rather the way they scramble that knowledge! Specifically, it’s the randomness with which digits appear in their expansions. Randomness will be essential for the construction of your universe, perhaps for generating life itself, as we’ll see ahead in our creation cycle.
Manil Suri is a distinguished university professor at the University of Maryland, Baltimore County. He is the author of The Big Bang of Numbers: How to Build the Universe Using Only Maths.
Excerpted with permission from The Big Bang of Numbers: How to Build the Universe Using Only Maths, Manil Suri, Bloomsbury.