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. Pythagorean

    Irrational Numbers Don't Exist: Worship Spiders

    don't exist! and spiders are your new gods, worship them...
  2. Orion1

    Irrational denominator limit derivative

    Can anyone provide some information for this formula? I tried rationalizing the denominator and cross multiplication and combining terms, and also multiplying by the conjugate of the demoninator, what am I doing wrong? f(x) = \frac{1}{\sqrt{x + 2}} \; \; \; \text{find} \; f'(a) f'(a) =...
  3. C

    Proof that q^2 is divisible by rm^2

    http://www.artofproblemsolving.com/Forum/weblog.php?w=564 Could someone help me with 2b? Thanks
  4. U

    Is the square root of a prime number always going to be irrational?

    is the square root of a prime number always going to be irrational? just a random question.
  5. S

    Are all irrational numbers rational?

    Since pie is the ratio of the circumference of the circle to its diameter, isn't it possible that there exist a fraction for all nonrepeating going on forever decimal values?
  6. R

    News Irrational fear infects 34; three hospitalized

    What is it with people's awful risk managment? Does a significant fraction of the populace have so little grasp of numbers that they're more afraid of venemous snakes and nuclear power plants than of driving on the interstates? Look what's just happened in the midwest; there are people so...
  7. MathematicalPhysicist

    Between two rationals there are at least one irrational.

    in courant's and fritz's calculus text I am given the assignment to show the above, but first in the same question I am given this task: 1) a) if a is rational and x is irrational then x+a is irrational and if a isn't 0 then ax is irrational too. well this task is ofcourse trivial. i thought...
  8. I

    Easy to state an irrational theory

    Its easy to state an irrational theory and base it on belief. I've stumbled upon many theories over the years. And I finally discovered a theory that made fun of it all. Its called the Flying Spaghetti Monster theory. Its an amazing parody of the irrational. Finally something to laugh about...
  9. B

    Question re proof of irrational nos.

    Hi everyone, I've been reading about the proof of irrational nos. and I often encounter this phrase: "decreasing sequence of positive integers must be finite". What does this actually mean? Can anyone explain or point me to a link. Here's a link of one proof I've read re proof of...
  10. B

    Help with gsce irrational number question

    please please help me quick! hi i was practisin a gcse maths paper and need some help with last question; x and y are two positive irrational numbers. x + y is rational and so it x times y. a) by writing the 1/x + 1/y as a single fraction explain why 1/x + 1/y is always rational. b)...
  11. U

    Cubic with three real irrational roots.

    Considering the case of cubic polynomials with integer coefficients and three real but irrational roots. Is it true that it's impossible that all three roots can be in the form of simple surd expressions like r+s \sqrt{n} (where r and s are rational and sqrt(n) is a surd). The argument is that...
  12. D

    Irrational Distances in Quantised Geometry

    Since distances have to be multiples of the quantum of length, how can there be irrational distances? Please provide a non-technical explanation if possible, or correct me if my assumption is wrong.
  13. B

    Irrational Number and the Borel Sets

    Took a test in my Analysis class today. One question asked us to prove that the set of Irrational numbers was a Borel Set. After working on the other problems for 90 minutes, I stared blankly at this one for what seemed life a long time. I eventually showed (I think) that the set of Rational...
  14. J

    Non-Negative Integer Binary Concatenation: Is This an Irrational Number?

    I am interested in the following number which is obtained by concatenting the binary representations of the non-negative integers: .011011100101110111... i.e. dot 0 1 10 11 100 101 110 111 ... This is a little bigger than .43 and I assume it irrational since no pattern of bits repeats...
  15. L

    Non integer square roots and pi = irrational?

    Since one can construct the length of a non-integer square root by drawing accurate triangles, and can draw a circle with a circumference of pi, then shouldn't one be able to plot corresponding non-integer square roots and pi on a number line? I know these numbers are supposedly irrational, but...
  16. V

    Prove by induction that 2^1/2 is irrational.

    Prove by induction that 2^1/2 is irrational.
  17. W

    Can Pi be Used as a Random Number Generator for Proving Normality?

    Just wondering, if you group decimal places of an irrational number, let's say into sequences of groups of 10, for example, if k is irrational 4.4252352352,3546262626,224332 (I made that up) they you group (.4252352352) (3546262626) (and so on) then my question is that the probability...
  18. E

    Checking if n is a Multiple of 2: Proving 3 (sqrt 2) is Irrational

    Hi can anyone help me check if I've approached this question correctly and offer any help on part b) of the question? Thanks! :smile: a) Prove that if n is an integer and n^3 is a multiple of 2 then n is a multiple of 2. Let n^3 be a multiple of 2 but suppose n is not a multiple of 2...
  19. M

    Is √5 + √3 Irrational by Contradiction?

    we can prove that √5 is irrational through contradiction and same applies for √3. but can we prove that √5 + √3 is irrational by contradiction?
  20. N

    Prove the Square Root of 2 is irrational

    This is Algebra 2 question... I have to prove that the square root of 2 is irrational... First we must assume that sqrt (2) = a/b I never took geometry and i don't know proofs... Please help me. Thank you.
  21. mattmns

    Proof: x is irrational => sqrt(x) is irrational

    Ok so I am to prove: If x is irrational, then \sqrt{x} is irrational. So I started by trying to prove the contrapositive: If \sqrt{x} is rational, then x is rational. So then \sqrt{x} = \frac{m}{n} For integers m and n, n\neq0 Then square both sides. x = \frac{m^2}{n^2} This is...
  22. mattmns

    Can the logarithm of a prime number be rational?

    I am to prove that \log_{2} 7 is irrational. So I started by saying that what if \log_{2} 7 is rational. Then it must be in the form of \frac{m}{n} where m and n are integers. So now \log_2 7 = \frac{m}{n} So I took the 2^ up of each and now 7 = 2^{\frac{m}{n}} Then 7 = \sqrt[n]{2^m} But...
  23. V

    Why Is the Number e Considered Irrational?

    My book does not make sense to me. Here is what it says: I know that: e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + ... + \frac{1}{n!} + \frac{\theta}{n!n}, 0 < \theta < 1 If e is rational then e = \frac{m}{n}; m, n \in Z :confused: And the greatest common factor of m, n is 1...
  24. S

    Explore Irrational Numbers: Find Out How Close You Can Get!

    Just curious about a thing I've been thinking of: It's true that that there are numbers that aren't rational... let's say x is such a number. Now we take two integers, a and b where a is the integer if x is rounded up, and b is the integer if x is rounded down. Forming their arithmetic...
  25. T

    Sme question about irrational numbers

    Some question about irrational numbers Our teacher showed us Cantor's second diagonal proof. He said that by this proof we can show that there are more irrational numbers than rational numbers. He also said that the cardinality of natural numbers or rational numbers has a magnitude...
  26. C

    Irrational digits countably infinite?

    Is the set of digits of an irrational number countably infinite? I suspect the answer has to do with long division.
  27. T

    Sequence of rationals that converge to irrational

    Find a sequence of rational numbers that converges to the square root of 2
  28. T

    Irrational + irrational = rational

    Can someone prove that there exists x and y which are elements of the reals such that x and y are irrational but x+y is rational? Certainly, there are an infinite number of examples (pi/4 + -pi/4 for example) to show this, but how would you prove the general case?
  29. C

    What is the Sum of Two Irrational Numbers?

    I have a question. I realize that two rational numbers added together equal a rational number and that a rational added to a irrational equal a irrational number; but how do I show what a irrational plus a irrational equal?
  30. C

    Proving the Irrationality of √3 and Other Non-Perfect Square Roots

    Hello all I encountered a few questions on irrational numbers. 1. Prove that \sqrt{3} is irrational [/tex]. So let l = \sqrt{3} . Then if l were a rational number and equal to \frac{p}{q} where p, q are integers different from zero then we have p^{2} = 3q^{2} . We can assume that...
  31. C

    Integral Involving Irrational values

    Hello all If we want to calculate the definite integral \int^b_a x^{\alpha} for any irrational value of \alpha where 0<a<b do we use the Mean Value Theorem? Would \alpha be represented as a limit of a sequence of rational numbers \alpha = \lim_{x\rightarrow \infty} \alpha_n and...
  32. L

    Rational and Irrational Numbers

    I need to show that a rational + irrational number is irrational. I am trying to do a proof by contradiction. So far I have: Suppose a rational, b irrational. Then a = p/q for p, q in Z. Then a + b = p/q + b = (p + qb) / q But I don't know where to go from here because I still have a...
  33. R

    Factoring irrational equations

    Ok, I hope there’s a math wiz out there who can help me. I have to factor: f (x)=3x^4-8x^3-5x^2+16x-5 Just from looking at it, you know that possible values for x are: ±1, ±5/3, ±1/3, ±5 However, if you plug in these real numbers, none of them work, therefore meaning that x is an...
  34. G

    Constructing Lengths with Irrational Numbers

    This might be a stupid question, but how can you construct something that has an irrational length? For example if you make a right triangle with the 2 sides=1 the hypotenuse is sqrt(2). How can sqrt(2) be a length if that number goes on for ever and never repeats?
  35. T

    What number is rational and irrational

    When is x rational and irrational? Also When is r positive and negative? :confused:
  36. C

    Schwarz Inequality and Irrational Numbers

    -------------------------------------------------------------------------------- Hello everyone. I have 2 questions. 1. Prove that the cube root (3) + sqrt (2) is irrational. My Solution Assume l is an irrational number of the form p/q where p and q are integers not equal to 0...
  37. C

    Schwarz Inequality and Irrational Numbers

    Hello everyone. I have 2 questions. 1. Prove that the cube root (3) + sqrt (2) is irrational. My Solution Assume l is an irrational number of the form p/q where p and q are integers not equal to 0. Then p^6 / q^6 = [(cube root(3) + sqrt (2))]^6 I concluded that it must be in the...
  38. M

    Pi miscalculated or not irrational?

    pi miscalculated or not irrational? I know that computers have calculated thousands of digits of pi, but does this mean that pi is an irrational number? How can we be so sure that it is irrational? And I have one more question. The circles we see in real life are not perfect circles. Does this...
  39. D

    An accurate representation of Irrational and rational numbers

    Is there an accurate way to write the value of an Irrational number? If there is no an accurate way to write the value of an Irrational number, then can we conclude that no irrational number has an exact place on the real line? And if there is an exact place to an irrational number on the...
  40. W

    Irrational numbers vs. Transcendental numbers

    It would seem that an irrational number would have to be a transcendental number. If a transcendental number is a number which goes on infinitely and never repeats, then all irrational numbers would have to be transcendental, because if they repeated then you could find a fraction doing the...
  41. N

    Understanding Irrational Numbers: Is it Possible to Exact Measure?

    Okay, I was thinking about irrational numbers, and I came to this conclusion: It is impossible exactly measure an irrational number.I am probably wrong, and that's why I posted this thread to check the validity of that statement. Here is my proof: If you wanted to cut a piece of paper...
  42. M

    Is the Sum of Square Roots of 2 and 3 Irrational? A Proof by Contradiction

    Yeh just having a problem seeing a way to prove that 6^(1/2) is irrational. Using this answer and proof by contradiction I need to prove that 2^(1/2) + 3^(1/2)is also irrational, however I sould be able to attempt this if I can get the above right. Any help much appreciated.
  43. K

    Is sqrt(5) Rational? Exploring the Proof by Contradiction Method

    Alright, heading says it all. This is a nice problem heh.. I can see how to prove sqrt(5) is irrational. I think this method works up to the points where the fact 5 is a prime is used, (ie prime lemma) on 5 which doesn't work so well on 6! hehe Was thinking of maybe using product of primes...
  44. J

    Irrational Numbers: Expressible as Infinite Summations?

    I apologise if this belongs in another place, but: Can all irrational numbers be expressed as infinite summations, ie like Pi and e? I'm looking for: provable, disprovable, or neither. This is essential to something else I am working on. sincerely, jeffceth
  45. M

    Death brings out the irrational in us

    When someone we love dies, we tend to believe in some sort of after life, or utopia that the person transfers to. At the least, we just don't want to think about it, and assume that there is an afterlife, without any reasoning whatsoever. Has anyone here ever survived the death of a loved one...
  46. U

    Looking for Easy proof of Pi Irrational

    Looking for "Easy" proof of Pi Irrational Hi, I just got to this forum after searching for an easy proof that Pi is irrational. The thread I found (google) was this one HERE. I wanted to reply, but since it is now “archived” I thought it would be better to post a new thread. Sorry if this...
  47. S

    Prove that sqrt2 + sqrt6 is irrational

    Prove that sqrt2 + sqrt6 is irrational. Where do I start with this?
  48. Iacchus32

    Is Irrational Thinking Unavoidable? Examining Society's Standards

    Does anyone on this forum believe reality should not be answered on an individual by individual basis? For indeed, we were all brought up under a diversity of circumstances, the very diversity which allows life to exist, and makes each one of us individuals. So who's to say one person is...
  49. K

    Integration of irrational function

    Here's the question that I got stuck: [inte]sqrt[x/(a-x)] dx .........(*) I tried to use the following substitution u=sqrt[x/(a-x)] and ........(1) dx = 2u(1-a)/(1+u2)2 du...(2) sub (1) and (2) into (*), after a few steps, I got (2-2a)[inte]du/(1+u2) - 2(1-a)[inte]du/(u2+1)2 The...
  50. N

    We know e (exponential) is a irrational number

    we know e (exponential) is a irrational number... how can we prove it??
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