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. chwala

    Find an irrational number that satisfies the given inequality

    I just came across this question and the ms indicates, Would ##31.5## be correct? ...i think it is rational as it can be expressed as ##31.5 = \dfrac{63}{2}##.
  2. M

    For ## n\geq 2 ##, ## \sqrt[n]{n} ## is irrational?

    Proof: Suppose for the sake of contradiction that ## \sqrt[n]{n} ## is rational for ## n\geq 2 ##. Then we have ## \sqrt[n]{n}=\frac{a}{b} ## for some ## a,b\in\mathbb{Z} ## such that ## gcd(a,b)=1 ## where ## b\neq 0 ##. Thus ## (\sqrt[n]{n})^{n}=(\frac{a}{b})^{n} ##...
  3. M

    Prove that ## \sqrt{p} ## is irrational for any prime ## p ##?

    Proof: Suppose for the sake of contradiction that ## \sqrt{p} ## is not irrational for any prime ## p ##, that is, ## \sqrt{p} ## is rational. Then we have ## \sqrt{p}=\frac{a}{b} ## for some ## a,b\in\mathbb{Z} ## such that ## gcd(a, b)=1 ## where ## b\neq 0 ##. Thus ## p=\frac{a^2}{b^2} ##...
  4. barryj

    Can a polynomial have an irrational coefficient?

    Is this a polynomial? y = x^2 + sqrt(5)x + 1 I was told NO, the coefficients had to be rational numbers. I this true? It seem to me this is an OK polynomial. I can graph it and use the quad formula to find the roots? so why or why not?
  5. e2m2a

    B Square Root of an Odd Powered Integer is Always Irrational?

    Is it always true that the square root of an odd powered integer will always be irrational?
  6. docnet

    Show that square root of 3 is an irrational number

    ##\sqrt{3}## is irrational. The negation of the statement is that ##\sqrt{3}## is rational. ##\sqrt{3}## is rational if there exist nonzero integers ##a## and ##b## such that ##\frac{a}{b}=\sqrt 3##. The fundamental theorem of arithmetic states that every integer is representable uniquely as a...
  7. T

    Lingusitics Question about irrationality and irrational language <moved>

    What part of the brain and/or mind does interpreting irrationality or irational language exersize/use? Hi, I couldn't find anything about this on nets and also went on a teachers forum and still haven't herd back from them for about 1 month or over a month now so I am positng this question here...
  8. e2m2a

    I Is the odd root of an even number always an irrational number?

    Is the odd root of an even number always an irrational number? For example the 7th root or the 11th root, etc. of an even number.
  9. N

    Irrational Number Raised To Irrational Number

    Can an irrational number raised to an irrational power yield an answer that is rational? This problem shows that the answer is “yes.” (However, if you study the following solution very carefully, you’ll see that even though we’ve answered the question in the affirmative, we’ve not pinpointed the...
  10. N

    Irrational Numbers a and b used in various expressions

    Give an example of irrational numbers a and b such that the indicated expression is (a) rational and (b) irrational. 1. a +b 2. a•b 3. a/b 4. a - b What exactly is this question asking for? Can someone rephrase the statement above? Thanks
  11. potatocake

    How to prove rational sequence converges to irrational number

    I attempted to solve it $$ x = \frac {1}{4x} + 1 $$ $$⇒ x^2 -x -\frac{1}{4} = 0 $$ $$⇒ x = \frac{1±\sqrt2}{2} $$ However, I don't know the next step for the proof. Do I need a closed-form of xn+1or do I just need to set the limit of xn and use inequality to solve it? If I have to use...
  12. greg_rack

    Studying non-differentiable points of an irrational function

    I calculated the derivative of this function as: $$\frac{6x^3-4x}{3\sqrt[3]{(x^3-x)^2}}$$ Now, in order to find and later study non-differentiable points, I must find the values which make the argument of the root equal to zero: $$x^3-x=0 \rightarrow x=0 \vee x=\pm 1$$ and then find the left and...
  13. bigchaka

    Sum of rational and irrational is irrational

    Summary:: i get a proof that sum of rational and irrational is rational which is wrong(obviously) let a be irrational and q is rational. prove that a+q is irrational. i already searched in the web for the correct proof but i can't seem to understand why my proof is false. my proof: as you...
  14. greg_rack

    Basic irrational equation problem

    For solving this equation I must take elevate to the square of each member, resulting in: $$(a-2)^2=a^2-4 \rightarrow a=2$$ Now, the thing I noticed and don't get is that if you simplify a ##(a-2)## factor, the equation becomes impossible: $$a-2=a+2$$ It must be a stupid thing which I'm missing...
  15. anemone

    MHB Challenge involving irrational number

    Let $x$ be an irrational number. Show that there are integers $m$ and $n$ such that $\dfrac{1}{2555}<mx+n<\dfrac{1}{2012}$.
  16. jedimath

    Algebra Book about irrational inequalities and other....

    Good evening, I have consulted several precalculus books, intermediate algebra but none of these lists irrational inequalities, trigonometric inequalities and more. In which book I can find them? Thank you :)
  17. LittleRookie

    I How to relate multiplication of irrational numbers to real world?

    I'm aware of the axioms of real numbers, the constructions of real number using the rational numbers (Cauchy sequence and Dedekind cut). But I can't relate the arithmetic of irrational numbers to real world usage. I can think the negative and positive irrational numbers to represent...
  18. R

    I Generating Irrational Ratios in Wave Simulations

    I am trying to write an algorithm that generates two random numbers in a given interval such that their ratio is an irrational number. I understand that all numbers stored on a computer are rational, so it is not possible to have a truly irrational number in a simulation. So, instead I am...
  19. N

    MHB Determine if (27/4)/(6.75) is a whole number, natural number, integer, rational or irrational.

    Let Z = set of real numbers Determine if (27/4)/(6.75) is a whole number, natural number, integer, rational or irrational. I will divide as step 1. 27/4 = 6.75 So, 6.75 divided by 6.75 = 1. Step 2, define 1. The number 1 is whole or natural. It is also an integer and definitely a rational...
  20. L

    I How do irrational numbers give incommensurate potential periods?

    I am trying to understand Aubry-Andre model. It has the following form $$H=∑_n c^†_nc_{n+1}+H.C.+V∑_n cos(2πβn)c^†_nc_n$$ This reference (at the 3rd page) says that if ##\beta## is irrational (rational) then the period of potential is quasi-periodic incommensurate (periodic commensurate) with...
  21. W

    MHB Can You Solve this Irrational Number Puzzle?

    Puzzle: 5.2.7.7 | 3.2 | 7.2.4 | {7}.7.2 hint #1) Irrational hint #2) 2 to 101 Can anybody help me figure this out? There are 2 hints. I am at a loss.
  22. Mr Davis 97

    Showing that tan(1) is irrational

    Homework Statement Prove that ##\tan (1^\circ)## is irrational. Homework EquationsThe Attempt at a Solution Suppose for contradiction that ##\tan (1^\circ)## is rational. We claim that this implies that ##\tan (n^\circ)## is rational. Here is the proof by induction: We know by supposition...
  23. V

    Proving something is irrational

    Homework Statement Prove that for all x ∈ ℝ, at least one of √3 - x and √3 +x is irrational. Homework EquationsThe Attempt at a Solution I understand how to prove that √3 is a irrational number by proof by contradiction. However I am not sure how to prove this one. Would I have to equate...
  24. R

    Proving the Irrationality of Square Root of 3

    Homework Statement Prove sqrt(3) is irrational Homework EquationsThe Attempt at a Solution (a/b)^2 = 3 assume a/b is in lowest form a^2 = 3b^2 so a^2 is of form 3n whenever n is even, a^2 will be even => a will be even whenever n is even so a is of form 2l whenever n is even => 4l^2 =...
  25. R

    Is a+b necessarily irrational?

    Homework Statement If a is rational and b is irrational, is a+b necessarily irrational? What if a and b are both irrational? Homework EquationsThe Attempt at a Solution The books answer: 1)Yes, for if a+b were rational, then b = (a+b) - a would be rational. This makes sense for me, but I...
  26. srfriggen

    B Is the square root of 945 irrational?

    Is the square root of 945 irrational? I feel it is rational because my TI-84 Plus converts it into 275561/8964, however, I am unsure whether the calculator is estimating. Can someone please advise. It can be broken down into 3√105, and again, my calculator is able to convert √105 into a...
  27. ranga519

    MHB Proving that solutions of a equation are irrational

    how to prove that solutions of the following equation are irrational x^3 + x + 1 = 0
  28. T

    I Proof that cube roots of 2 and 3 are irrational

    Proof by contradiction that cube root of 2 is irrational: Assume cube root of 2 is equal to a/b where a, b are integers of an improper fraction in its lowest terns. So the can be even/odd, odd/even or odd/odd. The only one that can make mathematical sense is even/odd. That is...
  29. sysprog

    How to distinguish the decimal expansions of irrational numbers from random numbers?

    How do we distinguish the decimal expansions of irrational numbers, and products thereof, from random sequences? Is an arbitrarily specified (not claimed to be perfectly randomly selected) numeric string, e.g. the 10^10th to 10^19th digits of the decimal extraction of the square root of 2.2...
  30. K

    MHB Why is Pi an irrational number?

    Pir2 (I am looking in the greek alphabet and geometry symbols and can not find the symbol for pi that looks anything like pi when in preview mode) Sorry. If Pi is the ratio of a circumference to the diameter of a circle and geometrically this gives the perfect measurement of a circumference...
  31. A

    B A Rational Game: Exploring the Paradox of Aligning Irrational Numbers

    This post is to set forth a little game that attempts to demonstrate something that I find to be intriguing about the real numbers. The game is one that takes place in a theoretical sense only. It starts by assuming we have two pieces of paper. On each is a line segment of length two: [0,2]...
  32. BWV

    I Rational powers of irrational numbers

    √2 is irrational but √22 is rational Is there any way to know if given some irrational number α, if αn is rational for some n? Or can it be proven that ∏n or en are irrational for all n?
  33. kaliprasad

    MHB Show that sin10∘ is irrational number

    Show that $\sin\,10^\circ$ is irrational
  34. M

    MHB How precise is (4/3)^4 compared to π?

    The value of the irrational number π, correct to ten decimal places (without rounding) is 3.1415926535. By using your calculator, determine to how many decimal places does the quantity (4/3)^4 agree with π.
  35. D

    B Irrational inequalities √f(x)>g(x) and √f(x)>g(x)

    So, I know that the inequality √f(x)<g(x) is equivalent to f(x)≥0 ∧ g(x)> 0 ∧ f(x)<(g(x))^2. However, why does g(x) have to be greater and not greater or equal to zero? Is it because for some x, f(x) = g(x)=0, and then > wouldn't hold? Doesn't f(x)<(g(x))^2 make sure that f(x) will not be...
  36. Delta2

    I Rational sequence converging to irrational

    In the textbook I have (its a textbook for calculus from my undergrad studies, written by Greek authors) some times it uses the lemma that "for any irrational number there exists a sequence of rational numbers that converges to it", and it doesn't have a proof for it, just saying that it is a...
  37. W

    MHB Unlocking An Irrational Location: Solving a Geocaching Puzzle

    This might not be the usual kind of question posted here, but I am trying to solve a geocaching puzzle. The puzzle is called "An Irrational Location", and the only information provided is more or less the following: ~~~~~ No rational person should attempt to visit the posted coordinates Cache...
  38. A

    How do you compute an exponent with irrational values?

    Homework Statement Let's say I want to compute ##2^{2.4134}##. We know that the base is a rational number and the power is an irrational number. Please keep in mind that I have not taken too many math classes yet and I am self-studying right now by making a calculator and respective algorithms...
  39. A

    Solving Irrational Equations: How to Discard Solutions

    I figured it out >_> I got a problem with discarding the second solution of this irrational equation: ##-\sqrt {x^2 - 1} + \sqrt {x^2 +3x} = 2## First I find the domain, which will end up being ##x\leq-3## v ##x\geq+1## since that's the common union of the domains of each square root. Then I...
  40. F

    I Use of irrational numbers for coordinate system

    Why should a person prefer irrational coordinate system over rational? My friend stated that its because most lines such as ##y=e## cannot be plotted on a rational grid system. But that cannot be true since ##e## does have a rational number summation ##2+1/10+7/100...## which can be utilised to...
  41. Albert1

    MHB We can find two irrational numbers x and y to make xy rational,true or false

    we can find two irrational numbers $x$ and $y$ to make $x^y$ rational,true or false statement? if true then find else prove it .
  42. Math Amateur

    MHB Sum of a rational number and an irrational number ....

    I am trying without success to provide a rigorous proof for the following exercise: Show that the sum of a rational number and an irrational number is irrational.Can someone please help me with a rigorous solution ...I am working from the following books: Ethan D. Bloch: The Real Numbers and...
  43. Math Amateur

    Sum of a rational number and an irrational number ....

    Homework Statement I am trying without success to provide a rigorous proof for the following exercise: Show that the sum of a rational number and an irrational number is irrational. Homework Equations I am working from the following books: Ethan D. Bloch: The Real Numbers and Real Analysis...
  44. Mr Davis 97

    I Proof that the square root of 2 is irrational

    Quick question: In the proof that the square root of 2 is irrational, when we are arguing by contradiction, why are we allowed to assume that ##\displaystyle \frac{p}{q}## is in lowest terms? What if we assumed that they weren't in lowest terms, or what if we assumed that ##\operatorname{gcd}...
  45. rumborak

    I Why are "irrational" and "transcendental" so commonly used to describe numbers

    (sorry, the thread title got mangled. It should be "why are irrational and transcendental so commonly used to describe numbers") Is this simply out of the most common ways of how one would try to describe a number? (e.g. first try ratios, then polynomials) Or is there a deeper reason for this...
  46. M

    MHB Irrational Raised to Irrational

    Precalculus by David Cohen 3rd Edition Chapter 1, Section 1.1. Question 66, page 6. Can an irrational number raised to an irrational power yield an answer that is rational? Let A = (sqrt{2})^(sqrt{2}). Now, either A is rational or irrational. If A is rational, we are done. Why? If A is...
  47. M

    MHB Exploring Rational and Irrational Numbers with Examples of Arithmetic Operations

    Give an example of irrational numbers a and b such that the indicated expression is (a) rational; (b) irrational. 1. a + b 2. a/b Must I replace a and b with numbers that create a rational and irrational number?
  48. M

    MHB Value of Irrational Number π (Part 2)

    The value of irrational number π, correct to ten decimal places (without rounding), is 3.1415926535. By using your calculator, determine to how many decimal places the following quantity (22/7) agrees with π. Extra notes from textbook: Archimedes (287-212 B.C.) showed that (223/71) < π <...
  49. M

    MHB Value of Irrational Number π (Part 1)

    The value of irrational number π, correct to ten decimal places (without rounding), is 3.1415926535. By using your calculator, determine to how many decimal places the following quantity [(4/3)^4] agrees with π. The value used for π in the Rhind papyrus, an ancient Babylonian text written...
  50. S

    MHB Integer Arithmetic for Precise Calculation of Irrational Numbers

    I have authored documents of 40 years of computer software development with a mind to collect them into a publication at some point. They have been built around several software topics but mathemetics is a favorite of mine. I find a point of inspiration and write a piece of software around it...
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