The Most Predictable Number That Looks Perfectly Random

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[00:00:00]Write down the digit zero, then a decimal point, then write all the positive integers in order glued together with no spaces between them.

[00:00:09]1 2 3 and so on forever. The result is a specific real number. After a few dozen digits, it starts to look essentially random. It is not. Every digit's position is determined by a simple rule.

[00:00:25]This number was defined in 1933 by a Cambridge undergraduate named David Champernowne. He was 20 years old. He published a single-page paper introducing the number and proving one fact about it. The fact was unexpected.

[00:00:40]The construction looks contrived. The digits are deterministic. Given the position, you can compute exactly what digit goes there.

[00:00:48]There is no randomness, no probability, no choice. It is the most predictable real number you can write down. And yet Champernowne proved it behaves exactly like a random sequence of digits would.

[00:01:01]The fraction of digits that are zero is 1/10. The fraction that are one is 1/10.

[00:01:07]Every pair of digits, every triple, every length K string appears with exactly the frequency you would expect if the digits had been generated by rolling a fair 10-sided die. This is called normality. And Champernowne's number is the first explicitly constructed example of a normal number.

[00:01:25]Four years later, Kurt Mahler proved something stronger. The number is transcendental. To understand the claim, let's define normality precisely.

[00:01:35]Pick a real number alpha between 0 and 1.

[00:01:38]Write its decimal expansion.

[00:01:40]Pick a digit, say three. Count what fraction of the first n digits of alpha equal three. If that fraction converges to 1/10 as n goes to infinity, alpha passes the first test.

[00:01:53]Now, pick a length two string, say 47.

[00:01:57]Slide a window of length two along the digits of alpha and count what fraction of starting positions begin with 47. If that fraction converges to 1 over 100, alpha passes the second test.

[00:02:10]For any length K, slide a window of length K.

[00:02:14]The number of possible length K strings is 10 to the K. If every one of them appears in alpha's expansion with frequency exactly 1 over 10 to the K, alpha is called normal in base 10.

[00:02:27]Borel proved in 1909 that almost every real number is normal. Almost every. In the precise sense, the set of real numbers in 0,1 that are not normal has Lebesgue measure zero.

[00:02:41]They are an infinitely small minority.

[00:02:44]But here is the strange part. Despite normal numbers being the vast majority, we have a hard time identifying any specific one.

[00:02:53]Is pi normal? We don't know. We have computed pi to 50 trillion digits.

[00:02:59]The digit frequencies look normal, but we have no proof.

[00:03:03]Is e normal? Don't know.

[00:03:06]Is root two normal? Don't know.

[00:03:09]Almost every real number is normal.

[00:03:11]We can name almost none of them.

[00:03:13]Champernowne 1933 was the first explicit construction of a number that we can prove is normal.

[00:03:20]The proof is short. The construction makes the counting tractable. We know exactly which integers contribute to which digit positions. For any length K window, we count how many of the first M digits are starting positions for that window.

[00:03:35]The integers up to 10 to the K contribute digits in a very structured way. As K stays fixed and M grows, the count of any specific length K string appearing in the first M digits is dominated by integers near a specific size, and these integers cycle through every possible digit pattern.

[00:03:52]The clean version of the counting argument. Every length K digit string from 000 up to 999 appears among the digits of the integers from 10 to the K minus 1 through 10 to the K minus 1.

[00:04:06]Because those integers are exactly the K digit numbers written in order, so all length K strings appear as substrings exactly once.

[00:04:15]Multiplying by K digits each and dividing by the total digit count gives a frequency of 1 over 10 to the K for every length K string. As K stays fixed and we look at integers further out, the fractional contribution converges to that frequency.

[00:04:31]This is the entire proof. Champernowne handled the edge cases, concatenation across integer boundaries, the contribution of integers of size different from 10 to the K, and the result is rigorous. The number is normal in base 10.

[00:04:46]It is the simplest known normal number.

[00:04:48]Normality is a frequency property. It does not mean unpredictability.

[00:04:53]Champernowne is deterministic. You can compute its nth digit for any n with a simple algorithm. Look up which integer the nth digit belongs to, then return the appropriate digit of that integer.

[00:05:06]Done. There is no randomness anywhere in the construction.

[00:05:10]And yet the resulting sequence of digits passes every frequency test you can run.

[00:05:15]Single digits uniform, pairs uniform, triples uniform.

[00:05:21]The chi squared test for digit independence passes, the runs test passes, the serial correlation test passes.

[00:05:29]Statistical tests that distinguish random sequences from non-random ones cannot distinguish Champernowne's digits from a random sequence.

[00:05:38]This is a paradox to anyone who first learns about randomness as unpredictability. Champernowne is maximally predictable. Given any prefix, you know exactly what comes next.

[00:05:49]But its digit distribution is indistinguishable from a fair die roll.

[00:05:53]The lesson: normality and randomness are different properties.

[00:05:58]A normal number has the right digit frequencies. A random number is unpredictable.

[00:06:04]The intersection of these properties is interesting, but they are independent.

[00:06:08]Champernowne is normal and not random.

[00:06:11]There are also numbers that are random but not normal. The Champernowne construction is the cleanest demonstration that behaves like random and is random are not the same thing. In 1937, the German mathematician Kurt Mahler proved something stronger than normality. Champernowne's number is transcendental.

[00:06:32]A transcendental number is one that is not algebraic. It is not the root of any polynomial with integer coefficients.

[00:06:39]The classical example is pi, proved transcendental by Lindemann in 1882.

[00:06:45]The construction of a transcendental number by hand was difficult. Liouville in 1844 was the first to do it.

[00:06:53]Mahler's method, applied to Champernowne, uses Liouville's theorem on rational approximation.

[00:06:59]The theorem says, if alpha is an algebraic number of degree d, then for any rational p over q, the inequality absolute value of alpha minus p over q is at least some constant divided by q to the d.

[00:07:13]Algebraic numbers cannot be approximated too well by rationals.

[00:07:17]The bound is sharp in d.

[00:07:20]A transcendental number is one that violates this bound. It admits rational approximations better than any polynomial in q.

[00:07:28]To show Champernowne's is transcendental, Mahler constructed an explicit sequence of rationals p sub k over q sub k with the property that the error term alpha minus p sub k over q sub k is bounded above by one over q sub k to the t for arbitrarily large t. If Champernowne's were algebraic of degree d, the error would be at least 1 over q sub k to the d.

[00:07:53]But Mahler's construction gives errors of 1 over q sub k to the k for k growing without bound. So, Champernowne's cannot be algebraic of any finite degree, hence transcendental. Where do the rational approximations come from?

[00:08:07]Champernowne's expansion has a specific repeating structure. The integers appear in increasing order, contributing more digits each time. After concatenating 1 through 9, you've laid down nine digits.

[00:08:20]After 1 through 99, you've laid down 9 + 90 * 2 = 189 digits.

[00:08:27]Pick a position right after the integer 10 to the k minus 1 has been laid down.

[00:08:32]That is immediately after the last k digit integer. Call the number of digits up to that position m sub k.

[00:08:39]The first m sub k digits of Champernowne form a rational number p sub k over 10 to the m sub k.

[00:08:47]This is the truncation.

[00:08:49]But notice what comes next in the expansion. Immediately after position m sub k, the digits of the integer 10 to the k are laid down 1 followed by k zeros. So, the next k plus 1 digits are 1 0 0 0 0. After those k plus 1 zeros, the expansion gives the digits of 10 to the k plus 1, which is 2 followed by k zeros. So, a lot of zeros in a row appear right after each integer boundary truncation.

[00:09:18]This means alpha is very close to the truncation. The error alpha minus p sub k over 10 to the m sub k is small, small enough to satisfy a strong rational approximation inequality. Specifically, the error is bounded above by 10 to the negative of some function of k, which is Q sub K to a negative power that grows with K. For any fixed D, there exists K such that K is greater than D.

[00:09:46]So, the approximation rate exceeds any algebraic bound.

[00:09:50]Champernowne is transcendental. Mahler's method generalizes immediately.

[00:09:56]In 1946, Paul Erdős showed that the number obtained by concatenating the squares, 0.149162536496481100, is also normal in base 10. The construction is the same idea applied to a different growing sequence of integers. In the same year, Copeland and Erdős proved that the constant obtained by concatenating the primes, 0.23571113171923, is normal in base 10.

[00:10:31]This is sometimes called the Copeland-Erdős constant. For the concatenation of squares, transcendence is known.

[00:10:39]For the Copeland-Erdős constant, the same Mahler-style argument applies, and transcendence follows from the rapid rational approximations available. The recipe: take any regular enough increasing sequence of positive integers, concatenate their digits, and you get a transcendental number that is normal in some base.

[00:10:59]The construction always works because the truncations at sequence boundaries give very good rational approximations.

[00:11:05]This is satisfying.

[00:11:06]We can construct transcendental normal numbers by hand. We have explicit examples. They are computable, deterministic, predictable in the strongest possible sense, and yet their digits cannot be told apart from genuinely random sequences by any statistical test. After Champernowne and Mahler, we know the following.

[00:11:27]There exists an explicit computable deterministic real number that is normal in base 10.

[00:11:33]That number is transcendental.

[00:11:35]We have its digits. We know its construction. In every measurable statistical sense, this number is random. In every constructive sense, it is the opposite of random.

[00:11:47]This dissolves a folk belief about what randomness means.

[00:11:51]A sequence of digits can be entirely determined by a simple rule and still be statistically indistinguishable from random. The Champernowne construction is the cleanest demonstration.

[00:12:03]The contrast with what we don't know is sharp. Pi, trillions of digits computed, statistical tests give every indication of normality. No proof.

[00:12:13]The Champernowne construction sidesteps this problem entirely. By choosing the construction carefully, we make the normality proof tractable.

[00:12:22]This is what mathematicians often do.

[00:12:25]We cannot answer questions about the numbers we want, pi, e, root 2.

[00:12:30]So, we ask the same questions about numbers we can construct. The constructed numbers turn out to be transcendental, normal, with all the properties we hoped for.

[00:12:40]The natural numbers remain mysterious.

[00:12:43]Champernowne's number is two things at once. It is a tool.

[00:12:47]The first example of a normal number used as a test case in measure theory and ergodic theory ever since.

[00:12:54]It is a lesson, a reminder that randomness in the statistical sense and randomness in the philosophical sense are not the same property. A 20-year-old undergraduate writing a single-page paper gave us both at once.