---
tag: SWJYG87QQH
kind: concept
name: Rational point
msc: [14G05, 11G35]
sources:
  - url: https://en.wikipedia.org/wiki/Rational_point
same_as:
  wikidata: Q3393476
  wikipedia: Rational point
---

## Summary

A rational point of a variety defined over a field $K$ is a point whose coordinates all lie in $K$.
Over $K = \Q$ these are the rational solutions of the defining equations, so rational points are the geometric form of [[Diophantine equations|08B1RTW7VZ]].
Deciding whether rational points exist, and describing them when they do, is the central problem of [[arithmetic geometry|824JAXQ5M6]].

## Explanations

### L1 human

The circle $x^2 + y^2 = 1$ has many points, but only some have both coordinates rational, such as $(3/5, 4/5)$.
These are its rational points over $\Q$.
Each one corresponds to a Pythagorean triple, here $3^2 + 4^2 = 5^2$.
The circle $x^2 + y^2 = 3$ has real points but no rational points at all.
Which curves and surfaces have rational points, and how many, is a deep question.