What is Irrational: Definition and 350 Discussions

Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. The term is used, usually pejoratively, to describe thinking and actions that are, or appear to be, less useful, or more illogical than other more rational alternatives.Irrational behaviors of individuals include taking offense or becoming angry about a situation that has not yet occurred, expressing emotions exaggeratedly (such as crying hysterically), maintaining unrealistic expectations, engaging in irresponsible conduct such as problem intoxication, disorganization, and falling victim to confidence tricks. People with a mental illness like schizophrenia may exhibit irrational paranoia.
These more contemporary normative conceptions of what constitutes a manifestation of irrationality are difficult to demonstrate empirically because it is not clear by whose standards we are to judge the behavior rational or irrational.

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  1. Pejeu

    Irrational numbers could they be more

    consistently thought of as actually emergent functions that take the desired accuracy as input? As them being numbers would imply the apparently paradoxical concept that infinite complexity can exist in a finite volume of space.
  2. B

    MHB Irrational numbers forming dense subset

    Hello. I have some problems with proving this. It is difficult for me. Please help me.:confused: "For arbitrary irrational number a>0, let A={n+ma｜n,m are integer.} Show that set A is dense in R(real number)
  3. C

    Irrational Numbers: Is It Possible?

    Is it possible to have an infinite string of the same number in the middle of an irrational number? For example could I have 1.2232355555555.....3434343232211 Where their was an infinite block of 5's. Then I was trying to think of ways to prove or disprove this. It does seem like it might...
  4. paulmdrdo1

    MHB Sums and Products of Rational and Irrational Numbers

    Explain why the sum, the difference, and the product of the rational numbers are rational numbers. Is the product of the irrational numbers necessarily irrational? What about the sum? Combining Rational Numbers with Irrational Numbers In general, what can you say about the sum of a rational...
  5. B

    How do irrational numbers play a role in physics?

    Hi, I have some theories about physical facts derived from the size of powers in physics, compared to the first fraction of an irrational number. I do not know if this is redundant with present day science, but I am curious about it. Regards, Justin
  6. J

    Differentproof there are more irrational numbers than rational numbers

    you can list and match up all rational numbers with irrational numbers this way.. lets say i have an irrational number 'c'. Rational->Irrational r1->cr1 r2->cr2 . . . rn->crn There exists an irrational number that is not on this matching, (not equal to any of the crx's) this...
  7. M

    Irrational number to an irrational power

    Homework Statement if a and b are irrational numbers, is a^b necessarily an irrational number ? prove it. The Attempt at a Solution this is an question i got from my first maths(real analysis) class (college) , and have to say, i have only little knowledge about rational number, i would like to...
  8. H

    Irrational Equation - I end up in a dead end

    Homework Statement This is the equation: 2/(2 - x) + 6/(x^2 - x - 2) = 1Homework Equations sqrt= square root ^ = to the power of The Attempt at a Solution First thing that comes to mind is to turn it into this: 2x^2 - 2x - 2 + 12 - 6x = (2 - 2)(x^2 -x -2) Then it gets real ugly: x^3 - x^2 -...
  9. M

    What Is The Root of 5? Irrational

    I wonder how do I find the root of 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, and so on? And why the circle and the curve looks so smooth in the computer graphic software such as AutoCAD, Adobe Illustrator, etc., if the root is can not be found? It should be looks rough. Thank you
  10. STEMucator

    Prove if a < b, there is an irrational inbetween them

    Homework Statement No giving up :biggrin:! The question : http://gyazo.com/08a3726f30e4fb34901dece9755216f3 Homework Equations A lemma and a theorem : http://gyazo.com/f3b61a9368cca5a7ed78a928a162427f http://gyazo.com/ca912b6fa01ea6c163c951e03571cecf The fact ##\sqrt{2}## and...
  11. L

    Is the Square Root of 2 an Irrational Number?

    SquareRoot 2 is Irrational? \sqrt{}2 I've attached an image of what I'm talking about. Tell me what you think.
  12. V

    DoubtProve the term is irrational

    Homework Statement Prove that any number with zeroes standing in all decimal places numbered 10^n and only in these places is irrational?(yeah,its the easiet question in my list,but I am still not sure about it) Homework Equations The Attempt at a Solution when i think about...
  13. 1

    (for fun) Any non-perfect square has an irrational 2nd root

    Homework Statement I'm trying to see if I can prove that any non-square number's square root is irrational. I'm using only what I already know how to do ( I like trying to prove things myself before looking up the best proof), so it's going to be round-about. Attempt#1 Eventually required me...
  14. B

    Proof of square root 3 irrational using well ordering

    The part I don't understand is how they show there exists a smaller element. They assume s=t√3 is the smallest element of S={a=b√3: a,b€Z} . Then what they do is add s√3 to both sides and get s√3-s=s√3-t√3. I don't get how they thought of that or why it works.I know there exists an element...
  15. R

    MHB Is a non-repeating and non-terminating decimal always an irrational?

    We can build 1/33 like this, .0303... (03 repeats). .0303... tends to 1/33 . So,I was wondering this: In the decimal representation, if we start writing the 10 numerals in such a way that the decimal portion never ends and never repeats; then am I getting closer and closer to some irrational...
  16. P

    Is the Square Root of Pi Irrational?

    A question in my book says to prove that pi is irrational, I found a proof which I'm happy with and found a similar one on the web however on the solutions they have done: assume √π is rational i.e \sqrt{\pi} = \frac{p}{q} p,q \in \mathbb{Z} \pi = \frac{p^2}{q^2}, p^2,q^2 \in \mathbb{Z} ∴...
  17. R

    Irrational circles about the orgin

    I recall a post previously where the Op was wondering if any circle about the orgin having an irrational radius could pass through a rational point. The answer then was if the irrational radius was the square root of the sum of two rational squares then of course. Now I am wondering what if...
  18. K

    Prove Continuous at Irrationals, Discontinuous at Rationals: Real Analysis

    Real Analysis--Prove Continuous at each irrational and discontinuous at each rational The question is, Let {q1, q2...qn} be an enumeration of the rational numbers. Consider the function f(x)=Summation(1/n^2). Prove that f is continuous at each rational and discontinuous at each irrational...
  19. B

    Why Must a/b Be in Lowest Terms to Prove sqrt(2) Is Irrational?

    In proofs like prove sqrt(2) is irrational using proof by contradiction it typically goes like-We assume to the contrary sqrt(2) is rational where sqrt(2)=a/b and b≠0 and a/b has been reduced to lowest terms. I understand that at the very end of the arrive we arrive at the conclusion that it...
  20. S

    Proving square root of 2 is irrational with well ordering principle?

    Homework Statement I know how to prove that square root of 2 is irrational using the well ordering principle but what I'm wondering is, how can we use the well ordering principle to prove this when the square root of two isn't even a subset of the natural numbers? Doesn't the well ordering...
  21. A

    Proving √n Irrational: A Proof by Contradiction

    The problem reads as follows: Let n be a positive integer that is not a perfect square. Prove that √n is irrational. I understand the basic outline that a proof would have. Assume √n is rational and use a proof by contradiction. We can set √n=p/q where p and q are integers with gcd(p,q)=1...
  22. L

    A naive question about irrational numbers

    I've been thinking about this recently and couldn't find the answer to my question (even though I assume it's a really simple one, so forgive me if it's too trivial). Let's say we have two rods of length 1 meter and we put them at right angles to each other. Then we cut a third rod just long...
  23. S

    Is sin(10) Irrational? Proving the Irrationality of sin(10) in Degrees

    Homework Statement Prove \sin{10} , in degrees, is irrational. Homework Equations None, got the problem as is. The Attempt at a Solution Im kinda lost.
  24. P

    Prove that 5^(2/3) is irrational

    Homework Statement Prove that 5^(2/3) is irrational Homework Equations The Attempt at a Solution I tried writing a proof but that is not getting me any where. This is what I did so far - Show that 52/3 is irrational Proof: Suppose that 52/3 is rational: 52/3 = a/b...
  25. C

    I solving a proof dealing with the set of irrational numbers.

    Homework Statement Let x,y,t be in the set of all real numbers (R) such that x<y and t>0. Prove that there exists a K in the set of irrational numbers (R\Q) such that x<(K/t)<y Homework Equations if x,y are in R and x<y then there exists an r in Q such that x<=r<y The Attempt at a...
  26. D

    Clarifications on the least upper bound property and the irrational numbers

    Hello everyone. I desperately need clarifications on the least upper bound property (as the title suggests). Here's the main question: Why doesn't the set of rational numbers ℚ satisfy the least upper bound property? Every textbook/website answer I have found uses this example: Let...
  27. I

    Why is \sqrt{2}+\sqrt{3} irrational?

    Demonstrate that \sqrt{2}+\sqrt{3} is irrational. Thanks
  28. H

    Proof must be integer or irrational?

    Homework Statement Suppose a, b ε Z. Prove that any solution to the equation x^3 +ax+b = 0 must either be an integer, or else be irrational. Homework Equations Not sure if this is right but x = m / n where m divides b and n divides 1 The Attempt at a Solution So far i think i...
  29. O

    Prove that sqrt of a prime is irrational

    Homework Statement If a is a prime number, prove that √a is not a rational number. (You may assume the uniqueness of prime factorization.) Homework Equations Per the text: A positive integer a is said to be prime if a > 1 and whenever a is written as the product of two positive...
  30. T

    News Why is European Union so irrational in regards to trade with China?

    The trade balance between EU and China is -156.3€ billions, yet today EU agreed with China (http://uk.reuters.com/article/2012/09/20/uk-eu-china-summit-idUKBRE88J0QR20120920) to avoid trade protectionist measures. They keep doing this because China keeps buying EU countries' bonds and has many...
  31. D

    MHB Irrational and rational numbers

    Show that there are infinitely many rational numbers between two different irrational numbers and vice versa. So I started as such: WLOG let $a,b$ be irrational numbers such that $a<b$. By theorem (not sure if there is a name for it), we know that there exist a rational number $x$ such that...
  32. L

    Prove Square Root of 15 is Irrational

    Homework Statement Prove Square Root of 15 is Irrational The Attempt at a Solution Here's what I have. I believe it's valid, but I want confirmation. As usual, for contradiction, assume 15.5=p/q, where p,q are coprime integers and q is non-zero. Thus, 15q2 = 5*3*q2 = p2...
  33. G

    Is this Proof of √3 Irrationality Flawed?

    Homework Statement Prove that there is no rational x such that x2=3 2. The attempt at a solution Suppose that there is a rational x=\frac{a}{b}=\sqrt{3} and that the fraction is fully simplified. (ie. a and b have no common factor) Then a2/b2=3 which means a2=b2.3 and it follows...
  34. P

    How to find 'self locating digits' in irrational numbers

    Let us take the most mainstream irrational out there, (Pi). Now write (Pi) as: 3. 14159265... Let us number the decimals of Pi. 0 gets paired with 1 1 gets paired with 4 2 gets paired with 1 . . . 6 gets paired with 6 Thus 6 is a self locating digit. My question is then...
  35. N

    Uncovering the Irrationality of Pi: Exploring Its Definition and Proof

    Here's a question. Pi is said to be the ratio of a circle's circumference to its diameter. If this is the case, what does it say about the circumference of a circle that pi is still irrational. I get that pi is also used in the calculation of a circumference in the first place. Since this is...
  36. M

    Proof that Sqrt[3] is irrational - Is my logic valid?

    I am self-studying elementary analysis and am learning how to prove things. I have come up with a proof that √3 is irrational, and I believe it is valid, but I am unsure of my logic, as I have not seen it proved in just this way, and I don't have a prof to ask! So if anyone could just take a...
  37. S

    Exploring the Irrationality of Space-Time

    It seems like human understanding of space can be no clearer than our understanding of time. I still don't understand time. On the one hand it is a discrete interval; but it is also continuous and infinite. All our science is based on an understanding of this time concept and its constructions...
  38. J

    MHB SE Class 10 Maths - Rational or Irrational Numbers: $\cos(1^0)$ and $\tan(1^0)$

    $\cos(1^0)$ and $\tan(1^0)$ are Rational or Irrational no. Where angle are in Degree help required
  39. K

    Can Rational Numbers Approximate Irrational Numbers Arbitrarily Closely?

    Prove the theorem comprising that an irrational number β can be described to any limit of accuracy with the help of rational. Attempt- Taking the β to be greater than zero and is expressed with an accuracy of 1/n For any arbitrary value of β, it falls between two consecutive integers which...
  40. M

    Proving a number is irrational

    I picked up a book by Stephen Abbott called "Understanding Analysis" and it begins talking about rational and irrational numbers then it goes on proving how √2 is irrational. The proof is easy to understand but I wanted to use the same exact proof on a number I knew was rational. Let (p/q)2...
  41. C

    If n is a positive integer n then sqrt(4n-2) is irrational.

    Homework Statement if n is a positive integer than √(4n-2) is irrational. Homework Equations The Attempt at a Solution √(4n-2) Assume is rational then by definition of rationality √(4n-2)=p/q for some integers p,q where q≠0 so √(2(2n-1))=p/q by factoring out the...
  42. T

    Square root of 3 is irrational

    I am trying to prove sqrt(3) is irrational. I figured I would do it the same way that sqrt(2) is irrational is proved: Assume sqrt(2)=p/q You square both sides. and you get p^2 is even, therefore p is even. Also q^2 is shown to be even along with q. This leads to a contradiction. However...
  43. C

    Is a + (1/√2)(b-a) Irrational?

    Homework Statement Prove that if a and b are rational numbers with a≠b then a+(1/√2)(b-a) is irrational. Homework Equations The Attempt at a Solution Assume that a+(1/√2)(b-a) is rational. then by definition of rationality a+(1/√2)(b-a) =p/q for some integers p&q...
  44. J

    Irrational Winding of the Torus

    I am trying to prove the following result: Fix a,b \in \mathbb{R} with a \neq 0. Let L = \{(x,y) \in \mathbb{R}^2:ax+by = 0\} and let \pi:\mathbb{R}^2 \rightarrow \mathbb{T}^2 be the canonical projection map. If \frac{b}{a} \notin \mathbb{Q}, then \pi(L) (with the subspace topology) is not a...
  45. G

    Solving Irrational Numbers: Exploring My Reasoning

    Can anyone explain what is wrong with my reasoning? Suppose x = \frac{p}{q} and let x = \sqrt 2 + \sqrt 3 . Also, let a,b,c \in {\Bbb Z} and assume a < xc < b. If I show that xc must be an integer, and I know there does not exist c such that \sqrt 2 c, or \sqrt 3 c is an integer. Then...
  46. C

    Dense orbits of irrational n-tuples in n-Torus

    Hey all, this is my first post! (Although I've found a lot of useful answers here during the past). I have been trying to prove this fact, which is widely stated in literature and relatively well-known, about density of orbits of irrational n-tuples in the n-torus. My question is this: If...
  47. T

    Does Planck length and irrational solutions mean time can't be reversed?

    If there is an irrational solution to an equation for where a particle should be, for example from an ODE, then what effect does Planck length have on that? Does the actual position of the particle get rounded to an a multiple of the Planck length? If it does, wouldn't that imply there is a loss...
  48. J

    Irrational Flow yields dense orbits.

    I have the folloring problem: Given the following flow on the torus (θ_1)' = ω_1 and (θ_2)' = ω_2, where ω_1 /ω_2 is irrational then I am asked to show that each trajectory is DENSE. So I need to prove that Given any point p on the torus, any initial condition q, and any ε > 0, then there...
  49. A

    Irrational power of an irrational number

    What kind of number is sqrt(2)^sqrt(2)? I have noted sqrt(2)^sqrt(2) = 2^(sqrt(2)/2) = 2^(1/sqrt(2)), i.e. a rational number to an irrational power. Now, 1/sqrt(2) is less than 1, but greater than zero. So, given that 2^x is an increasing function, 2^(1/sqrt(2)) is less than 2^1, but...
  50. T

    Showing a vector field is irrational on

    Homework Statement Let F = ( -y/(x2+y2) , x/(x2+y2) ) Show that this vector field is irrotational on ℝ2 - {0}, the real plane less the origin. Then calculate directly the line integral of F around a circle of radius 1.Homework Equations The Attempt at a Solution To show F is irrotational we...
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