# Dense subset

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| This article/section deals with mathematical concepts appropriate for a student in early high school. |

If Q is a subset of R (think of R as the real numbers), we say that Q is **dense** in R if, wherever you are in R, there is a point in Q "infinitely close" to you. Put another way, you can never get any finite distance away from points in Q.

A classic example of a set that is **not** dense is the integers. If you are at 1.414, there is no integer "infinitely close" to you— the nearest one is .414 away from you.

A classic example (in fact, **the** classic example) of a set that **is** dense is the set of rational numbers, that is, fractions of integers. There is a rational that exactly matches 1.414—. But how about a number that is not itself rational, like ? This is where the subtle aspects of "infinitely close" come into play. What this means is that, for any closeness that you want (but not exact equality), there is a rational number that close. If you want a rational number within one thousandth, will do. If you want a rational number within one trillionth, will do.

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## In multiple dimensions

Subsets can be dense in two (or more) dimensions. In the plane (the set of points given by two real numbers—the x-coordinate and the y-coordinate), the points with both coordinates rational are dense. If you are at , the rational point is within one-thousandth of where you are.

## More precise definition

This article/section deals with mathematical concepts appropriate for a student in late high school or early university. |

The precise definition of what we meant above by "as close as you want", uses the "" concept used in limits, which is a concept of calculus. Q is dense in R if, for any point x in R and any , there is a point in Q that is within of x.

## Metric spaces

In more advanced calculus treatments (this subject is often called analysis), denseness can be defined in any metric space. The notion of a point being "within of x" simply uses the metric.

## Countability

This article/section deals with mathematical concepts appropriate for a student in late university or graduate level. |

The fact that the rationals are both countable and dense is very important in mathematical analysis. There exist sets (not the reals, of course) that have no countable dense sets.

## Topology

In the language of topology, many of the above concepts are recast in different terms. Q is dense in R if the closure of Q is R.