# super simple math equation I forgot **URL:** \ **Category:** Factual Questions\ **Created:** [January 2, 2008, 3:52am UTC](https://boards.straightdope.com/t/super-simple-math-equation-i-forgot/431965 "2008-01-02T03:52:09Z")\ **Posts on this page:** 13\ **Page:** 1
**Author:** ![filmyak](https://avatars.discourse-cdn.com/v4/letter/f/94ad74/32.png) [@filmyak](https://boards.straightdope.com/u/filmyak)\ **Post date:** [January 2, 2008, 3:52am UTC](https://boards.straightdope.com/t/super-simple-math-equation-i-forgot/431965/1 "2008-01-02T03:52:09Z")
Sorry, been a long time since high school and I can’t remember this simple equation. I need to double the number one 64 times. In other words, 1x2 = 4. x2 = 8. x2 = 16. If I do that 64 times, what’s the total? And what’s the equation to figure it out? ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [January 2, 2008, 3:56am UTC](https://boards.straightdope.com/t/super-simple-math-equation-i-forgot/431965/2 "2008-01-02T03:56:35Z")
Um, 1 x 2 = 2. What you’re looking for is 2[sup]63[/sup]. 2[sup]0[/sup] = 1. 2[sup]1[/sup] = 2. 2[sup]2[/sup] = 4. 2[sup]63[/sup] is the 64th doubling. 9.22337204 × 10[sup]18[/sup] ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [January 2, 2008, 3:58am UTC](https://boards.straightdope.com/t/super-simple-math-equation-i-forgot/431965/3 "2008-01-02T03:58:02Z")
[QUOTE=filmyak] Sorry, been a long time since high school and I can’t remember this simple equation. I need to double the number one 64 times. In other words, 1x2 = 4. x2 = 8. x2 = 16. If I do that 64 times, what’s the total? And what’s the equation to figure it out? [/QUOTE] 2^64 … and 1x2 should be “2” in your example. ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [January 2, 2008, 3:59am UTC](https://boards.straightdope.com/t/super-simple-math-equation-i-forgot/431965/4 "2008-01-02T03:59:38Z")
[QUOTE=Exapno Mapcase] What you’re looking for is 2[sup]63[/sup]. 2[sup]0[/sup] = 1. 2[sup]1[/sup] = 2. 2[sup]2[/sup] = 4. 2[sup]63[/sup] is the 64th doubling. [/quote] 2[sup]64[/sup] would be the 64th doubling of 1. The first doubling is 1x2 =2, which is 2[sup]1[/sup]. ---
**Author:** ![totoismomo](https://avatars.discourse-cdn.com/v4/letter/t/59ef9b/32.png) [@totoismomo](https://boards.straightdope.com/u/totoismomo)\ **Post date:** [January 2, 2008, 3:59am UTC](https://boards.straightdope.com/t/super-simple-math-equation-i-forgot/431965/5 "2008-01-02T03:59:41Z")
Isn’t it 2[sup]64[/sup] = 18446744073709551616? First doubling is 2, second doubling is 4, so the nth doubling should be 2[sup]n[/sup]. ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [January 2, 2008, 4:01am UTC](https://boards.straightdope.com/t/super-simple-math-equation-i-forgot/431965/6 "2008-01-02T04:01:49Z")
[QUOTE=Exapno Mapcase] Um, 1 x 2 = 2. [/quote] Give him a break, it’s been even longer since elementary school. If by “total” you mean all the numbers added together, as in the old story about a man who wanted one grain of rice for the first square on a chessboard, two for the second, four for the third, etc. the total of 2[sup]0[/sup] + 2[sup]1[/sup] + 2[sup]2[/sup]… + 2[sup]63[/sup] = 2[sup]64[/sup] - 1 ---
**Author:** ![audiobottle](https://avatars.discourse-cdn.com/v4/letter/a/bc8723/32.png) [@audiobottle](https://boards.straightdope.com/u/audiobottle)\ **Post date:** [January 2, 2008, 4:15am UTC](https://boards.straightdope.com/t/super-simple-math-equation-i-forgot/431965/7 "2008-01-02T04:15:30Z")
[QUOTE=Bryan Ekers] If by “total” you mean all the numbers added together, as in the old story about a man who wanted one grain of rice for the first square on a chessboard, two for the second, four for the third, etc. the total of 2[sup]0[/sup] + 2[sup]1[/sup] + 2[sup]2[/sup]… + 2[sup]63[/sup] = 2[sup]64[/sup] - 1 [/QUOTE] Is that equation at the end correct? It seem somewhat counter-intuitive. ---
**Author:** ![filmyak](https://avatars.discourse-cdn.com/v4/letter/f/94ad74/32.png) [@filmyak](https://boards.straightdope.com/u/filmyak)\ **Post date:** [January 2, 2008, 4:16am UTC](https://boards.straightdope.com/t/super-simple-math-equation-i-forgot/431965/8 "2008-01-02T04:16:37Z")
[QUOTE=Bryan Ekers] Give him a break, it’s been even longer since elementary school. [/quote] LOL oops. Does it help if I blame a screaming baby who refuses to go to sleep in the next room? No? Oh well. > [@](#): > > If by “total” you mean all the numbers added together, as in the old story about a man who wanted one grain of rice for the first square on a chessboard, two for the second, four for the third, etc. the total of 2[sup]0[/sup] + 2[sup]1[/sup] + 2[sup]2[/sup]… + 2[sup]63[/sup] = 2[sup]64[/sup] - 1 Exactly what I am looking for! ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [January 2, 2008, 4:17am UTC](https://boards.straightdope.com/t/super-simple-math-equation-i-forgot/431965/9 "2008-01-02T04:17:11Z")
[QUOTE=audiobottle] Is that equation at the end correct? It seem somewhat counter-intuitive. [/QUOTE] Yes it is. For example, try it on four squares: 2^0+2^1+2^2+2^3 = 1+2+4+8 = 15, which is 2^4-1. ---
**Author:** ![SCSimmons](https://avatars.discourse-cdn.com/v4/letter/s/e495f1/32.png) [@SCSimmons](https://boards.straightdope.com/u/SCSimmons)\ **Post date:** [January 2, 2008, 6:29am UTC](https://boards.straightdope.com/t/super-simple-math-equation-i-forgot/431965/10 "2008-01-02T06:29:07Z")
[QUOTE=pulykamell] Yes it is. For example, try it on four squares: 2^0+2^1+2^2+2^3 = 1+2+4+8 = 15, which is 2^4-1. [/QUOTE] Well, the formula must be right then! :rolleyes: Maybe an actual proof will help the doubters? 1. 2^0 = 1 = 2-1 = 2^1-1 2. Suppose that for some n, 2^0 + 2^1 + … + 2^n = 2^(n+1) - 1 Then 2^0 + 2^1 + … + 2^n + 2^(n+1) = 2^(n+1) - 1 + 2^(n+1) = 2\*2(n+1)-1 =2(n+2)-1 3. Therefore, by induction, 2^0 + 2^1 + … + 2^n = 2^(n+1)-1 for all n. ---
**Author:** ![Malacandra](https://avatars.discourse-cdn.com/v4/letter/m/45deac/32.png) [@Malacandra](https://boards.straightdope.com/u/Malacandra)\ **Post date:** [January 2, 2008, 10:20am UTC](https://boards.straightdope.com/t/super-simple-math-equation-i-forgot/431965/11 "2008-01-02T10:20:36Z")
[QUOTE=audiobottle] Is that equation at the end correct? It seem somewhat counter-intuitive. [/QUOTE] Picture it in binary. You have a string of 64 consecutive 1s from 2[sup]0[/sup] all the way to 2[sup]63[/sup]. If you then added 1 to this you would flip every single bit as you carried 1 all the way from the right-hand end to the left, and a new 1 would appear at the left: 2[sup]64[/sup]. ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [January 2, 2008, 5:33pm UTC](https://boards.straightdope.com/t/super-simple-math-equation-i-forgot/431965/12 "2008-01-02T17:33:58Z")
[QUOTE=SCSimmons] Well, the formula must be right then! :rolleyes: [/QUOTE] Did I ever claim I had a proof? :rolleyes: Personally, I thought of the answer in Malacandra’s binary way, but I didn’t think that constitutes a proof, either, so I didn’t bother typing it. ---
**Author:** ![rowrrbazzle](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rowrrbazzle/32/414_2.png) [@rowrrbazzle](https://boards.straightdope.com/u/rowrrbazzle)\ **Post date:** [January 2, 2008, 7:58pm UTC](https://boards.straightdope.com/t/super-simple-math-equation-i-forgot/431965/13 "2008-01-02T19:58:49Z")
Without induction: ```auto x = 2^0 + 2^1 + ... + 2^(n-1) + 2^n 2x = 2^1 + 2^2 + ... + 2^n + 2^(n+1) 2x = x - 1 + 2^(n+1) x = - 1 + 2^(n+1) 2^0 + 2^1 + ... + 2^(n-1) + 2^n = 2^(n+1) - 1 ```