Real numbers Definition and 214 Threads

Homework Statement prove that (\forallx\inR) (\existsy\inR) : x < y Homework Equations The Attempt at a Solution x < y y= x+1 then x<x+1 which is correct but I'm kinda not sure of this answer...

bra - ket?? Hi, maybe a stupid question, but i would like to know if, if We have a real number, but we are i a vector space, and the operator is hermitian, is |a> is equal to < a |*? i assume this, because if a is the vector (1,0) (spin up), and only real entries. im trying to make...

Homework Statement Let x \in ℝ Prove that if 3x^{4}+1≤x^{7}+x^{3}, then x > 0 Homework Equations None The Attempt at a Solution Assume 3x^{4}+1≤x^{7}+x^{3} then 0 ≤ -3x^{4}-1≤x^{7}+x^{3} Then I assumed that each was greater than or equal to 0, which I thought gave the desired...

We define the cumulative hierarchy as: $V_0=\emptyset$ $V_{\alpha+1}=\mathcal{P}(V_\alpha)$ If $\lambda$ is a limit ordinal then $V_\lambda=\bigcup_{\alpha<\lambda} V_\alpha$ Then we have a picture of a big V where we keep building sets up from previous ones and each $V_\alpha$ is the class...

So the title says everything. Let's assume R is a set equipped with vector addition the same way we add real numbers and has a scalar multiplication that the scalars come from the field Q. I believe the dimension of this vector space is infinite, and the reason is we have transcendental numbers...

1. What can be said of the dimension of the basis of the Reals over the Irrationals 2. Homework Equations 3. I believe the basis is infinite because any real number can be made out of the combination of irrational vectors multiplied by the same irrational coefficient to make any real number...

how to prove that for any real number in r (0,1) there exist a natural number n in N such that rn > 1

Hi, I'm trying to write a short code in Mathematica that can generate random real numbers in - say 5 secs, and then plot this against any specified range I want. An additional complexity is that the function I'm generating the random numbers for is embedded in an integral. Here's an example...

I recently watched the following video on youtube: MF91: Difficulties with real numbers as infinite decimals I - YouTube The guy in this video is a mathematical finitist and he claims that there is some problem with the foundation of modern mathematics, in particular modern analysis, I think...

Hi all, I'm upgrading a legacy f77 code to f90 standards. I'm trying to make some of the global arrays as dynamic arrays so that at a later stage it would be easier for me to parallelize whole application. Coming to my problem, I need to read some numbers from a text file so that I can...

Homework Statement Prove that for every two distinct real numbers a and b, either (a+b)/2>a or (a+b)/2>bHomework Equations The Attempt at a Solution Proof: if two distinct numbers a and b then (a+b)/2>a Since a≠b and a,bεR, (a+b)/2>a=a+b>2a=b>a. Therefore (a+b)/2>a if b>a. and if two...

I couldn't really think of a good title for this question, lol. Is it possible that a real number exists that can only be expressed in exact form when that form must includes complex numbers? For example, the equation 2 \, x^{3} - 6 \, x^{2} + 2 = 0 has the following roots x_1 =...

How do you find all the values of "a" such that f is continuous on all real numbers? Find all values of a such that f is continuous on \Re f(x)= x+1 if x\leq a x^2 if x>a I tried solving but i do not even know where to start! Please help!

Just a quick question for you guys, I've been unable to find the answer to this. Can all real numbers be written as n^p, where p is a real number?

Homework Statement Factorise z^{8} -15z^{4} - 16 over the Complex numbers and Real numbers The Attempt at a Solution I factorised over the complex numbers, I'm not sure what they mean by over the real numbers. Do I substitute z = (x + iy) and then do it by expanding and separating...

Let \mathcal{S} = \{S_{i}:i \in \mathbb{R} \} where S_{i} is a set. Then \mathcal{S} is a set? Or, can this notation make sense in some way?

So far I have learned a bit about topological spaces, there has been several occasions regarding metric spaces where I had to invoke completeness (or at least uncountability) of real numbers R, which is itself a property of the usual metric space of R. For example, to show that any countable...

Hi there, In most books that I saw, the set of real numbers under the usual sum and product is considered as a Field and say that's by the field axioms. But I have surprised when I have seen, it is a theorem. The question, are these axioms? or can they be proved? Thank you very much.

Homework Statement In set theory, i have a two part question, the first is showing that the system S={set of all real numbers( \Re )}, #} where a#b=a+b-ab we have to show that it's not a group. and then find what c is so that the system = { \Re \cap\overline{c}, # } is a group.Homework...

Let ABC be a triangle. Prove that $sin^2\frac{A}{2}+sin^2\frac{B}{2}+sin^2\frac{C}{2}+2sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}=1$. Conversely, prove that if x, y and z are positive real numbers such that $x^2y^2+z^2+2xyz=1$, then there is a triangle ABC such that $x=sin\frac{A}{2}...

Find the necessary and sufficient conditions on the real numbers a,b,c for the matrix: \begin{bmatrix} 1 & a & b\\ 0 & 1 & c \\ 0 & 0 & 2 \end{bmatrix} to be diagonalizable. Attempt: Now for this one I also solved for the eigenvlues which were: λ1 = 1, λ2 = 1, λ3 = 2 So the problematic...

Homework Statement How to order a set of real numbers, from the smallest to the largest, without knowing any members of the set, but knowing that there are finite members, and that these members are all real numbers? Homework Equations - The Attempt at a Solution Tried Googling...

Homework Statement A vector space W over the real numbers is the set of all 2 x 2 Hermitian matrices. Show that the map T defined as: T(x,y,z,t) = [t+x y+iz] [y-iz t-x] from R4 to W is an isomorphism. Homework Equations The Attempt at a Solution I know that for the map...

Hello everybody, Yesterday I've read that there exist a real number r which cannot be defined by a finite number of words. This result, although quite awesome, is so strange that it lead Poincaré to doubt Cantor's work and state "never consider objects that can't be defined in finite number...

Homework Statement Show that the sequence of real numbers defined by x_{n + 1} = x_n + \frac{1}{x_n^2}, \, x_1 = 1 is not a Cauchy sequence. Homework Equations A sequence \{ p_n \} is Cauchy if and only if, for all \varepsilon > 0, there exists an N > 0 such that d(p_n, p_m) <...

I have a problem with Cantor's Diagonalization proof of the uncountability of the real numbers. His proof appears to be grossly flawed to me. I don't understand how it proves anything. Please take a moment to see what I'm talking about. Here is a totally abstract pictorial that attempts...

Homework Statement For which positive real numbers a does the series Ʃo→∞ a^log(n) converge. Here logarithms are to the base e Homework Equations Im afraid I'm not sure where to start, I'm not sure which topics would be applicable to this question. If someone could point...

Apparently there is an isomorphism between the additive group (ℝ,+) of real numbers and the multiplicative group (ℝ_{>0},×) of positive real numbers. But I thought that the reals were uncountably infinite and so don't understand how you could define a bijection between them?! Thanks for...

Homework Statement Find all subrings of \mathbb{R} which are discrete subsets Homework Equations For the purpose of our class, a ring is a ring with identity, not necessarily commutative. The Attempt at a Solution First suppose that S\subset \mathbb{R} is a subring of \mathbb{R}...

I was watching a lecture on computable real numbers, And in the lecture they talked about how this set of numbers is countable. And I could see this because the numbers that a computer generates would be listable, I could write them down on a list. For example pi is a computable real number...

I'm trying to use the MATLAB symbolic toolkit, but the "conj" term keeps annoying me. Here's an example: >>syms l d >>A=[l d ; d*l -l*l*d] >>A' ans = [ conj(l), conj(d)*conj(l)] [ conj(d), -conj(d)*conj(l)^2] How do I tell MATLAB I'm only interested in real numbers, and get rid...

Given any real numbers a and b such that a < b, prove that for any natural number n, there are real numbers x1, x2, x3, ... , xn such that a < x1 < x2 < x3 < ... < xn < b. The hint I was given says : Define xi recursively by x1 = (a+b)/2 and x(i+1) = (xi +b)/2. Prove that xi < xi + 1 < b...

Homework Statement Verify the rule that for two real numbers X and Y then |X+Y|≤|X|+|Y| Homework Equations The Attempt at a Solution 1. When all the variables are positive: |X+Y|=(X+Y) because (X+Y)>0 |X|=X because X>0 |Y|=Y because Y>0 So we got (X+Y)=X+Y 2...

A thought just struck me today after watching a lecture on the construction of the rational numbers. What is the definition of 'greater than (>)' and 'less than (<)' in the real number system. The only way I can think to describe it is to make some reference to the Euclidean distance between the...

Homework Statement Find all real numbers x ≠ −1 such that the series Ʃn=1∞ (1/n) . [(x-1)/(x+1)]n converges.. Homework Equations The Attempt at a Solution I really do not know where to start here. I assume you could first prove it converges sing a test but I'm really not...

Today I watched a lecture that talked about the extended reals, ℝ U {+infinity, -infinity}. Every course I've taken so far (differential calculus through multivariable calculus, linear algebra, a proofs writing course, etc) always defined the Reals with just they symbol ℝ and we always...

I was looking for some help on how to start this problem. I know that we must use the Mean-Value Theorem on |sin x| to get an f '(c). But I'm having a difficult time getting an initial start past that. Any hints and tips would be most useful. I also figure that we can let f(x) = |sin x| and g(x)...

I was wondering about the idea of representing real numbers as points on line , What is the basis of this assumptions , and as well the same question for Cartesian coordinates system ? All books I have read , express the idea of Cartesian Coordinates in an elementary way like spivak's , Apostol...

Homework Statement Let A be a nonempty set of real numbers which is bounded below. Let -A be the set of all real numbers -x, where x is in A. Prove that inf A = -sup(-A). Homework Equations Definitions of upper bound, lower bound, least upper bound, and least lower bound. The...

Could it be possible to come up with a formula or infinite series or continued fraction to generate the real numbers? I might have to change something in my formula to generate another real. I couldn't just change one of my numbers in the formula because then i would be saying there is a...

First, can all aleph 1 sets be generalized as sets of infinitely wide tuples? As in, let a_1a_2_a3 \ldots \in \Re map to (a_1, a_2, a_3, \ldots). Second, if countably infinite sets are n-tuples, aleph 1 sets are infinite tuples, can this pattern be generalized to even higher cardinality?

If the set of natural numbers is \aleph and when we write a real number we have 10 choices for each position 0-9 so can we say that there are 10^{\aleph} real numbers ?

(Title should be Connectedness of...) Hi. I'm trying to prove that if a set of Dedekind cuts is bounded, it has a least upper bound. We've defined a Dedekind cut, called E, to be a nonempty subset of Q (i) with no last point, (ii) an upper bound in Q, and (iii) the property that if x belongs...

With normal vectors i usually check there is the correct number of vectors i.e 3 for R3 2 for R2 etc and then just check for linear independence but reducing the matrix that results from c1v1+c2v2+..cnvn=0 and determining of unique solution or infinite solutions. There are the right number of...

Homework Statement For which real numbers  does the following system have a unique solution? 14x - 6y + 18z = 2\lambda z x = \lambda x 3x - 8y = -\lambda y Homework Equations The Attempt at a Solution hi, I rearranged the equations...

Homework Statement Let U and V both have the same cardinality as R (the real numbers). Show that U\cupV also has the same cardinality as R. Homework Equations The Attempt at a Solution Because U and V both have the same cardinality as R, I that that this means \exists f: R\rightarrowU that is...

I was looking at the construction of the real number system. I know dedekind cuts can be used(completely worthless in terms of understanding I think) and Cauchy sequences can be used (I wish my analysis book used them) but I would like to see a construction based on infinite decimal...

Homework Statement Give an example of a set of real numbers whose interior is empty but whose closure is all of the real numbers if it exists. Otherwise, explain why such example cannot be true. 2. The attempt at a solution For a set S ⊆ X, the closure of S is the intersection of all closed...

Question: What is the probability that a random variable X with domain all real numbers will take a value in the closed interval [a,b]? It seems to me that in order to answer this question you have to know how the real numbers are distributed. Given the appropriate distribution function, you...

Homework Statement Show that R^x/<-1> is isomorphic to the group of positive real numbers under multiplication. Homework Equations The Attempt at a Solution I know I need to show we have a homomorphism, and is one - to one and onto in order to be isomorphic. I know all...