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. Q

    If a, b are irrational, then is ##a^b## irrational?

    Homework Statement True or false and why: If a and b are irrational, then ##a^b## is irrational. Homework Equations None, but the relevant example provided in the text is the proof of irrationality of ##\sqrt{2}## The Attempt at a Solution Attempt proof by contradiction. Say ##a^b## is...
  2. CollinsArg

    I Irrational numbers aren't infinite. are they?

    Most than a question, I'd like to show you what I've got to understand and I want you to tell me what do you think about it. I'm not a math expert, I just beginning to study maths, and I'm reading Elements by Euclids, and I've been doing some research on immeasurable numbers. My statement is...
  3. PsychonautQQ

    Finding the minimal polynomial of an irrational over Q

    Homework Statement Let a = (1+(3)^1/2)^1/2. Find the minimal polynomial of a over Q. Homework EquationsThe Attempt at a Solution Maybe the first thing to realize is that Q(a):Q is probably going to be 4, in order to get rid of both of the square roots in the expression. I also suspect that...
  4. K

    MHB Prove that \sqrt{n+\sqrt{n}} is irrational for every natural number

    I am not sure if this is good, so I would like someone to help me a little and tell me if this is a good proof. I know how to prove that, for example, \sqrt{2} is irrational, so I tried to do something similar with this expression. So, let's assume otherwise, that \sqrt{n+\sqrt{n}} is not...
  5. Mr Davis 97

    B Solving Irrational Inequality: Why Square Root Matters

    So I am trying to solve a simple rational inequality: ##\sqrt{x} < 2x##. Now, why can't I just square the inequality and go on my way solving what results? What precisely is the reason that I need to be careful when squaring the square root?
  6. S

    B Proof that non-integer root of an integer is irrational

    I have been looking at various proofs of this statement, for example Proof 1 on this page : http://www.cut-the-knot.org/proofs/sq_root.shtml I'd like to know if the following can be considered as a valid and rigorous proof: Given ##y \in \mathbb{Z}##, we are looking for integers m and n ##\in...
  7. G

    Integration of irrational function

    Homework Statement Find the integral \int \frac{1}{(x-2)^3\sqrt{3x^2-8x+5}}\mathrm dx 2. The attempt at a solution I can't find a useful substitution to solve this integral. I tried x-2=\frac{1}{u},x=\frac{1}{u}+2,dx=-\frac{1}{u^2}du that gives \int \frac{1}{(x-2)^3\sqrt{3x^2-8x+5}}\mathrm...
  8. Kilo Vectors

    I How can any measure of a physical quantity be irrational?

    Hello Aren't all irrational numbers having an infinitely long decimal component? If so, how can any measure of a physical quantity be irrational? the decimal component is infinitely long..but the magnitude of the physical quantity surely isnt?
  9. Isaac0427

    Insights Complex and Irrational Exponents for the Layman - Comments

    Isaac0427 submitted a new PF Insights post Complex and Irrational Exponents for the Layman Continue reading the Original PF Insights Post.
  10. RJLiberator

    Prove that a^(1/n) is an integer or is irrational

    Homework Statement Let a and n be positive integers. Prove that a^(1/n) is either an integer or is irrational. Homework EquationsThe Attempt at a Solution Proof: If a^(1/n) = x/y where y divides x, then we have an integer. If a^(1/n) = x/y where y does not divide x, then a = (a^(1/n))^n =...
  11. TyroneTheDino

    Expressing the existence of irrational numbers

    Homework Statement Express the following using existential and universal quantifiers restricted to the sets of Real numbers and natural numbers Homework EquationsThe Attempt at a Solution I believe the existence of rational numbers can be stated as: ##(\forall n \in \Re)(\exists p,q \in...
  12. I

    Proof that e is irrational using Taylor series

    Homework Statement Using the equality ##e = \sum_{k=0}^n \frac{1}{k!} + e^\theta \frac{1}{(n+1)!}## with ##0< \theta < 1##, show the inequality ##0 < n!e-a_n<\frac{e}{n+1}## where ##a_n## is a natural number. Use this to show that ##e## is irrational. (Hint: set ##e=p/q## and ##n=q##)...
  13. E

    Simplifying square root of an irrational

    Homework Statement Find [(3 - 51/2)/2]1/2 Homework EquationsThe Attempt at a Solution My calculator says (-1 + √5)/2 I have no idea how. Rationalising doesn't really do much good. Just tell me where to start.
  14. Alpharup

    Spivak "root 2 is irrational number" problem

    Am using Spivak. Spivak elegantly proves that √2 is irrational. The proof is convincing. For that he takes 2 natural numbers, p and q ( p, q> 0)...and proves it. He defines irrational number which can't be expressed in m/n form (n is not zero). Here he defines m and n as integers. But in the...
  15. Chrono G. Xay

    Predict Digits of Irrational Numbers with Modular Arithmetic Summation?

    Would it be possible to write an equation utilizing a summation of a modular function of a Cartesian function, whose degree is dependent upon the index of the root, in that it predicts the digits less than 1 of the root, that when summed equals the computed value sqrt( n )? I already have what...
  16. J

    How to find the equation of this tangent?

    Mod note: Thread moved from Precalc section Homework Statement F(x)=sqrt(-2x^2 +2x+4) 1.discuss variation of f and draw (c) 2.find the equation of tangent line to (c) that passes through point A(-2,0) The Attempt at a Solution I solved first part I found the domain of definition and f'(x) and...
  17. J

    How to know if this irrational function has no asymptotes?

    1. The problem statement, all variables and given/known dat F(x)=x+1-3sqrt((x-1)/(ax+1)) For which value of a ,(c) has no asymptote? Homework EquationsThe Attempt at a Solution I know if a>0 then (c) will have 2 asymptote And if a<o then (c) will have 1 vertical asymptote. But I can't find...
  18. S

    Irrational Roots Theorems for Polynomial Functions

    Is any Irrational Roots Theorem been developed for polynomial functions in the same way as Rational Roots Theorems for polynomial functions? We can choose several possible RATIONAL roots to test when we have polynomial functions; but if there are suspected IRRATIONAL roots, can they be found...
  19. Curieuse

    Rational and irrational numbers

    Homework Statement Determine a positive rational number whose square differs from 7 by less than 0.000001 (10^(-6)) Homework Equations - The Attempt at a Solution Let p/q be the required rational number. So, 7> (p/q)^(2) > 7-(0.000001) ⇒ √(7) > p/q > √(7-.000001) ⇒√(7) q> p >...
  20. Grinkle

    Can one use an irrational number as a base?

    Is it sensible to consider a base pi number system? Can one make an irrational number rational by defining it as the unit of a counting system? I don't know what constitutes an mathematically consistent 'number line' - this question might not make sense. I'm just thinking that if I use pi as...
  21. Mr Davis 97

    Solving an irrational equation

    I have the irrational equation ##\sqrt{x - 1} + \sqrt{2 - x} = 0##, which has no real solutions. However, when I try to solve the equation, I get a real solution, that is: ##\sqrt{x - 1} + \sqrt{2 - x} = 0## ##\sqrt{x - 1} = -\sqrt{2 - x}## ##(\sqrt{x - 1})^{2} = (-\sqrt{2 - x})^{2}##...
  22. DiracPool

    Irrational numbers and Planck's constant

    [Mentor's note: this was originally posted in the Quantum Physics forum, so that is what "this section" means below.] ---------------------------------------------------- I wasn't sure whether to post this question in this section or the general math section, so I just decided to do it here...
  23. Fallen Angel

    MHB Proving $\sin\left(10^{\circ}\right)$ is Rational or Irrational

    Is $\sin\left(10^{\circ}\right)$ rational or not? Prove it.
  24. anemone

    MHB Prove cos (π/100) is irrational

    Prove that $\cos \dfrac{\pi}{100}$ is irrational.
  25. Math Amateur

    MHB Every interval (a,b) contains both rational and irrational numbers

    I am reading Chapter 1:"Real Numbers" of Charles Chapman Pugh's book "Real Mathematical Analysis. I need help with the proof of Theorem 7 on pages 19-20. Theorem 7 (Chapter 1) reads as follows: In the above proof, Pugh writes: " ... ... The fact that a \lt b implies the set B \ A contains...
  26. anemone

    MHB How Do You Solve This Complex Irrational Equation?

    Solve the irrational equation $x^4-9x^3+16x^2+15x+26=\dfrac{7}{\sqrt{x^2-10x+26}+\sqrt{x^2-10x+29}+\sqrt{x^2-10x+41}}$.
  27. davidbenari

    How calculators compute stuff (like irrational exponentiation)

    I'm just curious as to how a calculator does the following operation: ##5^{1/\pi}## I mean, it has to look for the number that raised to the power of pi, gives me 5. I think that's insane. How does it do that? How does a calculator store the value of pi? -- I guess that's a more boring...
  28. T

    In proof of SQRT(2) is irrational, why can't a,b both be even

    In the classic proof of irrationality of SQRT(2) we assume that it can be represented by a rational number,a/b where a, b are integers. This assumption after a few mathematical steps leads to a contradiction, namely that both a, b are even numbers. Why is that a contradiction? Well you can...
  29. M

    Prove that sqrt(2) is irrational using a specific technique

    Homework Statement Prove that √2 is irrational as follows. Assume for a contradiction that there exist integers a, b with b nonzero such that (a/b)2=2. 1. Show that we may assume a, b>0. 2. Observe that if such an expression exists, then there must be one in which b is as small as...
  30. Dethrone

    MHB Piecewise function - rational and irrational

    $$g(x)=\begin{cases}x^2, & \text{ if x is rational} \\[3pt] 0, & \text{ if x is irrational} \\ \end{cases}$$ a) Prove that $\lim_{{x}\to{0}}g(x)=0$ b) Prove also that $\lim_{{x}\to{1}}g(x) \text{ D.N.E}$ I've never seen a piecewise function defined that way...hints?
  31. anemone

    MHB What are the roots of a rational equation with given conditions?

    Find all irrational numbers $k$ such that $k^3-17k$ and $k^2+4k$ are both rational numbers.
  32. A

    My proof that the square root of 2 multiplied by r is irrational

    Here it is, for you to critique. This is a proof by contradiction. This is a good example of how I usually go about doing proofs, so if you give me tips on how to improve this particular proof, I'll be able to improve all my other proofs. I just learned how to do proof by contradiction...
  33. adjacent

    Is 3.62566 an Irrational Number?

    An irrational number is any real number which cannot be expressed as the ratio of two real numbers. Then is 3.62566 is also an irrational number? I thought all irrational numbers are uncountable. I am not sure that the above is an irrational number :confused:
  34. T

    Is there an irrational multiple of another irrational to yield Integer

    This question specifically relates to a numerator of '1'. So if I had the irrational number √75: 1/(x*√75) Could I have some irrational non-transindental value x that would yield a non '1', positive integer while the x value is also less than 1/√75? Caviat being x also can't just be a division...
  35. M

    Is sqrt(6) an Irrational Number? A Proof without Prefix

    Homework Statement Prove that \sqrt{6} is irrational.Homework Equations The Attempt at a Solution \sqrt{6} = \sqrt{2}*\sqrt{3} We know that \sqrt{2} is an irrational number (common knowledge) and also this was shown in the textbook. So, let's assume \sqrt{6} and \sqrt{3} are both rational...
  36. J

    If a irrational number be the basis of count

    In the comum sense, the number 10 is the base of the decimal system and is the more intuitive basis for make counts (certainly because the human being have 10 fingers). But, in the math, the number 10 is an horrible basis when compared with the constant e. You already thought if an irrational...
  37. 1

    Square root of a Mersenne Number is irrational

    Homework Statement A user on math.se wanted to prove that any Mersenne number m = 2^n - 1 has an irrational square root for n > 1. So, it can be proved rather easily that any non-perfect square has an irrational root, and that all of the numbers to be considered are not perfect squares...
  38. Seydlitz

    How Do Mathematicians Prove the Irrationality of Complex Number Combinations?

    The methods of proving irrational have always been bothering me in my study of proof. It seems that for each case a new method has to be invented out of the blue. I understand only the proof that ##\sqrt{k}## is irrational. But what will happen if I want to prove ##\sqrt{2}+\sqrt{5}## or...
  39. A

    How Do We Know If Irrational or Transcendental Numbers Repeat?

    Okay, so this is a problem I've been pondering for a while. I've heard from many people that pi doesn't repeat. Nor does e, or √2, or any other irrational or transcendental number. But what I'm wondering is, how do we know? If there truly is an infinite amount of digits, isn't it bound to...
  40. paulmdrdo1

    MHB Proving the Irrationality of $\sqrt{3}$

    prove that $\sqrt{3}$ is irrational. this is what I tried $\sqrt{3}=\frac{p}{q}$ whee p and q are integers in lowest terms. common factor of +\-1 only. squaring both sides $\frac{p^2}{q^2}=3$ $p^2=3q^2$ assuming that $3q^2$ is even then $p^2$ is even hence p is also even. $(3k)^2=3q^2$...
  41. C

    Is this proof valid? (e is irrational)

    Is this proof that e is an irrational number valid? e = ∑^{∞}_{n=0} 1/n! = 1 + 1/1! + 1/2! + 1/3! + ... + 1/n! +... Let e = a + b where a = Sn = 1 + 1/1! + 1/2! + 1/3! + ... + 1/n! b= 1/(n+1)! + 1/(n+2)! + 1/(n+3)! +... Multiply both sides by (n!) giving e(n!) = a(n!) + b(n!)...
  42. C

    Proof that the Square Root of 2 is Irrational.

    I am trying to prove that √2 is irrational using proof by contradiction. Here is my work so far: √2 = p/q where p & q are in their lowest terms. Where q is non-zero. 2=p2/q2 2q2 = p2 Which tells me that p2 is an even number, using the definition of an even number. We can use this definition...
  43. C

    New irrational number to develop transcendental operators

    Other than Bernouilli, Euler, and Lagrange, who else discovered an irrational number in which transcendental operators have been developed to simplify physics and geometry?
  44. K

    MHB Prove that if p and q are positive distinct primes, then log_p(q) is irrational.

    Prove that if p and q are positive distinct primes,then $\log_p(q)$ is irrational. Attempt: Proof by contradiction: Assume $\log_p(q)$ is rational.Suppose $\log_p(q) = \dfrac{m}{n}$ where $m,n \in \mathbb{Z}$ and $\gcd(m,n) = 1$. Then, $p^{\frac{m}{n}} = q$ which implies $p^m = q^n$.
  45. srfriggen

    Rational roots theorem to prove irrational

    Homework Statement Use the rational roots theorem to prove 31/2-21/3is irrational. The Attempt at a Solution My teacher strongly hinted to us that this problem had something to do with the fact that complex roots come in conjugate pairs, and all we had to do was, "flip the sign"...
  46. Mandelbroth

    Proving that \sqrt{p} is irrational

    I'm aware of the standard proof. What I'm wondering is why we can't just do the following. Given, I haven't slept well and I'm currently out of caffeine, so this one might be trivial for you guys. Suppose, by way of contradiction, that ##\sqrt{p}=\frac{m}{n}##, for ##m,n\in\mathbb{Z}##...
  47. K

    If p is prime, then its square root is irrational

    Homework Statement Im trying to prove that if p is prime, then its square root is irrational. The Attempt at a Solution Is a proof by contradiction a good way to do this? All i can think of is suppose p is prime and √p is a/b, p= (a^2)/ (b^2) Is there any property i can...
  48. K

    Show that these numbers are irrational

    Greetings , Im taking an online course on mathematical thinking, and this question has me stumped. r is irrational: Show that r+3 is irrational Show that 5r is irrational Show that the square root of r is irrational. Im sorry if i posted this in the wrong forum, but I am not sure...
  49. G

    Are there any almost irrational numbers that have deceived mathematicians?

    Does any known rational number look irrational at first glance but when calculated to 100s or 1000s of digits actually resolve into a repeating sequence? Have they deceived mathematicians?
  50. J

    Confirmation of irrational proof

    Homework Statement Prove ##\sqrt n## is irrational Homework Equations The Attempt at a Solution Assume ## p^2/q^2 = n ## is an irreducible fraction. If ##p^2 = nq^2##, then q is a multiple of n. Call this ##p' = nq## substituting this for our original equation. We get...
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