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. J

    The cardinality of the set of irrational numbers

    Homework Statement Suppose \mathbb{Q},\mathbb{R} are the set of all rational numbers and the set of all real numbers, respectively. Then what is |\mathbb{R} \backslash \mathbb{Q}|?Homework Equations |\mathbb{Q}| = |\mathbb{Z^{+}}| < |P(\mathbb{Z^{+}})| = |\mathbb{R}|The Attempt at a Solution I...
  2. M

    Calculus theory proof- Suppose a is irrational, prove√(1+a) is irrational.

    Homework Statement Suppose a is irrational, prove√(1+a) is irrational. Homework Equations A number is rational if it can be expressed as p/q, p,q integers with q≠0 The Attempt at a Solution I can reason through it intuitively but not sure how to demonstrate it formally. Any...
  3. H

    Square Root of an Irrational Number is Irrational

    Homework Statement Let a be a positive real number. Prove that if a is irrational, then √a is irrational. Is the converse true? Homework Equations So, an irrational number is one in which m=q/p does not exist. I understand that part, but then trying to show that the square root of an...
  4. T

    Is (2^1/2 + 7^1/3)^1/2 irrational?

    Hi, So I proved that 71/3 is irrational, and I know 21/2 is irrational, but how can I show if (21/2 + 71/3)1/2 is irrational? Hint of where to start?
  5. T

    Finding Positive Integers for Irrational Number Interval

    someome please help me with this problem: "Any real numbers x and y with 0 < x < y, there exist positive integers p and q such that the irrational number s =( p√2)/q is in the interval (x; y)."
  6. L

    Prove that x is irrational unless it is an integer.

    Homework Statement This is taken from an answer book that I have. I don't understand the bolded step. Can someone explain it to me?Suppose x = p/q where p and q are natural numbers with no common factor. Then: pn/qn + an-1pn-1/qn-1 + ... + ao = 0 and multiplying both sides by...
  7. L

    Two types of irrational numbers

    Non-repeating patterns in decimal expansions of irrational numbers seem to have two forms. I am wondering if there is any theory about the two. First - the decimal expansion is ultimately random - unpredictable Second - The decimal expansion follows an algorithm e.g. .01001000100001 ...
  8. F

    Contrapositive proof of irrational relations

    I'm confused with a question and wondered if anyone could help explain where I need to go... let x ε R. Prove that x is irrational thenI'm confused with a question and wondered if anyone could help explain where I need to go... let x ε R. Prove that x is irrational then ((5*x^(1/3))-2)/7)...
  9. C

    Proof about irrational numbers.

    Homework Statement Prove that \sqrt{6} is irrational. The Attempt at a Solution Would I just do a proof by contradiction and assume that \sqrt{6} is rational and then get that 6q^2=p^2 which would imply that p is even so I put in p=2r and then multiply it out. then this would imply...
  10. C

    Proof about an irrational number.

    Homework Statement Prove that \sqrt{3} is irrational. The Attempt at a Solution SO I will start by assuming that \sqrt{3} is rational and i can represent this as 3=\frac{b^2}{a^2} and I assume that a and b have no common factors. so now I have 3b^2=a^2 but this is not possible...
  11. B

    Prove If x^2 is irrational then x is irrational

    Homework Statement Prove If x^2 is irrational then x is irrational. I can find for example π^2 which is irrational and then π is irrational but I don't know how to approach the proof. Any hint?
  12. C

    Proving a number is irrational.

    Homework Statement Prove that log_2(3) is irrational. The Attempt at a Solution This is also equivalent to 2^x=3 from the definition of logs. Proof: For the sake of contradiction let's assume that x is rational and that their exists integers P and Q such that x=P/Q . so now we have...
  13. Seydlitz

    Integration Method for Irrational Root?

    Homework Statement I need to evaluate this particular integral and I'm confused on what method to use. I'm currently learning integration calculus and I tried doing some introduction on electromagnetic field. Quite unexpectedly the integral turned to be heavy. \int_{-a}^a...
  14. S

    Shrinking down a^b mod m when b is huge and a is irrational?

    Is there a good way to do this? I am trying to figure out a good way to calculate a^b mod m, but the problem is that b is huge and a is irrational, and therefore I am getting inaccurate values because too much precision is required. I'm trying to find a smaller, "equivalent" ^b mod m to use...
  15. J

    Is 2.7177117771117777 irrational?

    Is 2.71771177711177771111... irrational? Homework Statement I'm student teaching 8th graders Numbers and Operations. This is from an 8th grade activity I inherited with no "answer key." Is this decimal (with a pattern but not a repeating pattern) irrational? I am guessing it is, but I want to...
  16. B

    Show that sqrt(n) is irrational

    Let n>2. Where n is integer show that sqrt(n!) is irrational. I am supposed to use the Chebyshev theorem that for n>2. There is a prime p such that n<p<2n. So far I am up to inductive hypothesis. Assume it holds for k then show it holds for k+1. Well if k! is irrational==> k!=...
  17. A

    Operations on irrational numbers

    Heres two problems from an A Level related paper: prove that if pq is irrational then atleast one of p or q is irrational. Also prove that if if p + q is irrational then atleast one of p or q is irrational. These two proofs are trivial proof by contradiction problems but it got me thinking more...
  18. C

    Is log4(18) an Irrational Number?

    Homework Statement log418 rational numbers are in form x/y Homework Equations logab = logcb / logca The Attempt at a Solution log218 / log218 = x/y (b) log218 = (a) log218 log218b = log218a Then I am stuck.
  19. M

    F(x) = x if x is rational, 0 if x is irrational.

    Homework Statement F(x) = x if x is rational, 0 if x is irrational. Use the δ, ε definition of the limit to prove that lim(x→0)f(x)=0. Use the δ, ε definition of the limit to prove that lim(x→a)f(x) does not exist for any a≠0. Homework Equations lim(x→a)f(x)=L 0<|x-a|<δ...
  20. T

    Need help with proving a number is irrational

    Prove that if x satisfies 'xn +an-1xn-1+ ... a0=0' for some integers an-1,..., a0, then x is irrational unless x is an integer. My main question is that I don't quite understand what the question is trying to ask me prove. I'm fairly new with this so pardon me if this question is really basic...
  21. A

    Discrete Math- Irrational numbers, proof or counterexample

    Homework Statement Determine if the statement is true or false. Prove those that are true and give a counterexample for those that are false. If r is any rational number and if s is any irrational number, then r/s is irrational. Homework Equations A rational number is equal to the...
  22. A

    Discrete Math irrational and rational numbers proof

    Homework Statement Prove by contradiction. Your proof should be based only on properties of the integers, simple algebra, and the definition of rational and irrational. If a and b are rational numbers, b does not equal 0, and r is an irrational number, then a+br is irrational. Homework...
  23. C

    Rational and Irrational numbers

    Homework Statement Let f be the function defined on the real line by f(x)= \begin{cases} \frac{x}{3} & \text{if $x$ is rational } \\ \frac{x}{4} &\text{if $x$ is irrational.} \end{cases} Let D be the set of points of discontinuities of f. What is D? Homework Equations None...
  24. B

    Finding an Irrational not in the Union.

    Hi, All: This is an old problem I never solved, and I recently saw somewhere else: We are given an enumeration {q_1,q_2,..,q_n,...} of the rationals in the real line. We construct the union : S:=\/ (q_i+[e/2^(i+1)] , q_i-[e/2^(i+1)] ) for i=1,2..,n,.. i.e., we want the...
  25. C

    Why do irrational numbers result in uneven divisions?

    This has been aggravating me for years. Call it "IDP" as a placeholder name for now, if you will. How come irrational numbers keep propelling forward for particular divisions? My inquiry applies for both repeating and non-repeating irrational numbers. "Just is" or "You're thinking too much into...
  26. ArcanaNoir

    Proof: between 2 reals is an irrational

    I'd like to know if this indeed proves that between any 2 reals is an irrational. Choose two reals A and B, B>A. There are two cases of B: B is irrational or B is rational. Assume B is irrational. Then B- \frac{1}{n} (n is a natural number) is irrational. You can get as close as you like...
  27. rcgldr

    Computers rational and irrational numbers

    I think this needs it's own thread. e and pi are transcendental numbers: http://en.wikipedia.org/wiki/Transcendental_number The square root of 2 is n irrational number: http://en.wikipedia.org/wiki/Irrational_number 1/3 is a rational number...
  28. W

    Conversion of Irrational roots for cubic functions and higher

    I have been seeing a few during in my practice questions which leaves me worrying. If it is a quadratic function, the irrational numbers can be easily obtained using the equation. However, I got a question today which eventually took this form: 28D3+36D2-41D2+4 = 0 (I reevaluated...
  29. W

    Vauled-ness of a complex number to an irrational power

    Homework Statement For z complex: a.) is z\sqrt{2} a multi-valued function, if so how many values does it have? b.) Claim: z\sqrt{2}=e\sqrt{2}ln(z)=e\sqrt{2}eln(z)=ze\sqrt{2} Since \sqrt{2} has 2 values, z\sqrt{2} is 2 valued. Is this correct? If not, correct it. Homework...
  30. C

    Irrational sequence that converges to a rational limit

    Hi. I found some rational sequences that converge to irrational limits, but am not having any luck going the other direction, i.e., an irrational sequence that converges to a rational limit. Any suggestions?
  31. L

    Proving Irrational Numbers: Even Natural Numbers & Prime Products

    Prove that: 1-If n^2 (n is a natural number) is even then n is even too . 2-Product of infinit number of primes bigger than 2 is not even. Please do not "google it for me" :biggrin: .
  32. R

    Computation of Continued Fractions of irrational numbers

    In this field, computer algorithms may produce false continued fraction expansions because of the limited accuracy in the floating point arithmetic used. Who knows more?
  33. B

    Deducing Irrational Identity - a1=a2 & b1=b2?

    How do you deduce that a1 - \sqrt{N} b1 = a2 - \sqrt{N}b2 to be a1=a2 and b1=b2?
  34. B

    Is √N Always Irrational for Non-Square Integers?

    I came across this question. How do you show that √N is irrational when N is a nonsquare integer? Cheers.
  35. S

    How irrational are employers on the topic of related disciplines?

    It would be foolish to try and predict how employers will react -- having few job experiences, currently. So I’m here to ask: How much trouble will I have attempting to be hired as a Software Developer, after graduating with a BS in mathematics, and a minor in computer science? Are interviews...
  36. K

    Algebraic and Transcendental irrational numbers

    It's my understanding that algebraic numbers are the roots of polynomials with rational (or equivalently integer) coefficients. I know all surds have a simple repeating continued fraction representation Is it also the case that all simple repeating continued fractions are algebraic numbers...
  37. L

    Irrational numbers in real life.

    So I was thinking about numbers like pi. If you were to measure the area or circumference of a sphere in real life, you would get a never ending decimal. How can this exist in real life? How can an actual physical object have a circumference that is an irrational number?
  38. E

    Explore Irrational Exponents: {e^(2ki)|k=intiger}

    I am playing around with the set {e^(2ki)|k=intiger}. all of these numbers when raised to the pi power give one (at least as one possible value). In other words, they can all be thought of as values of 1^(1/pi). There are a countable infinity of them, and I believe that these numbers are...
  39. S

    Where are the irrational numbers?

    Rational numbers are those that can be represented as a/b. It is simple (I think) to demonstrate that the series of rationals is continuous, since, for any two rational numbers, X=a/b, and Y=c/d, you can always find at least one rational number between them. \frac{X+Y}{2} = \frac{ad+bc}{2bd}...
  40. T

    Prove an Infinite Series is Irrational

    Is it possible and is there a general method?
  41. B

    Number Theory: nth root of n is irrational

    1. For n ≥2, n^(1/n) is irrational. Hint provided: Use the fact that 2^n > n2. This is probably familiar to many. By contradiction, n = a^n/b^n --> a^n = n(b^n) --> n|a^n --> n|a Am I trying to force the same contradiction as with 2^1/2 is rational, that is, that a/b are not in lowest terms? Or...
  42. E

    Proof that irrational numbers do not exist

    Any number c in the real numbers has the form x.{c_1}{c_2}...{c_n}, in which x is an integer and 0 \le {c_n} \le 9 is a natural number. From the way that we have enumerated the decimal places, clearly number of decimal places is countable. Then there is a bijection from the indexes of the...
  43. I

    How do you simplify irrational exponents?

    It is clear that 10^2 can be simplified to 10*10=100. But what about say, 10^0.5? I have been thinking about this for days and can't figure out how it simplifies. 10^1 is 10, 10^0 is 1, so 10^0.5 should be under 1, but it is 3.16, so I don't get it. Same with 10^-1 is 0.1. How exactly are...
  44. P

    Proving Root n is Irrational: Perfect Square Affects Proof

    How to prove that root n is irrational, if n is not a perfect square. Also, if n is a perfect square then how does it affect the proof.
  45. P

    Cube root of 6 is irrational. (please check if my proof is correct)

    Prove that cube root of 6 irrational. Solution: I am trying to prove by contradiction. Assume cube root 6 is rational. Then let cube root 6 = a/b ( a & b are co-prime and b not = 0) Cubing both sides : 6=a^3/b^3 a^3 = 6b^3 a^3 =...
  46. A

    Proving sqrt of 3 is irrational

    Homework Statement So, I'm trying to prove that the square root of 3 is irrational. 2. Attempt at a Solution 2x can be any even number and 2x+1 can be any odd number. Since an irrational number is any number that can't be expressed as a ratio of two integers, I just have to show that...
  47. icystrike

    Exploring the Mysterious World of Irrational Numbers

    They can fit into number lines but not marked on a sewing thread ? I love to think of between 2 infinity small rational numbers there is a infinity deep hole that you can always pick a different irrational number out of it. (Is it a safe idea? )
  48. 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 \,\,\,\,\...
  49. 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
  50. 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...
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