# Math fun: Express a number with specific digits. How about Euler's Number? **URL:** \ **Category:** Miscellaneous and Personal Stuff I Must Share\ **Created:** [May 16, 2018, 9:27am UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273 "2018-05-16T09:27:43Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [May 16, 2018, 9:27am UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/1 "2018-05-16T09:27:43Z")
A popular way to have fun with arithmetic is to find a way to express a number in terms of other numbers. For example, here’s 69 expressed with the digits 1,2,3,4,5 in order: 69 = 1·√2 [sup]3·4[/sup] + 5 I just watched a Numberphile YouTube that demonstrated an impressive equation. Express Euler’s Number (2.7182818…) using each of the digits 1,2,3,4,5,6,7,8,9 one time: e = (1+9[sup]-4[sup]6·7) [sup] (3[sup]2[sup]85[/sup][/sup])[/sup][/sup][/sup] Wait! Didn’t Charles Hermite prove that Euler’s Number is transcendental? This number, however ridiculous, isn’t transcendental. And, sure enough, the “equation” is only an approximation. … **But it approximates _e_ [accurately to 18 trillion _trillion_ digits!](https://curiosity.com/topics/theres-a-formula-that-gives-the-first-18-trillion-trillion-digits-of-the-constant-e-curiosity/)** Isn’t that close enough for you? ---
**Author:** ![Richard\_Pearse](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/richard_pearse/32/13144_2.png) [@Richard\_Pearse](https://boards.straightdope.com/u/Richard_Pearse)\ **Post date:** [May 16, 2018, 10:56am UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/2 "2018-05-16T10:56:14Z")
That is cool. I’ve become a bit of a Numberphile and Mathologer fan lately. ---
**Author:** ![74westy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/74westy/32/3950_2.png) [@74westy](https://boards.straightdope.com/u/74westy)\ **Post date:** [May 16, 2018, 3:43pm UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/3 "2018-05-16T15:43:54Z")
> [@septimus](#): > > sure enough, the “equation” is only an approximation. … > **But it approximates _e_ [accurately to 18 trillion _trillion_ digits!](https://curiosity.com/topics/theres-a-formula-that-gives-the-first-18-trillion-trillion-digits-of-the-constant-e-curiosity/)** Isn’t that close enough for you? Close enough for an Engineer maybe. ---
**Author:** ![Rysto](https://avatars.discourse-cdn.com/v4/letter/r/ecccb3/32.png) [@Rysto](https://boards.straightdope.com/u/Rysto)\ **Post date:** [May 16, 2018, 3:49pm UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/4 "2018-05-16T15:49:18Z")
Does anyone else find the term “trillion trillion” to be really confusing? I suppose it means 10[sup]24[/sup]? (Assuming we’re talking short trillions and not long trillions, _sigh_) ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [May 16, 2018, 4:07pm UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/5 "2018-05-16T16:07:55Z")
> [@74westy](#): > > Close enough for an Engineer maybe. You win the thread. 🙂 ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [May 16, 2018, 10:44pm UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/6 "2018-05-16T22:44:26Z")
> [@Rysto](#): > > Does anyone else find the term “trillion trillion” to be really confusing? I suppose it means 10[sup]24[/sup]? If it’s confusion with numbers you seek, learn Thai! พันพัน /phan phan/ (literally ‘thousand thousand’) means ‘several thousands’, and หมื่นหมื่น /muen muen/ (literally ‘ten-thousand ten-thousand’) means ‘several tens of thousands’, but, since there are no specific words for numbers bigger than a million, ล้านล้าน /la:n la:n/ (literally ‘million million’) can mean _either_ ‘several millions’ or ‘trillion.’ I suppose ambiguity is often avoided since even the brothers who own CP Group (the vast Thai conglomerate which dominates pork and poultry here, owns the telecom giant True, has expanded into China and other countries, and owns the Thailand franchises for KFC, 7-Eleven and Heineken beer) are not quite baht trillionaires yet. > [@Rysto](#): > > (Assuming we’re talking short trillions and not long trillions, _sigh_) Even the Brits have quietly given up on their long trillion. First they lost the Empire where the Sun never set. Then they lost their super-sized billion and trillion. (The quaint notion that a penny represented a penny-weight of silver was lost long ago: in 1464 King Edward IV was minting almost 39 shillings in pennies from a troy pound of silver, in contrast to the 31-shilling pound of his predecessor.) ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [May 17, 2018, 1:42am UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/7 "2018-05-17T01:42:58Z")
I’m dubious that e has been calculated to a trillion trillion digits. The last I’d heard (2010) I believe the 500 billionth digit was determined. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [May 17, 2018, 2:03am UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/8 "2018-05-17T02:03:09Z")
There must be a way to estimate the error. Using a calculator just now I see that (1 + 10^-100)^(10^100) agrees with e to a hundred decimal digits and (1 + 10^-200)^(10^200) equals e to 200 decimal digits. Apparently this extrapolates in the obvious way, because it appears 3^2^85 ~= 10^{18.4 trillion trillion}. What makes the weird expression work, of course, is that 9^4^(6\*7) = 3^2^85 exactly. ---
**Author:** ![Mr\_Zox](https://avatars.discourse-cdn.com/v4/letter/m/9e8a1a/32.png) [@Mr\_Zox](https://boards.straightdope.com/u/Mr_Zox)\ **Post date:** [May 17, 2018, 2:26am UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/9 "2018-05-17T02:26:19Z")
153 = 1^3 + 5^3 + 3^3. It is the smallest number that is equal to the sum of the cubes of its digits. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [May 17, 2018, 2:35am UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/10 "2018-05-17T02:35:06Z")
> [@septimus](#): > > There must be a way to estimate the error. Using a calculator just now I see that (1 + 10^-100)^(10^100) agrees with e to a hundred decimal digits and (1 + 10^-200)^(10^200) equals e to 200 decimal digits. Apparently this extrapolates in the obvious way, because it appears 3^2^85 ~= 10^{18.4 trillion trillion}. Now I’m going to be thinking about this all night. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [May 17, 2018, 2:37am UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/11 "2018-05-17T02:37:17Z")
Yeah, they didn’t calculate both to 10^24 digits and compare them-- Even if we could calculate either that precisely, the comparison alone would take far too long. They calculated the error directly. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [May 17, 2018, 2:47am UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/12 "2018-05-17T02:47:56Z")
> [@Andy\_L](#): > > Now I’m going to be thinking about this all night. I’ve got a notion on how to estimate the error - I’ll do the math tomorrow morning (if no one else has done it before). ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [May 17, 2018, 3:39am UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/13 "2018-05-17T03:39:13Z")
e - (1 + 1/n)[sup]n[/sup] = e(1/2n - 11/24n[sup]2[/sup] + O(n[sup]-3[/sup])) ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [May 17, 2018, 4:15am UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/14 "2018-05-17T04:15:44Z")
> [@Andy\_L](#): > > Now I’m going to be thinking about this all night. I used to have that problem; even Lorazepam didn’t put me to sleep. But I found a way to solve the problem! I moved my laptop _outside_ our air-con bedroom; the last two (hot) evenings I’ve lain down for a brief rest in the air-con room … and woken up eight hours later! 🙂 ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [May 17, 2018, 12:45pm UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/15 "2018-05-17T12:45:09Z")
> [@DPRK](#): > > e - (1 + 1/n)[sup]n[/sup] = e(1/2n - 11/24n[sup]2[/sup] + O(n[sup]-3[/sup])) Thanks! ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [May 17, 2018, 12:48pm UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/16 "2018-05-17T12:48:36Z")
> [@Mr\_Zox](#): > > 153 = 1^3 + 5^3 + 3^3. > > It is the smallest number that is equal to the sum of the cubes of its digits. As opposed to 1? ETA OK, it’s the second-smallest. Then comes 370. ---
**Author:** ![John\_DiFool](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_difool/32/19543_2.png) [@John\_DiFool](https://boards.straightdope.com/u/John_DiFool)\ **Post date:** [May 17, 2018, 1:27pm UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/17 "2018-05-17T13:27:57Z")
> [@septimus](#): > > A popular way to have fun with arithmetic is to find a way to express a number in terms of other numbers. For example, here’s 69 expressed with the digits 1,2,3,4,5 in order: > 69 = 1·√2 [sup]3·4[/sup] + 5 So, if walk up to a female math PhD, and ask her, “Hey baby, want to 1·√2 [sup]3·4[/sup] + 5?” She’ll know what I mean? ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [May 17, 2018, 2:44pm UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/18 "2018-05-17T14:44:26Z")
> [@DPRK](#): > > e - (1 + 1/n)[sup]n[/sup] = e(1/2n - 11/24n[sup]2[/sup] + O(n[sup]-3[/sup])) So the number of digits in n is proportional to the number of correct digits in the expression - which is consistent with what septimus was seeing. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [May 17, 2018, 4:26pm UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/19 "2018-05-17T16:26:47Z")
Nitpick: Since the digits of e are unknown beyond the first trillion or so, it is possible that, at the key point, e = 2.7…400000000000000000000000000…0001… where there are a septillion or more consecutive zeros, with the approximation in OP then having a septillion consecutive nines: 2.7…39999999999999999… In such a low-probability scenario, the approximation would be correct to only 17 trillion trillion digits instead of 18 trillion trillion. [/nitpick] ---
**Author:** ![octopus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/octopus/32/3716_2.png) [@octopus](https://boards.straightdope.com/u/octopus)\ **Post date:** [May 17, 2018, 4:44pm UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/20 "2018-05-17T16:44:03Z")
> [@DPRK](#): > > As opposed to 1? > > ETA OK, it’s the second-smallest. Then comes 370. What about 0? [Next page](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273.md?page=2)