A number NOBODY has thought of - Numberphile

More links & stuff in full description below ↓↓↓

Order Tony's book Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity

NUMBERPHILE

Video by Brady Haran and Pete McPartlan

[RickSanchez-dv]

A ten digit number could just be somebody's phone number

[mphayes]

I think Tony mis-stated the question and I think that's why Brady was so confused. So he stated that if you pick a number larger than 10^67, there is a 99% chance that it has never been thought of before. But in the mathematics, he then shows that there is a 99% chance that NOT A SINGLE number above 10^67 has EVER been thought of before when thinking of numbers according to that 1/n^1.3 distribution. That's a big difference. The 99% isn't a probability for that exact number, it's the probability that every number ever thought of following that distribution is less than that number.

At 10:58 he states this. But then he says "so if you go farther, there's a 99% chance you'll find a number that's never been thought of before." But really what the math means is "if you go farther, there's a 99% chance that you'll never again come across a number that has been thought of by following the distribution"

[dumnor]

I like that Brady doesn't just accept 99% probability at face value.

[trentgraham]

Seems like there is a really big difference between "no one has ever thought of this number" and "all the numbers that have ever been thought of are less than this number". I think the latter is an amazingly loose bound on the former.

[alansmithee]

5:35
But then the question you asked at the start is different to the one you just answered.
Start: "How big a number do you have to generate for it to be likely to be *different to* any other number ever thought of?"
Answered: "How big a number do you have to generate for it to be likely to be *bigger than* any other number ever thought of, ignoring anomalous occurrences?"

[stephenandrusyszyn]

10^67 is a huge over estimate. Consider a 30 digit number. To have a greater than 1% chance of repeating a 30 digit number, then more than 10^28 of those 10^30 numbers would have to been "thought of". With a total population of 10^11 people, that would mean that every person would have to have thought of 10^17 30 digit numbers in their lifetime. So if everyone lived for 80 years, then you would have to come up with 40 million 30 digit numbers every second of your life (60 million if you want some sleep).

[gtziavelis]

the shape in the thought bubble at 4:16 is a Calabi Yau manifold. some physics theories postulate at least 10^500 different ones of those (a number with 501 digits).

[Hyproxious]

For those that wondering, the spike on the graph at 7:09 is 2004, the year the data was gathered. The reason 2003 isn't as high, is many pages are updated to the current year

[lucromel]

I agree with Brady.

If picking a random 67 digit number gives a 99% chance of having a new number, doesn't that imply that 1% of all 67 digits numbers have already been thought of? Which would be many MANY orders of magnitude more numbers than humanity has ever thought of...

[EpicGamerScout]

I think I've solved this in a fairly reasonable method, starting from the same assumptions and formula that Tony used. I might be slightly bodging the explanation but this should explain the ideas for anyone that wants to quickly replicate my math:
With N being his 1.5e18 estimate, and p(x) calculating the percentage of numbers above x
And 'amount of numbers' meaning some defined portion of N because I don't want to write 'amount of numbers thought of by humanity' every time

Calculating the amount of numbers BELOW some N* is simply the total N minus the fraction of N above N*, so N-(p(10)*N)
So below 10 would be N-(p(10)*N)

With the ability to calculate the number above and below some range of N*, you can then get the amount of numbers in that range by subtracting the number below the minimum and the number above the maximum from the original N
Or Calculate the number above the minimum, and subtract the amount above the maximum. This is how I actually implemented it in the test spreadsheet.

And the 'normal size' of any range its maximum-minimum.

So I think it's fair to propose that you can roughly estimate the likeliness of a random number in some range being unique by simply dividing the 'amount of numbers' in a range into the 'size' of that range. Of course to be REALLY precise about this you'd want to do some further stats to account for the birthday-paradox type errors that my simple estimation leaves in.

Using a spreadsheet to test these formulas on power of 10 ranges(0-10, 10-100, 100-1000, etc) gives fairly intuitive results:
For a 15 digit number the amount of numbers that land there is 47.2 trillion, and the size of that range is 900 trillion, so you'd be at ~5% odds that your number has never occurred in that N dataset before.
For a 16 digit number the amount of numbers that land there is 23 trillion, and the size of that range is 9 quadrillion, so you'd be at ~.2% odds that your number has never occurred in that N dataset before.
17 digit - .013%
20 digit -


So Brady's 10 digits is definitely an underestimation, especially considering other commenter's examples such as ip addresses and phone numbers. But you definitely don't have to go too much further to reach near-certainty even from these assumptions.

[cruxofthecookie]

2:38 The shadow moves across the Earth is rotating in the wrong direction! But correct at 3:50 and subsequently.

[prhobo]

im with brady on this one. I think hes tricking himself with his own math and confirmation bias. Brady's argument about 66 digit numbers makes sense if you instead say how many numbers of length 66 have been thought of? which surely is waay less than a majority so you should have good odds of picking a random number that if within that set. not to mention decimals.

[JouvaMoufette]

Everybody gets one MASSIVE thing wrong about life expectancy of the past: They don't take into account that the average was SKEWED.

The way we calculate life expectancy is just a mean average. But when you have a high rate of infantile deaths, the average age gets WAY skewed towards lower numbers. If you made it into adult hood, you'd probably be making it into your 60s. But the "life expectancy" would be 40 because SO MANY children died at young ages compared to now.

[mathmachine]

There's a huge gaping problem with Tony's assumptions: he's trying to find how large a number must be such that there's only a 1% chance humanity has thought of a larger number. That is, the cumulative distribution function up to that point is 0.99. The original problem, however, wasn't that no human has thought of a bigger number. It's that no human has ever thought of that EXACT SPECIFIC number. He needs to evaluate the probability mass function. He needs to find a number such that the probability of that number is no more than 1%, and the probability of any individual number after it is no more than 1%. Not combined probabilities, mind you, individual probabilities.

[mebamme]

This kind of reminds me of a question I keep thinking about: what's the largest known prime consecutively (i.e. where we know all primes up to that point)?
Sure, there's all the Mersenne primes with millions of digits, but that's skipping a lot of primes inbetween.

[overestimatedforesight]

The Doomsday argument always makes me laugh. Imagine a particularly intelligent cro-magnon figuring it out and concluding that humans will go extinct in the next few hundred years. It's basically "well if we're roughly in the middle then we must be roughly in the middle."

[martinkarsai]

The fact that there is a 99 percent chance that nobody has ever thought of a bigger number doesn't matter. According to his assumptions if the humanity has thought of 10^17 numbers then the chance you think of a unique number is huge very quickly. Let's say you pick an 18 digit number. Even if every number ever thought of had 18 digits that would still be only 1 tenth of all the 18 digit numbers so you already have at least 90% chance. If you choose 19 digits than it's at least 99%. In reality it's likely much higher as they mentioned as well most numbers thought of are tiny.

[theprofessionalfence-sitter]

I'm sceptical of his reasoning. There is no point in looking for some number that, with some probability, is larger than every number ever thought of (which is easily invalidated, anyway, by a single person thinking of a larger number). It is, indeed, far simpler: Take a (uniformly) random number less than or equal to 1.5*10^20. Even if all numbers ever thought of are distinct and less than 1.5*10^20 and his ridiculous estimate of 1.5*10^18 is anywhere near accurate, there are only those 1.5*10^18 numbers below 1.5*10^20 that have ever been chosen, making the probability that yours is one of them only about 1.5*10^18/(1.5*10^20)=1%. In fact, you can likely go far lower, yet: you really just need to find some n, such that there are at most n*(1-sqrt(0.99)) distinct random numbers less than or equal to n with probability at least sqrt(0.99) which is really awful to actually work out so I won't even attempt to do so, here.

[xyz.ijk.]

For encryption purposes, we use super large numbers — primes! - all the time! But I suppose no one will think of them if they are larger than 10^73.

[agentdarkboote]

Wait a second! He's calculating something different though. He's looking at the probability that nobody has thought of a LARGER number, NOT that the number is unique. Notice that his probability equation he uses is to calculate "what's the probability that a randomly chosen number from this distribution is LOWER than the number n* ?" (See timestamp 7:39 )

And then his number clearly fails, because of all of the obvious examples... Tree(3), Graham's number, etc.

For unique, you need to look at the finite differences of the CDF, and calculate (1-FDCDF)^ (1.5*10^18) >= 0.99. I get about 1.16*10^28 which seems a lot more reasonable. Still a huge overestimate from going on his assumptions of how many numbers have been considered though. I'm sure you could confidently bring it down a few more orders of magnitude.