Irrational Definition and 353 Threads

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. H

    Is an irrational root of a real number imaginary or real?

    We can easily comment the result of a root operation just by the information if the degree of the root is odd or even. But what if the degree of the root (or power) is irrational? For example; -64 ^ \frac{1}{2} \, = \, j8 \,\,\,\,\, (imaginary) -64 ^ \frac{1}{3} \, = \, -4 \,\,\,\,\...
  2. T

    Natural constants: are they irrational numbers?

    Do we have at present any knowledge whether our natural constants (gravity constant, Planck's constant, ...) are rational or irrational numbers? Thanks, Trinitiet
  3. P

    Physical representation of irrational numbers

    My question relates to a specific example, namely the square root of two. If one forms a right isosceles triangle with the hypotenuse equal to 2 (be it metres, centimetres or whatever) then the other two sides must equal the square root of 2. But the square root of 2 is an irrational number. If...
  4. V

    Can a Circle with an Irrational Center Have More Than Two Rational Points?

    How many rational points can be there on a circle which has an irrational centre? (rational point is a point which have both x,y as rational numbers) how to proceed?? answer is: atmost 2
  5. S

    Proof Involving Continuity, Irrational Numbers From Elementary Proof Class

    Homework Statement Let f be a non-zero continuous function. Prove or disprove that there exists a unique, real number, x, such that the integral from 0 to x of f(s) w.r.t. s = pi. Homework Equations If any exist, please let me know. The Attempt at a Solution...
  6. C

    Simplifying √(3+2√(2)): Find the Integer and Irrational

    Homework Statement √(3+2√(2)) can be simplified into a fairly simple sum of two number: one is an integer and one is an irrational number . Find them and show you are correct by squaring both sides of the "equation" Homework Equations The Attempt at a Solution
  7. B

    Number system with an irrational base

    Could you provide a link to a 'number system with an irrational base'? I only found this link http://www.jstor.org/pss/3029218 The link shows a small part of this number system ... I would to know more about it.
  8. J

    Prove that if a>b, there is an irrational number between a and b

    Homework Statement I can't figure this part out Homework Equations In the previous part of this problem, I proved that there is a rational number between a and b. The Attempt at a Solution Maybe 1 < √2 < 2 ---> a < √2 a < 2a ---> a < √2 a < 2b ... then somehow morph that into a < q < b...
  9. J

    Proving √2 is Irrational: A Brief Explanation

    Homework Statement As the title says. Homework Equations Rational number: a/b for some integers a, b Even number: 2k for some integer k Odd number: 2j+1 for some integer j The Attempt at a Solution Assume √2 is a rational number. Then it can be expressed a/b for some integers...
  10. J

    Why do irrational numbers appear in quantum physics?

    This is a question I've had for some time, but didn't think to ask whenever I was around someone who might have been able to answer it. If energy and matter are made of quanta, then why is quantum physics coming up with so many irrational results instead of integral ones?
  11. A

    Proving that n-th root of n is irrational

    Homework Statement if n>1, prove that \sqrt[n]{n} irrational Homework Equations n/a The Attempt at a Solution so suppose it is rational, so i know \sqrt[n]{n} must be integer, say p then n=p^n and how do i prove this is not true?? i can prove if p>1 then p^n>n for all n>1 by induction...
  12. M

    Set of irrational numbers between 9 and 10 are countable

    Homework Statement The set of irrational numbers between 9 and 10 is countable. Homework Equations The Attempt at a Solution My belief is that I can prove by contradiction. first, i must prove by contradiction using diagonalization that the real numbers between 9 and 10 are...
  13. J

    How can the ratio of those two numbers be irrational?

    if the diameter of a circle is a finite value, and the circumference of a circle is a finite value, how can the ratio of those two numbers be irrational? a new team, yee and kondo, just determined the value of pi to 5 trillion digits. (wow) it takes only 39 digits of pi to make a circle...
  14. A

    Is the Square Root of 2 Rational?

    sorry, I am using phone, owho i hope you can get it. suppose its rational. in the form a/b gcd(a,b)=1 and so and so and so and then they concluded that a^2=2 is a contradiction. but i cannot see what it conradict the assumption. help
  15. M

    Sign Analysis of irrational functions

    Hi, I am from Belgium so i apologize for my English, I have encountered a problem, i don't understand how to set up the sign analysis of irrational functions, so i decided to find some information on the internet, but the problem is that I am not sure if "sign analysis of irrational functions"...
  16. M

    Is \sqrt{2} + \sqrt{3} + \sqrt{5} irrational?

    Homework Statement Is \sqrt{2} + \sqrt{3} + \sqrt{5} rational? Homework Equations If n is an integer and not a square, then \sqrt{n} is irrational For a rational number a and an irrational number b, a + b is irrational a * b is irrational if a is not equal to 0 The Attempt at a Solution...
  17. K

    Show f(x) = { x/2 if x rational , x if x irrational is not differentiable at 0

    Homework Statement Show that the function f(x) = { x/2 if x is rational { x if x irrational is not differentiable at 0 Homework Equations If f is differentiable at 0 then for every e > 0 there exists some d > 0 such that when |x| < d, |(f(x)-f(0))/x - L | < e...
  18. G

    What Are Some Interesting Properties of Pi?

    So if Pi is an irrational number, and therefore has an infinite line of numbers after the decimal point; my intuition tells me it would take an infinite amount of time to determine its exact value. a) Do calculators and computers somehow know Pi's exact value or is it just an estimate? b)...
  19. M

    Can irrational numbers exist on the numberline?

    This may be an elementary question, but I've been thinking about it a little bit and wondering what other people thought. First, let me say that I'm talking about a number line not as a set but in the more literal sense, like a partitioned line that might exist as the axis of a graph. So...
  20. A

    Find Irrational Exponents of 2^π

    How can we find irrational exponents of a real or imaginary number?? say 2 raise to the power pi
  21. n.karthick

    Irrational number approximation by a rational number

    Is there a way ( a theorem ) to find a rational number for a given irrational number such that it is an approximation to it to the required decimal places of accuracy. For example 22/7 is an approximate for pi for 2 decimal places.
  22. K

    Smallest positive irrational number

    Homework Statement Prove that there is no smallest positive irrational number Homework Equations The Attempt at a Solution I have no idea how to do this, please help walk me through it.
  23. H

    0 when irrational, 1/q in lowest terms with rational not differentiable.

    Homework Statement Hi, I have this function: f(x ) = 0 (x is irrational) or f(x) = 1/q for rational p/q in lowest terms. show that this function is not differentiable anywhere The Attempt at a Solution This is the answer from the solutions book: consider [(f(a+h) - f(a ) ) /...
  24. B

    Rational and irrational numbers proof

    Hey all, I'm new here so I'm a little noobish at the formatting capabilities of PF. Trying my best though! :P Homework Statement Let a, b, c, d \in Q, where \sqrt{b} and \sqrt{d} exist and are irrational. If a + \sqrt{b} = c + \sqrt{d}, prove that a = c and b = d. Homework...
  25. K

    Prove square root of 3 is irrational

    I have no idea how to go about this, except that we are supposed to use proof by contradiction to show that the square root of 3 is an irrational number. Any help or tips is appreciated. Thanks.
  26. W

    Irrational numbers - incomprehnsible?

    Hi, I am having an issue with irrational numbers and the term irrational. Main Entry: 1ir·ra·tio·nal Pronunciation: \i-?ra-sh(?-)n?l, ?i(r)-\ Function: adjective Etymology: Middle English, from Latin irrationalis, from in- + rationalis rational Date: 14th century : not rational: as a...
  27. Mentallic

    Proving \sqrt{3} is Irrational

    Homework Statement I need to prove that \sqrt{3} is irrational. The Attempt at a Solution The prior problem was to show \sqrt{2} is irrational and the solution had to do with a contradiction that each number must be even or something (frankly, I didn't understand it too well). But I...
  28. G

    Irrational numbers in infinite list of integers

    Is it safe to assume that the absolute value of sin x is greater than zero for all positive integer values of x? I have no real experience in number theory, and I don't know if you can say that there are no irrational numbers in an infinite list of integers.
  29. S

    How to prove √X is irrational number

    when X is even number,it's easy to prove but how about the condition which X is odd number? I have no idea of this
  30. J

    Irrational Numbers and Real Life I need 6 answers

    Once again my professor asked us to ask 6 people the following question and see how they answer it so if you could respond and give an answer, I would really appreciate it. And if possible can you also tell me a little bit about your mathematics background? We are supposed to write up what...
  31. O

    Question about irrational numbers

    Let p and q be distinct primes. Prove that \sqrt{p/q} is a irrational number.
  32. C

    For every rational number, there exists sum of two irrational numbers

    Homework Statement Prove: For every rational number z, there exists irrational numbers x and y such that x + y = z. Homework Equations by definition, a rational number can be represented by ratio of two integers, p/q. The Attempt at a Solution Is there a way to do this by...
  33. Char. Limit

    Is Every Root of 2 Higher Than 1 Irrational?

    I was thinking on the square root of 2 being irrational proof... and I got the idea that you could use the same idea for every root higher than two. The cube root, the quartic root, the quintic root, etcetera. (Obviously assuming the roots are natural numbers.) As a reassurance I'm not crazy...
  34. C

    Prove in any interval there exists an irrational z

    Homework Statement Prove that in any interval there exists an irrational z. Homework Equations The Attempt at a Solution My professor wrote this for me when trying to explain how to prove this: a \notin Q, \epsilon rational [r, s]\in a l([r, s])<\frac{\epsilon}{2}...
  35. icystrike

    Irrational Numbers: Infinite Numerical Values Explained

    Irrational Numbers are contained by infinite numerical values?
  36. P

    Prove the following is irrational

    Prove that http://img705.imageshack.us/img705/2408/aaa12.png is irrational. A user on another forum suggested the following: http://img130.imageshack.us/img130/1352/abcde.png I follow that up to the last sentence. Can anyone clarify for me how to show this proof?
  37. Mentallic

    What is the value of an irrational infinite sum with a unique pattern?

    I'm curious to answer (or at least reasonably understand) what the answer to: \sqrt{a}+\sqrt{a+\sqrt{a}}+\sqrt{a+\sqrt{a+\sqrt{a}}}+... might be, where a>0. It doesn't follow any ordinary pattern, such as an arithmetic or geometric progression. Also, if there is for any reason an easily...
  38. M

    Is (x+sqrt(2)) or (-x+sqrt(2)) rational?

    Homework Statement Prove that for each real number x, (x+sqrt(2)) is irrational or (-x+sqrt(2)) is irrational. Homework Equations We have already proven sqrt(2) is irrational and a rational+an irrational=irrational. The Attempt at a Solution Proof by contradiction...
  39. G

    Prove the set of irrational numbers is uncountable.

    Homework Statement Prove the set of irrational numbers is uncountable. Homework Equations The Attempt at a Solution We proved that the set [0,1] is uncountable, but I'm not sure how to do it for the irrational numbers.
  40. S

    Rational and irrational numbers. (semi- )

    Rational and irrational numbers. (semi-urgent) I need to figure this out by tomorrow =/ Homework Statement a. If a is rational and b is irrational, is a+b necessarily irrational? What if a and b are both irrational? b. If a is rational and b is irrational, is ab necessarily irrational...
  41. U

    Is Planck's Constant Irrational?

    My maths teacher was talking about irrational numbers and I asked if Planck's constant was one, but he said no. However, I don't understand how this can be as it does not seem to terminate. Can anyone help? Thanks, Jamie
  42. F

    Sequence of rationals converging to an irrational limit

    Homework Statement If (p_n/q_n)_{n \in \mathbb{N}} is a sequence of rationals in lowest terms converging to the irrational number x > 0, then \lim_{n \to \infty} q_n = +\infty. Homework Equations The Attempt at a Solution I'm quite clueless about this. But heuristically, I...
  43. N

    Integral of product of irrational powers

    Hello, I'm trying to integrate a product of irrational "polynomials" that have arisen through dimensional regularization. As a warmup, I'm trying to understand how to integrate: (w-a)^b (w-c)^d, where b and d are arbitrary numbers, possibly irrational. Mathematica can solve this...
  44. S

    Is sqrt{30} Irrational? Proving its Irrationality through Unique Factorization

    Well, there is a problem, i have solved/proved it, but i am not sure whether it is correct. THe problem is this: Using unique factorization into primes prove that there are no integers a and b such that a^2=30b^2, and thus show that sqrt{30} is irrational. Proof:using unique factorization...
  45. C

    Irrational Numbers: Proving a Number is Irrational

    What is a proof that a number is irrational. For instance, how do we know the PI goes on forever without a pattern?
  46. P

    Is Gamma Irrational? Investigating the Irrationality of Pi and e

    I've been thinking about pi^e lately, and trying to prove that it is irrational. By rewriting e as 1+1+1/2+1/3!+...+1/n! I got it to pi^2*pi^(1/2)*pi^(1/3!)*...*pi^(1/n!), and proved that each of these terms is irrational. I'm stuck when it comes to showing that multiplied together these numbers...
  47. Y

    Prove sq root of 2 + sq root of 3 is irrational

    hi,i read the post of sq root of 2 + sq root of 3.i understand tat i should use contradiction to solve it.yet,i stuck halfway when i tried to solve it. sq root 2 + sq root 3 = p/q 2 + 2*sq root 6 +3=p^2/q^2 5+2*sq root 6 = p^2/q^2 wat should i do after this?i hav to prove that the addition...
  48. C

    Apostol's Analysis 1.11- Irrational Numbers Proof

    Homework Statement Given any real x > 0, prove that there is an irrational number between 0 and x. Homework Equations I'm not sure if the concepts of supremums or upper bounds can used. The Attempt at a Solution Take an irrational number say Pi. We can always choose a number n such...
  49. N

    Irrational numbers and repeating patterns

    Hi, I want to show that an irrational number (let's say pi) can never have an (infinitely) repeating pattern (such as 0.12347 12347 12347 ...). Is it possible to 'proof' (or just make it more acceptable, I don't need a 100% rigorous proof) this easily, without using too much complicated math...
  50. J

    Show that 2^(1/3) + 3^(1/3) is irrational.

    Homework Statement Show that 2^(1/3) + 3^(1/3) is irrational. Hint: show that [x][/0] = 2^(1/3) + 3^(1/3) is algebraic by constructing an explicit polynomial f(x) with integer coefficients such that f([x][/0]) = 0. Then prove that f(x) has no rational roots. Note:[x][/0] means x subscript...
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