---
tag: 61CDKWSXV4
kind: concept
name: Discrete calculus
msc: [39A12]
sources:
  - url: https://en.wikipedia.org/wiki/Discrete_calculus
same_as:
  wikidata: Q20968946
  wikipedia: Discrete calculus
---

## Summary

Discrete calculus studies change and accumulation for functions on discrete sets, such as the integers or the vertices and cells of a grid.
Difference quotients take the place of derivatives, and finite sums take the place of integrals.
Its results mirror those of ordinary calculus, often with exact formulas instead of limits, and it includes the [[calculus of finite differences|FT9JQA58KJ]].
It is used in difference equations, numerical analysis, combinatorics, and in discrete models of geometry and physics.

## Explanations

### L1 human

Ordinary calculus lets the step size shrink to zero; discrete calculus keeps a fixed step.
A table of daily prices has no derivative, but it has daily changes, and adding the changes gives the total change.
For example, the daily changes of the values $1, 3, 7, 15$ are $2, 4, 8$, and their sum $14$ is the change from $1$ to $15$.
This is the discrete form of the fundamental theorem of calculus.