# How to explain Real Big Numbers?

Mathematicians, and esp. number theorists, are used to working with big numbers. I have noted on several occasions that lots of people don't have a clear understanding of big numbers as far as the real world is concerned. I recall a request for a list of all primes of less than 500 digits.

Another example is homeopathic dilutions. I understand they use dilutions like 200C, which is 1 in $10^{400}$. An absurd number in view of the fact that the total number of particles in the universe is estimated (safe margin) to be less than a googol.

How would you give people insight in big numbers? I'm not talking about Skewes' Number or Graham's Number; for most practical purposes $10^{20}$ is equal to infinity.

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To whoever voted me down: if you vote this down, please also tell me why. Thanks

• Stuff like "real world", (effectiveness of) "homeopathy" and "practical purposes" sounds more like engineering than mathematics so I don't think this is on topic. Although I quite liked this poster. – anon Aug 28 '10 at 11:43
• This question seems too vague as written. I would appreciate a more focused question. – Qiaochu Yuan Aug 28 '10 at 12:18
• If you really want to feel small (in the context of numbers), watch this short. – Tom Stephens Aug 28 '10 at 13:19
• Here is one disconnect I see between laypeople and those with scientific training: for scientists, one merely needs to show the number in scientific notation, and only a look at the exponent is needed to appreciate the magnitude. The lay, on the other hand, needs (or seems to need) to have all the number's digits written in full, zeroes and all, just to grasp how big or tiny a quantity is. – J. M. is a poor mathematician Aug 28 '10 at 15:10
• The sci.math link was very amusing :) – user641 Aug 28 '10 at 15:38

Very few (if any) mathematicians have significant insight regarding huge natural numbers (cf. various ultrafinitism arguments). Perhaps the only exceptions are logicians who work with esoteric ordinal notations. This is one of the few ways one can gain any insight into arbitrarily large numbers - using various complicated inductions to show that some property holds for all naturals - thus lifting our intuition up from small naturals to arbitrarily large naturals. For example, see the Goodstein sequence (or, more graphically: the Hercules vs. Hydra game) which encodes the ordinals below $\epsilon_0 = \omega^{\omega^{\omega^{\cdot^{\cdot^\cdot}}}} \;$ into huge natural numbers.

• +1: In particular, as a working number theorist, I do not often encounter "Real Big Numbers". Usually I am interested in properties that hold for all / all sufficiently large / infinitely many positive integers, but that's not the same. Maybe the OP is thinking of notoriously bad explicit bounds in analytic number theory, but (in my opinion) this arises only because analytic number theorists have a long tradition of being interested in explicit bounds and not for any deeper reason inherent to the subject. – Pete L. Clark Aug 29 '10 at 7:12
• @PeteL.Clark What about the fast growing hierarchy? :-) – Simply Beautiful Art Apr 14 '17 at 22:51

Though I don't quite know that you actually want to hear - what kind of numbers do you want to give people insight in, whom and why? - I'll give a few thoughts.

I) Real cases

That's just understanding of natural sciences - numbers of particles in the universe, number of cells in a body ... Try to first of all break down the number by using smaller parts of the example - e.g. count bacteria in a drop of water and not in a whole lake.

II) Thought experiments (explaining probabilities, complexity etc.)

Extremely big numbers arise when you try to visualize probabilities or complexities, especially when exponential growth is involved. What about getting the jackpot ten times successively or trying to solve a TSP for 100 cities.

When you know people aren't comfortable with that big numbers, decide:

• Is it really important to know the number? Maybe, extremely long or extremely improbable is just the important fact.

• Can you find an easier to grasp example (special units)? Longer than the universe is old is better than insert giant amount of milliseconds.

• Can you describe the growth differently? If your problem with 999 cities can be solved in a certain amount of time and you take one additional city, you'll need 1000 times longer

III) Data

Especially in the context of CS / cryptography, numbers can often most accurately be explained as some data you can calculate with.

E.g. RSA (as in your link) is of course a mathematical, number-based algorithm, but in fact, you're encrypting data, so why not say a 500 char key instead of explaining the giant number involved there.

• IMO your RSA example is exactly the problem: at least some people seem to think they can grasp 10^500. I was wondering how to explain the magnitude of such a number such that they might have an aha-experience (or maybe uh-oh). – stevenvh Aug 28 '10 at 12:57
• For that purpose I like Bruce Schneier's example in his Applied Cryptography that shows that all the energy the sun will put out over its lifetime (not just what hits Earth) is not enough to even count to 2^256. The argument is by its very nature physical rather than mathematical, but it gives a good feel for why bruteforcing e.g. 256-bit AES is completely hopeless (of course cryptanalysis, dictionary attacks and rubberhose cryptanalysis are still very real issues) – kahen Jan 22 '11 at 5:54
• @stevenvh It'll take a human a quadrillion years to crack it if they were doing it by hand? – Mateen Ulhaq Jan 22 '11 at 6:08

I like using real world examples to demonstrate big numbers. One great example is Poincaré's recurrence theorem applied to our Universe.

The volume of our Universe is roughly $4\cdot10^{32}$ light years cubed. Its big.

In laymen's terms, Poincaré's recurrence theorem calculates how much time must past before the quantum state of a system returns to its initial state. Said another way, it is the amount of time until the system repeats itself. Physicist Don Page wrote a paper which he calculated that time applied to our Universe. He estimated Poincaré's recurrence of our Universe would happen in:

$10^{10^{10^{10^{10^{1.1}}}}} years$

This number is ridiculously big. The Universe would sooner experience heat death before we reach Poincaré's recurrence. Heat death is when all energy in the Universe reaches equilibrium. There would be no more thermodynamic free energy to sustain processes that increase entropy. Heat death is estimated to happen in:

$10^{10^{10^{56}}} years.$

One side fact about Poincaré's recurrence is that we can't ever observe it. Hope that helps.

• I don't think this really helps understand big numbers . . . – Noah Schweber Oct 19 '16 at 19:35
• Yeah, after I posted it I realized that it didn't really help. I think the best bet for helping people understand big numbers is taking them through a progression for how to build them. Maybe walking them through the fast growing hierarchy? Successorship, addition, multiplication, exponentiation, tetration, etc.. It does a good job of showing the magnitude of each jump in hyperoperation. You get a sense for the power of the function by the one that came before it. Use something like n=3 for the examples. – Josh Kerr Oct 19 '16 at 19:39