Real numbers Definition and 214 Threads

Homework Statement Homework Equations character equation The Attempt at a Solution Should I set a = ax2 b= bx c =c in the character equation?

Homework Statement Please see questions (c) and (e) on the attachement 2.Relevant Equations The Attempt at a Solution So long story short, these two questions were given out as a challenge in one of our Swedish lessons to see if we could remember our high school calculus, which I shamefully...

Why is it that the distance between two real numbers ##a## and ##b## in an ordered interval of numbers, for example ##a<x_{1}<\ldots <x_{n-1}<b##, is given by $$\lvert a-b\rvert$$ when there are in actual fact $$\lvert a-b\rvert +1$$ numbers within this range?! Is it simply that, when measuring...

Homework Statement Show, using the axiom of completeness of ##\mathbb{R}##, that every positive real number has a unique n'th root that is a positive real number. Or in symbols: ##n \in \mathbb{N_0}, a \in \mathbb{R^{+}} \Rightarrow \exists! x \in \mathbb{R^{+}}: x^n = a## Homework...

I know that there are several models of the real numbers, some where the Continuum Hypothesis holds, others where it does not. I would like to understand why the usual definition of the reals, limits of Cauchy sequences of rational numbers under the usual absolute value norm, isn't one of these...

is the function x²+3 surjective for real numbers. how do you test for surjectivity in general?

For any $a \in \mathbb{R}$, let $a^3$ denote $a \cdot a \cdot a$. Let $x, y \in \mathbb{R}$. 1. Prove that if $x < y$ then $x^3 < y^3$. 2. Prove that there are $c, d \in \mathbb{R}$ such that $c^3 < x < d^3$.

Hello guys I am trying to write a code which is below; But my results seems to be fairly wrong. I noticed some of my real numbers are not what I assigned them. For example Ks shows on the watch window as 9.9999999E-5. How can I fix such situation? program hw1 REAL:: G,DVIS,Ks...

Call a nonempty (finite or infinite) set $A\subseteq\Bbb R$ complete if for all $a,b\in\Bbb R$ such that $a+b\in A$ it is also the case that $ab\in A$. Find all complete sets.

For positive real numbers $a,\,b,\,c$, prove the inequality: a + b + c ≥ \frac{a(b + 1)}{a + 1} + \frac{b(c + 1)}{b + 1}+ \frac{c(a + 1)}{c + 1}

If $a$ and $b$ are two positive real, and that $a^3+b^3=a-b$, prove that $2\left(\sqrt{2}-1\right)a^2-b^2\lt 1$.

I need to calculate some quantiles for a sample of 108 real numbers with unknown mean and unknown variance. I currently store and sort those numbers, but I would try a streaming method where the numbers are not stored. In a paper is written: "If the size of the input stream, N is known, then the...

I know that 1. To show the relation is reflexive, we need to show that for any x, using the definition of R, we have xRx. The definition of R means that we must have |x - x| is even.2. To show that R is symmetric, we would have to show that if xRy then yRx. In the context of the definition...

Cardinality of the set of binary-expressed real numbers This article gives the cardinal number of the set of all binary numbers by counting its elements, analyses the consequences of the found value and discusses Cantor's diagonal argument, power set and the continuum hypothesis. 1. Counting...

Let $a,\,b,\,c,\,d$ be different positive real numbers such that $a+\dfrac{1}{b}=b+\dfrac{1}{c}=c+\dfrac{1}{d}=d+\dfrac{1}{a}=x$. Find $x$.

Let $x,\,y,\,m,\,n$ be positive real numbers such that $m^2-m+1=x^2$, $n^2+n+1=y^2$ and $(2m-1)(2n+1)=2xy+3$. Prove that $m+n=xy$.

I was able to prove a), but I am unsure how to prove b. Is there some sort of geometric interpretation I should be aware of?

Let $x,\,y,\,z$ be positive real numbers such that $xy+yz+zx=3$. Prove the inequality $(x^3-x+5)(y^5-y^3+5)(z^7-z^5+5)\ge 125$.

Hi! (Smile) We define the set $U=\mathbb{Z} \times (\mathbb{Z}-\{0\})$ and over $U$ we define the following relation $S$: $$\langle i,j \rangle S \langle k,l \rangle \iff i \cdot l=j \cdot k$$ $$\mathbb{Q}=U/S=\{ [\langle i, j \rangle ]_S: i \in \mathbb{Z}, j \in \mathbb{Z} \setminus \{0\}...

How many real numbers $x$ satisfy $\sin x=\dfrac{x}{100}$?

Q4) Let a and b be real numbers with a < b. 1) Show that there are infinitely many rational numbers x with a < x < b, and 2) infinitely many irrational numbers y with a < y < b. Deduce that there is no smallest positive irrational number, and no smallest positive rational number. 1) a < x <...

Hey! :o I have to find an open and dense subset of the real numbers with arbitrarily small measure. Since the set of the rational numbers is dense, could we use a subset of the rationals?? (Wondering) How could I find such a subset, that the measure is arbitrarily small?? (Wondering)

Question: Show that the set of all functions of the form f(x) = ax+b, with a and b real numbers forms a vector space, but that the same set of functions with a > 2 does not. Equations: the axioms for vector spaces Attempt: I think that the axiom about the zero vector is the one I need to use...

I am confused, since some claims about the first Godel incompleteness theorem and real numbers seem mutually contradictory. In essence, from one point of view it seems that the Godel theorem applies to real numbers, while from another point of view it seems that the Godel theorem does not apply...

I think its fairly obvious to most people why a number being rational (or not) is extremely important. But I honestly do not see why being transcendental is as interesting of a property (though its clearly somewhat interesting). What interesting applications are there of knowing a number is...

If $a,b\in \mathbb{R}^{+}.$ Show that $a>b\implies a^{-1}<b^{-1}.$

Find the sum of all real numbers $a$ such that $5a^4-10a^3+10a^2-5a-11=0$.

Definition/Summary Let \mathbb{R} be the set of all real numbers. We can extend \mathbb{R} by adjoining two elements +\infty and -\infty. This forms the extended real number system. In notation: \overline{\mathbb{R}}:=\mathbb{R}\cup \{+\infty,-\infty\} The extended real numbers are...

$x,y\in R$ $if: \sqrt {3x+5y-2-m}+\sqrt {2x+3y-m}=\sqrt {x-200+y}\,\times\sqrt {200-x-y}$ $find:\,m=?$

Solve in real numbers the equation $\sqrt[3]{a-1}+\sqrt[3]{a}+\sqrt[3]{a+1}=0$

Homework Statement Is there a real number c such that the series: ∑ (e - (1+ 1/n)^n + c/n), where the series goes from n=1 to n=∞, is convergent? The Attempt at a Solution I used the ratio test by separating each term of the function as usual to find a radius of convergence, but that doesn't...

I understand the definition of real numbers in set theory. We define the term "Dedekind-complete ordered field" and prove that all Dedekind-complete ordered fields are isomorphic. Then it makes sense to say that any of them can be thought of as "the" set of real numbers. We can prove that a...

The set of the real numbers is closed. For me this is nearly trivial (*) but perhaps I miss something; a colleagues insists that there are some deeper considerations why this is far from trivial - but I don't get his point (*) A) A set is closed if its complement is open; the complement...

Homework Statement . Prove that ##\{x_n\}_{n \in \mathbb N} \subset \mathbb R## doesn't have any convergent subsequence iff ##lim_{n \to \infty} |x_n|=+\infty##. The attempt at a solution. I think I could correctly prove the implication ##lim_{n \to \infty} |x_n|=+\infty \implies## it...

A) I understand that complex numbers come in the form z= a+ib where a and b are real numbers. In the special case that b = 0 you get pure real numbers which are a subset of complex numbers. I read that both real and imaginary numbers are complex numbers so I am a little confused with notations...

Hi MHB, This problem has given me a very hard time because I have exhausted all the methods that I know to figure out a way to find for the values for both a and b but no, there must be a trick to this problem and I admit that it is a question that is out of my reach... Could you show me...

Homework Statement Let a and b be real numbers with a < b. a. Derive a formula for the distance from a to b. Hint: Use 3 cases and a visual argument on the number line. b. Use your work in part (a) to derive a formula for the distance between (a,c) and (b,c) in a plane. c. Use the...

Let $a, b, c, d$ be real numbers such that a=\sqrt{4-\sqrt{5-a}}, b=\sqrt{4+\sqrt{5-b}}, c=\sqrt{4-\sqrt{5+c}} and d=\sqrt{4+\sqrt{5+d}}. Calculate $abcd$.

1.show that there is no axiom for set union that correspond to "Existence of additive inverses" for real numbers, by demonstrating that in general it is impossible to find a set X such that $A\cup X=\emptyset$. what is the only set $\emptyset$ which possesses an inverse in this sense? 2. show...

in the following exercises, assume that x stands for an unknown real number, and assume that $x^2=x\times x$. which of the properties of real numbers justifies each of the following statement? a. $(2x)x=2x^2$ b. $(x+3)x=x^2+3x$ c. $4(x+3)=4x+4\times 3$ my answers a. distributive property b...

Find the real numbers $c$ for which there is a straight line that intersects the curve $y=x^4+9x^3+cx^2+9x+4$ at four distinct points?

in this problem we drop the use of parentheses when this step is justified by associative axioms. thus we write $\displaystyle x^2+2x+3\,\,instead\,\,of\,\,\left(x^2+2x\right)+3\,or\,x^2+\left(2x+3\right)$. tell what axioms justify the statement: 1. $\displaystyle...

justify each of the steps in the following equalities. i don't know where to start. what i know is i have to use properties of real numbers. please help! 1. $\displaystyle \left ( x+3 \right )\left(x+2\right)\,=\,\left ( x+3 \right )x+\left ( x+3 \right )2\,=\,\left ( x^2+3x \right )+\left (...

The real numbers x and y satisfy x^3-3x^2+5x-17=0 and y^3-3y^2+5y+11=0. Determine the value of x+y.

Homework Statement An eight digit number is a multiple of 73 and 137. If the second digit from the left of the number is seven, find the 6th digit from the left of the number. Homework Equations N.A. The Attempt at a Solution I don't know any clear method for solving this problem...

Homework Statement The question : http://gyazo.com/7eb4b86c61150e4af092b9f8afeaf169 Homework Equations Sup/Inf axioms Methods of constructing sequences ##ε-N## ##lim(a_n) ≤ sup_n a_n## from question 5 right before it. I'll split the question into two parts. The Attempt at a...

Find all real numbers a such that the equation |x-|x-|x-4||| =a has exactly three real solutions.

Can you help me to construct a 1-1 mapping from real numbers onto non-integers? thanks

I have to prove that the cardinality of the set of infinite sequences of real numbers is equal to the cardinality of the set of real numbers. So: A := |\mathbb{R}^\mathbb{N}|=|\mathbb{R}| =: B My plan was to define 2 injective maps, 1 from A to B, and 1 from B to A. B <= A is trivial, just...

Homework Statement Show that |a-b|<= |y-a|+|x-y|+|x-b|, for all x,y in ℝ Homework Equations The Attempt at a Solution |a-b| <= |y-a+x-y+x-b| (correct? Not sure about this one...is it not part of the triangle rule?) |a-b| <= |2x-b-1| |a-b+2x-2x| <= |2x-b-1| |a-2x| + |2x-b| <=...