# What is Discrete Mathematics? **URL:** \ **Category:** Factual Questions\ **Created:** [February 26, 2003, 2:23pm UTC](https://boards.straightdope.com/t/what-is-discrete-mathematics/157631 "2003-02-26T14:23:21Z")\ **Posts on this page:** 8\ **Page:** 2
**Author:** ![Shalmanese](https://avatars.discourse-cdn.com/v4/letter/s/45deac/32.png) [@Shalmanese](https://boards.straightdope.com/u/Shalmanese)\ **Post date:** [February 28, 2003, 1:01am UTC](https://boards.straightdope.com/t/what-is-discrete-mathematics/157631/21 "2003-02-28T01:01:23Z")
A bit part of discrete is working with modular arithmetic. ie, for working in mod n, you have 0+1 = 1, 1+1 = 2… n-1+1 = n, n+1 = 0. This is especially useful in computing as computers can only store integers mod n, not the entire number line. ---
**Author:** ![Derleth](https://avatars.discourse-cdn.com/v4/letter/d/b9e5f3/32.png) [@Derleth](https://boards.straightdope.com/u/Derleth)\ **Post date:** [February 28, 2003, 1:29am UTC](https://boards.straightdope.com/t/what-is-discrete-mathematics/157631/22 "2003-02-28T01:29:28Z")
**Master of Ceremonies** : It’s spelled `kudos’. [Kudos at Dictionary.com](http://dictionary.reference.com/search?q=kudos&r=67) > [@](#): > > Usage Note: Kudos is one of those words like congeries that look like plurals but are etymologically singular. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [February 28, 2003, 1:45am UTC](https://boards.straightdope.com/t/what-is-discrete-mathematics/157631/23 "2003-02-28T01:45:01Z")
> [@](#): > > \*Originally posted by Shalmanese \* > \*\*A bit part of discrete is working with modular arithmetic. ie, for working in mod n, you have 0+1 = 1, 1+1 = 2… n-1+1 = n, n+1 = 0. This is especially useful in computing as computers can only store integers mod n, not the entire number line. \*\* Minor nitpick: Working mod n, you only have 0 through n - 1. ---
**Author:** ![iamthewalrus\_3](https://avatars.discourse-cdn.com/v4/letter/i/258eb7/32.png) [@iamthewalrus\_3](https://boards.straightdope.com/u/iamthewalrus_3)\ **Post date:** [February 28, 2003, 4:37am UTC](https://boards.straightdope.com/t/what-is-discrete-mathematics/157631/24 "2003-02-28T04:37:13Z")
> [@](#): > > _Originally posted by Achernar_ > There’s even a discrete calculus! Although, I don’t know how useful it is… It’s incredibly useful to those of us who are trying to analyze algorithms and need to find a closed form for some series. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [February 28, 2003, 4:53am UTC](https://boards.straightdope.com/t/what-is-discrete-mathematics/157631/25 "2003-02-28T04:53:25Z")
This brings to mind an interesting question. Integration is the inverse of differentiation. Summation is (more or less) the inverse of the finite difference operator. First off, is there an analogue of the fundamental theorem of calculus using the discrete operations? Secondly, as we all know, summation is a special case of (Lebesgue) integration. Is there some general theory that absorbs traditional derivatives and finite differences? ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [February 28, 2003, 2:14pm UTC](https://boards.straightdope.com/t/what-is-discrete-mathematics/157631/26 "2003-02-28T14:14:17Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\*This brings to mind an interesting question. > > Integration is the inverse of differentiation. Summation is (more or less) the inverse of the finite difference operator. > > First off, is there an analogue of the fundamental theorem of calculus using the discrete operations? > > Secondly, as we all know, summation is a special case of (Lebesgue) integration. Is there some general theory that absorbs traditional derivatives and finite differences? \*\* Of course there is a fundamental theorem for finite differences and sums. If s\_n = sum {a\_i : 0 LE i LE n}, then obviously (Delta s)\_n = a\_n. This is too trivial to be worth calling a theorem. Now you probably meant the other version of the fundamental theorem and that is also valid. As for the general theory, the answer there is also yes. It is called measure theory and applied to lebesgue measure it reduces to lebesgue integration, while applied to discrete measure spaces (the measure of a set is the number of elements, if it is finite. If it is infinite, that is also its measure (it usually makes sense to allow only finite or countable sets). ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [February 28, 2003, 3:45pm UTC](https://boards.straightdope.com/t/what-is-discrete-mathematics/157631/27 "2003-02-28T15:45:36Z")
> [@](#): > > \*Originally posted by Hari Seldon \* > \*\*As for the general theory, the answer there is also yes. It is called measure theory and applied to lebesgue measure it reduces to lebesgue integration, while applied to discrete measure spaces (the measure of a set is the number of elements, if it is finite. If it is infinite, that is also its measure (it usually makes sense to allow only finite or countable sets). \*\* Would you mind expanding a bit on how measure theory relates to differentiation and finite differences? ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [February 28, 2003, 4:37pm UTC](https://boards.straightdope.com/t/what-is-discrete-mathematics/157631/28 "2003-02-28T16:37:17Z")
Good question. Simple answer is that they don’t. There is a lebesgue differentiation operator, but while it uses lebesgue measure to define, it is not really part of measure theory. I guess the simple answer is that I was thinking of integration and insofar as differentiation is a left inverse to integration anything about integration is relevant to differentiation too. But I would have to think a lot about this to come up with a real answer. So let my reply be restricted to integration. However, note that a form of the Leibniz theorem (d(fg) = fd(g) + gd(f)) is also valid for finite differences. However, the difference is not at the same n; one is at n and the other at n-1, IIRC. In calculus, this difference is infinitesimal and is ignored. [Previous page](https://boards.straightdope.com/t/what-is-discrete-mathematics/157631.md?page=1)