# What Is Adding the Digits of a Number Called? **URL:** \ **Category:** Factual Questions\ **Created:** [June 11, 2004, 2:59am UTC](https://boards.straightdope.com/t/what-is-adding-the-digits-of-a-number-called/249763 "2004-06-11T02:59:44Z")\ **Posts on this page:** 9\ **Page:** 1
**Author:** ![Jim\_B](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jim_b/32/4338_2.png) [@Jim\_B](https://boards.straightdope.com/u/Jim_B)\ **Post date:** [June 11, 2004, 2:59am UTC](https://boards.straightdope.com/t/what-is-adding-the-digits-of-a-number-called/249763/1 "2004-06-11T02:59:44Z")
1, 341, 594, 999, 225, 585, 315, 963, 396, 882 The above number is **over an octillion** , yet I know that it is _exactly divisible by 9_. How do I know this? Easy. When you add the digits together, starting with the first 4 and then each group of 3 after that, you get 9+9+9+… And by random experiments I have conducted with calculators (and something I once had in grade school math IIRC), I know that when adding digits together, and you get all 9s, the _whole number itself must be divisible by 9_. (Another interesting application of this technique seems to happen when you add the digits of a number and get 3, thus showing it must be perfectly divisible by 3. E.g., 12, 201, 102, etc. are all evenly divisible by 3…) Anyways, this is not MPSIMS, and my thread was not intended for MPSIMS. I simply want to know if there is a mathematical term for this: adding digits of a number together. And if there is a mathematical word or phrase for it, what is it? As you can see, I have known of this phenomenon since I was a child. But I never knew (or perhaps forgot) what the technical name for it is. 🙂 ---
**Author:** ![Pasta](https://avatars.discourse-cdn.com/v4/letter/p/ecccb3/32.png) [@Pasta](https://boards.straightdope.com/u/Pasta)\ **Post date:** [June 11, 2004, 4:32am UTC](https://boards.straightdope.com/t/what-is-adding-the-digits-of-a-number-called/249763/2 "2004-06-11T04:32:37Z")
A “digit sum”. I’m not aware of any fancier name. ---
**Author:** ![wolf\_meister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolf_meister/32/15202_2.png) [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)\ **Post date:** [June 11, 2004, 5:13am UTC](https://boards.straightdope.com/t/what-is-adding-the-digits-of-a-number-called/249763/3 "2004-06-11T05:13:52Z")
I’ve heard the expression “finding the digital root”. ---
**Author:** ![tim314](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tim314/32/3468_2.png) [@tim314](https://boards.straightdope.com/u/tim314)\ **Post date:** [June 11, 2004, 6:07am UTC](https://boards.straightdope.com/t/what-is-adding-the-digits-of-a-number-called/249763/4 "2004-06-11T06:07:25Z")
I’ve also only heard “digit sum.” Or the technical term “adding up the digits.” 😃 Let me see if I can give a better proof for what you’re saying than “random experiments with calculators.” Suppose you have a number X who’s digits are x0, x1, x2, . . . where xn is the digit in the 10^n place. (e.g. 145 has x0 = 5, x1 = 4, x2 = 1, and all others equal to zero.) We can write X = x0 \* 10^0 + x1 \* 10^1 + x2 \* 10^2 + . . . 10^n can be written as 1 + (an n-digit number made of only nines). e.g. 100 = 1 + 99 Thus, X = x0 \* (1) + x1 \* (9+1) + x2 \* (99+1) + . . . Rearranging this, we have X = A + B, where A = x0 + x1 + x2 + x3 + . . . B = x1_9 + x2_99 + x3\*999 + . . . Clearly B is divisible by 9 (since 9 = 1_9, 99 = 11_ 9, etc. and thus B is a sum of numbers divisible by 9). Thus, if A is divisible by 9 as well then we know X is divisible by 9 (since it’s a sum of numbers divisible by 9). But A is just the sum of the digits, so this proves that if the sum of the digits is divisible by 9, then the number itself is divisible by 9 as well. (Note that B is also divisible by 3, so if A is divisible by 3 then X must be as well.) ---
**Author:** ![chaoticbear](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@chaoticbear](https://boards.straightdope.com/u/chaoticbear)\ **Post date:** [June 11, 2004, 12:21pm UTC](https://boards.straightdope.com/t/what-is-adding-the-digits-of-a-number-called/249763/5 "2004-06-11T12:21:32Z")
> [@wolf\_meister](#): > > I’ve heard the expression “finding the digital root”. Yup, go with that… we had a Knowledge Master question with this a while back and we all stared at the screen like 😕… We went and looked it up later lol. ---
**Author:** ![chrisk](https://avatars.discourse-cdn.com/v4/letter/c/6de8d8/32.png) [@chrisk](https://boards.straightdope.com/u/chrisk)\ **Post date:** [June 11, 2004, 12:41pm UTC](https://boards.straightdope.com/t/what-is-adding-the-digits-of-a-number-called/249763/6 "2004-06-11T12:41:06Z")
I know that in his scientific american articles, Martin Gardner has used the term ‘casting out nines’ in reference to this procedure, or maybe more properly its result. I still like the term though and have used it to describe the process because it sounds so cool and old-fashioned mathematicky. 😉 ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 11, 2004, 4:00pm UTC](https://boards.straightdope.com/t/what-is-adding-the-digits-of-a-number-called/249763/7 "2004-06-11T16:00:23Z")
In base n, this trick works for divisibility by n - 1, or any number that divides n - 1. But there’s no fancy name for it. ---
**Author:** ![wolf\_meister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolf_meister/32/15202_2.png) [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)\ **Post date:** [June 11, 2004, 8:11pm UTC](https://boards.straightdope.com/t/what-is-adding-the-digits-of-a-number-called/249763/8 "2004-06-11T20:11:39Z")
Go to this link: [http://www.cut-the-knot.org/arithmetic/rapid/defensive.shtml](http://www.cut-the-knot.org/arithmetic/rapid/defensive.shtml) go to #4 and I think the term “ **digital root** ” is just what Jim B is describing. ---
**Author:** ![Schnitte](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/schnitte/32/9033_2.png) [@Schnitte](https://boards.straightdope.com/u/Schnitte)\ **Post date:** [June 12, 2004, 3:46pm UTC](https://boards.straightdope.com/t/what-is-adding-the-digits-of-a-number-called/249763/9 "2004-06-12T15:46:45Z")
I looked up the German term for it (_Quersumme_) in a dictionary and it gave me the terms “cross sum” and “cross total.”