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—Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1414}{1000}} . But how about a number that is not itself rational, like Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{2}} ? 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, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1414}{1000}} will do. If you want a rational number within one trillionth, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1414213562373}{1000000000000}} will do.
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\sqrt{2}, \sqrt{3})} , the rational point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\frac{14142}{10000}, \frac{17321}{10000})} is within one-thousandth of where you are.
More precise definition
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx} \sin x=?\,} | This article/section deals with mathematical concepts appropriate for late high school or early college. |
The precise definition of what we meant above by "as close as you want", uses the "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon\,} " concept used in limits, which is a concept of calculus. Q is dense in R if, for any point x in R and any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon > 0\,} , there is a point in Q that is within Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon\,} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon\,} of x" simply uses the metric.
Countability
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi_1(S^1)=?\,} | 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.