What does it mean for rational numbers to be "dense in the reals?" I can't seem to find a decent explanation online...
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2http://en.wikipedia.org/wiki/Dense_set is the "dense" involved. – Dan Uznanski Nov 18 '14 at 18:55
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2As Michael Hardy pointed out, it means that there is a rational between any two reals. It turns out that there is also an irrational between any two rationals: https://www.khanacademy.org/math/algebra/rational-and-irrational-numbers/proofs-concerning-irrational-numbers/v/proof-that-there-is-an-irrational-number-between-any-two-rational-numbers – mitchus Mar 04 '18 at 14:49
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- $\mathbb{Q}$ is dense in $\mathbb{R}$ means if you draw a ball around any point in $\mathbb{Q}$ you will have point in R too in this open ball.
– Konstantin Burlachenko Jun 24 '20 at 09:52 -
- The more formal definition that first thing you defined what closure means - is's a set of all limit points of the set. For example $cl (0,1] = [0,1]$. Q is dense in R $\iff cl(Q) \subset R$
– Konstantin Burlachenko Jun 24 '20 at 09:56
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It means that between any two reals there is a rational number. The integers, for example, are not dense in the reals because one can find two reals with no integers between them.
That definition works well when the set is linearly ordered, but one may also say that the set of rational points, i.e. points with rational coordinates, in the plane is dense in the plane. Then it must be defined differently: it means that every open set in the plane intersects the set of all rational points. No matter how small you make an open disk in the plane, it cannot avoid containing some rational points; so the set of all rational points is dense in the plane. In the case of the line, saying that every open interval contains some rationals amounts to the same thing as saying that between any two reals there is some rational number between them.
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It means that you can well-approximate any real number using a rational number. By well-approximate I mean that you can find a fraction arbitrarily close to any real number.
To make this precise, for any real number $x$ and any 'arbitrary closeness' $\varepsilon>0$, there exists a fraction $\displaystyle q=m/n\in\mathbb{Q}$ such that
$$\left|x-\frac{m}{n}\right|<\varepsilon.$$
Another example of density is that the set of polynomials is dense in the set of continuous functions on a closed interval. That is given any continuous function $f:[a,b]\rightarrow \mathbb{R}$ and 'arbitrary closeness', there is a polynomial $$p(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n$$ such that $\|f-p\|<\varepsilon$. This '$\|\cdot\|$' is the distance between $f$ and $p$ and is given as $$\|f-p\|=\sup_{x\in [a,b]}|f(x)-p(x)|\underset{[a,b]\text{ is compact }}{=}\max_{x\in [a,b]}|f(x)-p(x)|.$$
JP McCarthy
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Between any two rational numbers there exist another rational number. For example 1/2 and 1/4 are two rational numbers, but there exist another rational number 1/3 between the two above.In the case of other subsets of numbers in real numbers for instance,integers,there cannot exist another integers between any two.
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4To say that the rationals are dense in the reals does not simply mean there is a rational between any two rationals; rather it means there is a rational between any two reals. That is a stronger statement, and takes more work to prove. Finding a rational between two rationals is just arithmetic computation; one only needs to compute their average. – Michael Hardy Mar 04 '18 at 15:55
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1Kudos for leaving your answer despite downvotes (we can learn a lot from incorrect answers as long as someone points out what is wrong) and thanks Michael for clarifying. – APR123 Oct 06 '22 at 01:03