# 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:** 9\ **Page:** 2
**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, 5:10pm UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/21 "2018-05-17T17:10:18Z")
> [@octopus](#): > > What about 0? What about -370 ? (Minus 370) Or -407 for that matter. ---
**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, 7:09pm UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/22 "2018-05-17T19:09:52Z")
> [@septimus](#): > > What about -370 ? (Minus 370) Or -407 for that matter. I thought about the negative numbers but the minus sign isn’t a digit is it? So when you cube the digits and then sum them you aren’t cubing a minus sign each time correct? So cubing the digits of -370 is still 3^3, 7^3, and 0^3 correct? Not -3^3, -7^3, and 0^3. ---
**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, 9:30pm UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/23 "2018-05-17T21:30:55Z")
Mea culpa iterum. ---
**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, 10:47pm UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/24 "2018-05-17T22:47:47Z")
> [@DPRK](#): > > e - (1 + 1/n)[sup]n[/sup] = e(1/2n - 11/24n[sup]2[/sup] + O(n[sup]-3[/sup])) I’m getting something closer to 1/2n - 11/24n[sup]2[/sup] + O(n[sup]-3[/sup]) - is the e needed there? ---
**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, 11:30pm UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/25 "2018-05-17T23:30:44Z")
> [@Andy\_L](#): > > I’m getting something closer to 1/2n - 11/24n[sup]2[/sup] + O(n[sup]-3[/sup]) - is the e needed there? Short answer: yes. Here are a few more terms: e(1/2n - 11/24n² + 7/16n³ - 2447/5760n⁴ + …) Let n = 100. Then e - (1 + 1/n)ⁿ = 0.01346799903751914… while 1/2n - 11/24n² + 7/16n³ = 0.00495460416… (just shy of 1/200), while e(1/2n - 11/24n² + 7/16n³) = 0.01346801047… A good exercise is, use your preferred method (e.g., binomial theorem, Taylor series, some tricks with limits, whatever) to calculate just the order-1/n term. ---
**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, 11:49pm UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/26 "2018-05-17T23:49:44Z")
> [@DPRK](#): > > Short answer: yes. > > Here are a few more terms: > > e(1/2n - 11/24n² + 7/16n³ - 2447/5760n⁴ + …) > > Let n = 100. Then e - (1 + 1/n)ⁿ = 0.01346799903751914… > while 1/2n - 11/24n² + 7/16n³ = 0.00495460416… (just shy of 1/200), while > e(1/2n - 11/24n² + 7/16n³) = 0.01346801047… > > A good exercise is, use your preferred method (e.g., binomial theorem, Taylor series, some tricks with limits, whatever) to calculate just the order-1/n term. Thanks. I think I see what I was missing (I saw something that looked like it was working without the e, without carrying the calculation far enough). ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [May 18, 2018, 12:01am UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/27 "2018-05-18T00:01:49Z")
Also, perhaps think about exp(1 + Bx + Cx² + O(x³)) = e(1 + Bx + ((B² + 2C)/2) x² + …) [I don’t recall the name of this formula, at least in this form; please remind me!], so starting from (log(1+u))/u = 1 - u/2 + u²/3 - …, you get (1+u)[sup]1/u[/sup] = e(1 - u/2 + 11u²/24 - …); either way, the “e” should be there (consider the order-0 term!) ---
**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 18, 2018, 12:08am UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/28 "2018-05-18T00:08:56Z")
> [@DPRK](#): > > Also, perhaps think about exp(1 + Bx + Cx² + O(x³)) = e(1 + Bx + ((B² + 2C)/2) x² + …) [I don’t recall the name of this formula, at least in this form; please remind me!], so starting from (log(1+u))/u = 1 - u/2 + u²/3 - …, you get (1+u)[sup]1/u[/sup] = e(1 - u/2 + 11u²/24 - …); either way, the “e” should be there (consider the order-0 term!) Thanks. ---
**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 18, 2018, 12:11am UTC](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273/29 "2018-05-18T00:11:26Z")
> [@DPRK](#): > > Short answer: yes. > > Here are a few more terms: > > e(1/2n - 11/24n² + 7/16n³ - 2447/5760n⁴ + …) > > Let n = 100. Then e - (1 + 1/n)ⁿ = 0.01346799903751914… > while 1/2n - 11/24n² + 7/16n³ = 0.00495460416… (just shy of 1/200), while > e(1/2n - 11/24n² + 7/16n³) = 0.01346801047… > > A good exercise is, use your preferred method (e.g., binomial theorem, Taylor series, some tricks with limits, whatever) to calculate just the order-1/n term. My mistake was to think that I could get the order 1/n term with just the first few terms of the binomial expansion. (and maybe some algebra mistakes too) [Previous page](https://boards.straightdope.com/t/math-fun-express-a-number-with-specific-digits-how-about-eulers-number/814273.md?page=1)