What is Real numbers: Definition and 211 Discussions

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line (or alternatively, a quantity that can be represented as an infinite decimal expansion). The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the real transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more. The set of real numbers is denoted using the symbol R or




R



{\displaystyle \mathbb {R} }
and is sometimes called "the reals".Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, where each consecutive digit is measured in units one-tenth the size of the previous one. The real line can be thought of as a part of the complex plane, and the real numbers can be thought of as a part of the complex numbers.

These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers—indeed, the realization that a better definition was needed—was one of the most important developments of 19th-century mathematics. The current standard axiomatic definition is that real numbers form the unique Dedekind-complete ordered field (




R



{\displaystyle \mathbb {R} }
; + ; · ; <), up to an isomorphism, whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences (of rational numbers), Dedekind cuts, or infinite decimal representations, together with precise interpretations for the arithmetic operations and the order relation. All these definitions satisfy the axiomatic definition and are thus equivalent.
The set of all real numbers is uncountable, in the sense that while both the set of all natural numbers and the set of all real numbers are infinite sets, there can be no one-to-one function from the real numbers to the natural numbers. In fact, the cardinality of the set of all real numbers, denoted by





c




{\displaystyle {\mathfrak {c}}}
and called the cardinality of the continuum, is strictly greater than the cardinality of the set of all natural numbers (denoted






0




{\displaystyle \aleph _{0}}
, 'aleph-naught').
The statement that there is no subset of the reals with cardinality strictly greater than






0




{\displaystyle \aleph _{0}}
and strictly smaller than





c




{\displaystyle {\mathfrak {c}}}
is known as the continuum hypothesis (CH). It is neither provable nor refutable using the axioms of Zermelo–Fraenkel set theory including the axiom of choice (ZFC)—the standard foundation of modern mathematics. In fact, some models of ZFC satisfy CH, while others violate it.

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  1. Dale

    I Number systems in science

    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...
  2. Jiketz

    A Determining rationality of real numbers represented by prime digit sequence

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

    POTW Inequality of Positive Real Numbers: Proving (1-1/x^2)(1-1/y^2)≥9 with x+y≤1

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

    POTW Proving Inequality between Positive Real Numbers a and b: a(a+b) ≤ 2

    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##.
  5. anemone

    POTW Find the Sum of Two Real Numbers | Solve a + b with Given Equations

    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.
  6. chwala

    Find the maximum value of the product of two real numbers

    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.
  7. E

    I Are real numbers Countable?

    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?
  8. N

    What Are the Different Types of Numbers and How Can You Determine Them?

    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...
  9. G

    B Real numbers and complex numbers

    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.
  10. anemone

    MHB Inequality involving positive real numbers

    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 $z$.
  11. anemone

    MHB Absolute value of real numbers

    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 $p+q$.
  12. mcastillo356

    B Inequalities in the real numbers

    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...
  13. anemone

    MHB Inequality of positive real numbers

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

    MHB Inequality with positive real numbers a and b

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

    Vector space of functions from finite set to real numbers

    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...
  16. f9CSERS

    B Radian measure and real numbers

    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...
  17. dontknow

    A Explore Carroll's Theory on Dual Space and Real Numbers

    "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...
  18. GodfreyHW

    I Courant and Fritz, Construction of the real numbers

    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...
  19. E

    MHB Can we prove that (m+1)/(n+1) > m/n if n>m>0 using synthetic proof?

    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
  20. e2m2a

    A Euclid's formula and real numbers

    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...
  21. G

    B Understanding Dual Space: Mapping Vector Space to Real Numbers

    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...
  22. LittleRookie

    I How to relate multiplication of irrational numbers to real world?

    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...
  23. Math Amateur

    I Real Numbers & Sequences of Rationals .... Garling, Corollary 3.2.7 ....

    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...
  24. Math Amateur

    MHB Understanding Garling's Corollary 3.2.7 on Real Numbers and Rational Sequences

    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...
  25. T

    Showing ##\sqrt{2}\in\Bbb{R}## using Dedekind cuts

    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...
  26. evinda

    MHB Eigenvalues are real numbers and satisfy inequality

    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...
  27. A

    Fortran Print correct value of Real number with gfortran?

    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)...
  28. Z

    I Confusion over countability

    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...
  29. G

    I Need a practical example of E=mc^2 with real numbers

    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...
  30. T

    Regarding Real numbers as limits of Cauchy sequences

    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...
  31. L

    Proof of uniqueness of limits for a sequence of real numbers

    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...
  32. Mr Davis 97

    Show Cardinality of Real Numbers and Complements

    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 ##...
  33. Mr Davis 97

    Proofs involving real numbers

    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...
  34. Math Amateur

    MHB Ordering on the Set of Real Numbers .... Sohrab, Exercise 2.1.10 (c) ....

    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...
  35. Math Amateur

    MHB Ordering on the Set of Real Numbers .... Sohrab, Exercise 2.1.10 (b) ....

    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...
  36. Math Amateur

    MHB Ordering on the Set of Real Numbers .... Sohrab, Exercise 2.1.10 (1) ....

    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...
  37. Math Amateur

    MHB Ordering on the Set of Real Numbers .... Sohrab, Exercise 2.1.12 ....

    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...
  38. Math Amateur

    Ordering on the Set of Real Numbers .... Sohrab, Ex. 2.1(1)

    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...
  39. L

    I How can we consider a complex number as two separate real numbers for in X and Y plane?

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

    I Maybe all reals can be listed?

    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...
  41. M

    B Find the Minimum Value of Expression Involving Positive Real Numbers

    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}...
  42. I

    MHB Imaginary numbers and real numbers?

    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?
  43. G

    MHB Find x and y if x, y are members of real numbers and: (x+i)(3-iy)=1+13i

    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.
  44. anemone

    MHB Find Real Numbers a, b and c

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

    Suppose a, b, c are three real numbers such that

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

    Finding Real Numbers: Questions (c) & (e) Solutions

    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...
  47. F

    B Distance between real numbers

    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...
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