reals

1. Count members of all sets of interchanged reals

What is the cardinality of members of unique sets representing all bijective interchanges among the real numbers?
2. Induction of reals

Can be some cases of induction used on real numbers? Why? or Why not?
3. Limits' and reals' cardinality

Does the limit function relate to a maximum cardinality? Does the set of real numbers, as they are an "absolute" continuum? Can one map the set of real numbers onto a finite surface? By bijection?
4. Find a basis for the vectors over the reals:

the vectors are (1+i) (1-i) (2+3i) where the vectors are complex numbers that span the reals. Do i just have to set up a system of equations and row reduce it to find the basis vectors and find the rank of the system? Or could i just put coefficients infront of the vectors and set it equal to 0...
5. Non-computable Reals

Question 1: Can a non-computable real $x$ be created using the following method? Let $s \in ( \,0,1) \,$ be a computable number that is irrational. A turing machine can therefore encode the binary expansion of $s$ to any desired number of bits: s = s_1s_2s_3s_4\dots\text{, where each }...
6. Reals

Let $R$ be a finite commutative ring with no zero divisors then A) $R$ is a field B) $R$ has unity C) Characteristic of $R$ is prime number D) None of the above Option A & B are true and I got confused with Option C. I know that every field has characteristic of either $0$ or prime and...
7. The Completeness Axiom Of the Reals

Hi, I've been taught the completeness axiom for the real numbers in my calculus lectures, but whilst reading an analysis book I discovered an axiom with the same name, but different in its presentation: (lecture definition) every convex subset of the real numbers is an interval. (book) every...
8. Countability of the Reals without "Infinity"

Associate with each n-place decimal in [0,1) a natural number by removing the decimal (.035->35). Let Sn be the set of natural numbers so created. The real numbers correspond to the case S has no maximum, ie, Sn=N, the set of natural numbers. By definition, the reals are countable.
9. Reals are uncountable - without Cantor diagonal argument

Gist of proof: length of unit interval = 1, if the reals are countable, the length = 0, contradiction. Proof: Arrange reals into countable list. Let x be an arbitrarily number > 0. Cover first point on list by interval of length x/2, second point by interval of length x/4, third point by...
10. Power Set of the Reals is Countable

1) Reals in [0,1) correspond to the natural numbers.* 2) Power Set of natural numbers is countable.** * http://mymathforum.com/topology/331962-cantor-s-diagonal-argument-binary-sequences.html ** http://mymathforum.com/topology/333500-power-set-natural-numbers-countable.html
11. Countability of the Rationals and Reals

List the binary fractions 0 \leq x < 1 and assign natural numbers as follows: .1 -> 1 .01 -> 2 .11 -> 3 .001 -> 4 .101 -> 5 .011 -> 6 .111 -> 7 .0001 -> 8 ........ ........ Every real number appears in the list because the list includes all possible combinations of 0,1 (ending in 1) for any...
12. Cantors Diag Argument Proves Reals Countable

Let T be the set of all infinite binary series. List all the elements of T and assume the list is uncountable. Use Cantor's Diagonal Argument to show there is a member of T not in the list. Contradiction. Therefore T is countable.
13. Cantor: The reals are uncountable

The following proof is incorrect From: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument -------------------------------------------------------------------------------------- In his 1891 article, Cantor considered the set T of all infinite sequences of binary digits (i.e...
14. The reals can be placed in countable order

This post is meant to replace the OP of http://mymathforum.com/topology/326513-counting-irrational-numbers.html which is more intuitive than precise. It also provides a precise simple clear proof to comment on. I also include the proof that the reals are countable for easy reference and...
15. Messing Around - Relating Sets of Dyadics to Reals

I haven't seen any new threads here for a while. As always, I promise this is Fields Medal material ;), but perhaps someone will find it interesting anyways. To start, in my hopeless quest to find a discrete uniform distribution over $\mathbb{N}$, I was thinking in terms of base 2 on the...
16. Cardinality of integers equals cardinality of reals

pi squared is an integer because its last digit in base 2 is 0. The surface hypervolumes (or hyperareas/hypervolumes on alternating indexs) are an integer multiple of eachother while having sometimes different and sometimes same exponent of pi. So theres really no important difference between a...
17. Approach reals by 3-smooth numbers

Hello everyone, I am not a mathematician but I am looking for a theorem. I want to know if it is possible to approximate any real number by the quotient of two 3-smooth numbers. In practice that means: can any real number be approached arbitrary close by 2^m3^n where m and n are in \mathbb{Z}...
18. Independence of Reals

Well, I think this worths asking, that's why I thought to post it here : In a recent sequence (not yet approved) on OEIS, I defined independence of two reals by the fact that if x and y are independent over a subset of R if there is no polynomial with coefficients in the specified subset of R of...
19. Prove a set is a subring of the reals

Prove that \{x +\sqrt{3}y : x,y \in \mathbb{Z}\} is a subring of the ring \mathbb{R} I know a nonempty subset S of a ring R is a subring of R if and only if S is closed under subtraction and multiplication. Is this all I need to show that is closure under subt. and mult.? Or do I have to...
20. Integration over the reals? finding R(dx)

I am trying to find a function R(dx) in a paper by Rosinski "Tempering Stable Processes" which has the following theorem Theorem 2.3. The Levy measure M of a tempered alpha stable distribution can be written in the form M(A) = \int_{R^d} \hspace{7mm} \int_0^{\infty}...