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Roughly speaking, rational numbers can form real numbers that's why they are more densed than real numbers. For example, if A is a subset of some set X & every point x of X belongs to A then A is densed in X. Also Cauchy sequence is the best example of it in which every bumber gets close to each other hence makes a real number.
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∙ 11y ago
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The densest subset of real numbers is the set of fractions?
Your question is ill-posed. I have not come across a comparison dense-denser-densest. The term "dense" is a topological property of a set: A set A is dense in a set B, if for all y in B, there is an open set O of B, such that O and A have nonempty intersection. The rational numbers are indeed dense in the set of real numbers with the standard topology. An open set containing a real number contains always a rational number. Another way of saying it is that every real number can be approximated to any precision by rational numbers. There are denser sets, if you are willing to consider more elements. Suppose you construct a set consisting of the rational numbers plus all algebraic numbers. The set of algebraic numbers is also countable, but adding them, makes it obviously easier to approximate real numbers. Can you perhaps construct a set less dense than the set of rational numbers? Suppose we take the set of rational numbers without the element 0. Is this set still dense in the real numbers? Yes, because 0 can be approximated by 1/n, n>1. In fact, you can remove finite number of rational numbers from the set of rational numbers and the resulting set will still be dense in the set of the real numbers.
What is the formula of density in numbers?
Numbers are infinitely dense. Between any two rational or real numbers, no matter how close, there are infinitely many numbers.
What rational numbers are real numbers?
All rational numbers are real numbers.
Are some rational numbers are not real numbers?
No. Rational numbers are numbers that can be written as a fraction. All rational numbers are real.
What is the relationship between real number and rational number?
Rational numbers form a proper subset of real numbers. So all rational numbers are real numbers but all real numbers are not rational.
Are all real numbers a rational numbers?
Yes. Rational numbers are numbers that can be written as a fraction. All rationals are real.
What are example of both real and rational numbers?
All rational numbers are examples of numbers which are both rational and real.
How can you use rational numbers to represent real world problems?
You cannot. The diagonal of a unit square cannot be represented by a rational number. However, because rational numbers are infinitely dense, you can get as close to an irrational number as you like even if you cannot get to it. If this approximation is adequate than you are able to represent the real world using rational numbers.
What are the rational numbers between 1 and 10?
Rational numbers are infinitely dense and that means that there are infiitely many rational numbers between any two numbers.
Why the rational number 10 is also a real number?
Rational numbers are a proper subset of real numbers so all rational numbers are real numbers.
If a number is a real number then is it a rational number?
Not necessarily. All rational numbers are real, not all real numbers are rational.
Real numbers are not rational numbers?
Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction. NOt all real numbers are rational.
