Real numbers Definition and 213 Threads

I feel silly for asking, since I have accepted this always as true, but I don't have a reference for what ##0^p## equals when ##p## is a positive real number. This dawned on me when trying to show the positive definiteness of the ##p##-norm for ##x\in\mathbb R^n##, that is, $$x=0\iff...

In Folland's real analysis text, twice (so far) he drops a very similar statement about base-##b## expansions of nonnegative reals. The first time is when discussing the proof of ##\operatorname{card}(\mathcal{P}(\mathbb N))=\mathfrak c##, where he says that every positive real number has a...

I know that the definition of multiplication for integers is just repeated addition. For example, 5 times 3 means 5 + 5 + 5, but what about if we want to extend this definition to real or complex numbers ? Like for example, what does pi times e mean ? How are we supposed to add pi to itself e...

A couple of weeks ago we had an interesting thread where a tangent developed discussing whether real-valued measurements were possible. I would like to generalize that discussion a bit in this one and discuss all scientific purposes, not just measurements. 1) What is a measurement anyway? Is it...

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I would like to know if my answer is correct and if no ,could you correct.But it should be right I hope:

Let ##x,\,y>0## and ##x+y \leq 1##. Prove that ##(1-\frac{1}{x^2})(1-\frac{1}{y^2})\geq 9##.

Let a and b be positive real numbers such that ##a^5+b^3 \le a^2+b^2##. Prove that ##a(a+b) \le 2##.

If a and b are two real numbers such that ##a^3-3a^2+5a=1## and ##b^3-3b^2+5b=5##, evaluate a + b.

Using the inequality of arithmetic and geometric means, $$\frac {x+y}{2}≥\sqrt{xy}$$ $$6^2≥xy$$ $$36≥xy$$ I can see the textbook answer is ##36##, my question is can ##x=y?##, like in this case.

I think that real number is countable. Because there is one to one correspondence from natural numbers to (0,1) real numbers. 0.1 - 1 0.2 - 2 0.3 - 3 ... 0.21 - 12 ... 0.123 - 321 ... 0.1245 - 5421 ... I think that is a one-to-one corresepondence. Any errors here?

Determine whether the number is a natural number, an integer, a rational number, or an irrational number. (Some numbers fit in more than one category.) The following facts will be helpful in some cases: Any number of the form sqrt{n} where n is a natural number that is not a perfect square, is...

To find √(-2)√(-3). Method 1. √(-2)√(-3) = √( (-2)(-3) ) = √(6). Method 2. √(-2)√(-3) = √( (-1)(2) )√( (-1)(3) ) = √((-1)√(2)√(-1)√(3) = i√(2)i√(3) = (i)(i)√(2)√(3) = -1√( (2)(3) ) =-√6. Why don't the two methods give the same answer? Thanks for any help.

Prove that $\dfrac{y^2z}{x}+y^2+z\ge\dfrac{9y^2z}{x+y^2+z}$ for all positive real numbers $x,\,y$ and $z$.

Reals $x,\,y$ and $z$ satisfies $3x+2y+z=1$. For relatively prime positive integers $p$ and $q$, let the maximum of $\dfrac{1}{1+|x|}+\dfrac{1}{1+|y|}+\dfrac{1}{1+|z|}$ be $\dfrac{q}{p}$. Find $p+q$.

How do we get ##\epsilon(2p+\epsilon)<\epsilon(2p+1)<2-p^2## from ##0<\epsilon<1## and ##\epsilon<\dfrac{2-p^2}{2p+1}##? Answer: As we have ##\epsilon<1##, we've got ##2p+\epsilon<2p+1##; therefore, ## \epsilon(2p+\epsilon)<\epsilon(2p+1) ##; -as we have ##\epsilon<\dfrac{2-p^2}{2p+1}##, we...

If $x$ and $y$ are positive real numbers, prove that $4x^4+4y^3+5x^2+y+1\ge 12xy$.

Let $a$ and $b$ be positive real numbers such that $a+b=1$. Prove that $a^ab^b+a^bb^a\le 1$.

Summary:: Problem interpreting a vector space of functions f such that f: S={1} -> R Hello, Another question related to Jim Hefferon' Linear Algebra free book. Before explaining what I don't understand, here is the problem : I have trouble understanding how the dimension of resulting space...

Formula used : arc length = radius × angle (in radian). I interpreted this as: •Taking a unit circle, we get "angle (in radian) = arc length". This means radian measure of an angle is arc length, which can be represented on a real number line. Hence, it is a real number. Is this way to...

"The dual space is the space of all linear maps from the original vector space to the real numbers." Spacetime and Geometry by Carroll. Dual space can be anything that maps a vector space (including matrix and all other vector spaces) to real numbers. So why do we picked only a vector as a...

In chapter 1, page 10, real numbers are found by confining them to an interval that shrinks to "zero" length (we consider subintervals ##I_0,\,I_1,...,\,I_n##). Basically, if ##x## is between ##c## and ##c+1##, then we can divide that interval into ten subintervals, and we can, then, have...

Let m,n be real numbers. Prove that if n>m>0 , then (m+1)/(n+1) > m/n I'm currently confuse in this one help will be very much needed

Recently I created a spreadsheet that generates Phythagorean triples. Curious, instead of using only positive integers for the values of m and n, I found that as long as m>n, the sides 2mn, msq + nsq, msq - nsq, still form the sides of a right triangle even though m and n are non-whole...

I understand that the Dual Space is composed of elements that linearly map the elements of the Vector Space onto Real numbers If my preamble shows that I have understood correctly the basic premise, I have one or two questions that I am trying to work through. So: 1: Is there a one to one...

I'm aware of the axioms of real numbers, the constructions of real number using the rational numbers (Cauchy sequence and Dedekind cut). But I can't relate the arithmetic of irrational numbers to real world usage. I can think the negative and positive irrational numbers to represent...

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ... I am focused on Chapter 3: Convergent Sequences ... ... I need some help to fully understand the proof of Corollary 3.2.7 ...Garling's statement and proof of...

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ... I am focused on Chapter 3: Convergent Sequences I need some help to fully understand the proof of Corollary 3.2.7 ...Garling's statement and proof of...

1. The problem statement, all variables and given Prove that ##\sqrt{2}\in\Bbb{R}## by showing ##x\cdot x=2## where ##x=A\vert B## is the cut in ##\Bbb{Q}## such that ##A=\{r\in\Bbb{Q}\quad \vert \quad r\leq 0 \quad\lor\quad r^2\lt 2\}##. I believe that I have to show ##A^2=L## however, it...

Hello! (Wave) Let $A$ be a $n \times n$ complex unitary matrix. I want to show that the eigenvalues $\lambda$ of the matrix $A+A^{\star}$ are real numbers that satisfy the relation $-2 \leq \lambda \leq 2$. I have looked up the definitions and I read that a unitary matrix is a square matrix...

The following program is printing wrong value of RWTSED. How can I print correct value?? program inpdat c IMPLICIT NONE REAL RHOMN,RWTSED,VOLSED VOLSED = 17424.0 RHOMN = 2.42 !0000076293945 RWTSED= VOLSED*RHOMN*1.0E6 2000 FORMAT(/,3F40.5)...

Hello experts, Full disclosure: I am a total layman at math, nothing in my training aside from high school courses and one college calculus class. I'm sure a week doesn't pass without someone posting a question about or challenge to Cantor. I am not here to challenge anything but rather to...

Need a practical example of E=MC2 with real numbers Ok, so I understand that Energy = Mass of an Object * Speed of light square, if we must convert this to numbers, how can this be presented for let’s say 1,000 hydrogen atoms? Energy = 1.008 (Hydrogen mass) * 1,000 (hydrogen atoms) * speed of...

Homework Statement Let ##x\in\Bbb{R}## such that ##x\neq 0##. Then ##x=LIM_{n\rightarrow\infty}a_n## for some Cauchy sequence ##(a_n)_{n=1}^{\infty}## which is bounded away from zero. 2. Relevant definitions and propositions: 3. The attempt at a proof: Proof:(by construction) Let...

Homework Statement [/B] The proposition that I intend to prove is the following. (From Terence Tao "Analysis I" 3rd ed., Proposition 6.1.7, p. 128). ##Proposition##. Let ##(a_n)^\infty_{n=m}## be a real sequence starting at some integer index m, and let ##l\neq l'## be two distinct real...

Homework Statement ##\mathbb{R} \setminus C \sim \mathbb{R} \sim \mathbb{R} \cup C##. Homework EquationsThe Attempt at a Solution I have to show that all of these have the same cardinality. For ##\mathbb{R} \cup C \sim \mathbb{R}##, if ##C = \{c_1, c_2, ... c_n \}## is finite we can define ##...

Homework Statement (1) Prove that there exists no smallest positive real number. (2) Does there exist a smallest positive rational number? (3) Given a real number x, does there exist a smallest real number y > x? Homework EquationsThe Attempt at a Solution (1) Suppose that ##a## is the...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Exercise 2.1.10 Part (c) ... ... Exercise 2.1.10 Part (c) reads as follows:I am unable to make a meaningful start on...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Exercise 2.1.10 Part (b) ... ... Exercise 2.1.10 Part (b) reads as follows: I am unable to make a meaningful start on...

Ordering on the Set of Real Numbers ... Sohrab, Exercise 2.1.10 (a) ... I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Exercise 2.1.10 Part (a) ... ... Exercise 2.1.10...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Exercise 2.1.12 Part (1) ... ... Exercise 2.1.12 Part (1) reads as follows: I am unable to make a meaningful start on...

Homework Statement I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Exercise 2.1.12 Part (1) ... ... Exercise 2.1.12 Part (1) reads as follows: I am unable to make a...

How is it possible to ignore the addition sign and imaginary number without contradicting fundamental Mathematics? I find it difficult to understand.

Firstly, thanks to everyone who participated in my last thread. It helped a lot! This will be the only other topic I can think of posting in physics forums, because, honestly, I don't know very much. I remember sitting down one time and thinking I was quite brilliant when I started to make a...

if ## a,b,c,d,e ## are positive real numbers, minimum value of (a+b+c+d+e)( \frac{1}{a} +\frac{1}{b} +\frac{1}{c} +\frac{1}{d} +\frac{1}{e} ) (A) 25 (B) 5 (C) 125 (D) cannot be determined My approach : expanding the expression , i get 5+a( \frac{1}{b} +\frac{1}{c} +\frac{1}{d} +\frac{1}{e}...

z is either a real, imaginary or complex number, and z^12=1 and z^20 also equals 1. What are all possible values of z? I know 1 and -1 are them, and I think its also i and -i?

Find x and y if x, y are members of real numbers and: (x+i)(3-iy)=1+13i I first expanded it to give: 3x-yix+3i+y=1+13i Then I equaled 3x+y=1 and -yx+3=13 But afterwards I do other steps and get the wrong answer.

Determine all three real numbers $a, \,b$ and $c$ which satisfies the conditions $a^2+ b^2+ c^2= 26$, $a + b = 5$ and $b + c\ge 7.$