What is Irrational numbers: Definition and 92 Discussions

In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers which are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.
Among irrational numbers are the ratio π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational.
Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end with a repeating sequence. For example, the decimal representation of π starts with 3.14159, but no finite number of digits can represent π exactly, nor does it repeat. Conversely, a decimal expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems, and are not used as definitions in mathematics.
Irrational numbers can also be expressed as non-terminating continued fractions and many other ways.
As a consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational.

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  1. Jiketz

    A Determining rationality of real numbers represented by prime digit sequence

    I would like to know if my answer is correct and if no ,could you correct.But it should be right I hope:
  2. 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
  3. 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...
  4. 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...
  5. 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...
  6. 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...
  7. 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?
  8. 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...
  9. 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...
  10. 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 .
  11. 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...
  12. 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...
  13. 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...
  14. 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...
  15. P

    Can Rational Numbers Fully Represent Pi?

    If pi is a part of the area of a perfect circle, which I assume we can construct. Why does it have uncertainty? Can we just measure the area of circle and assign a perfect value for pi. If the answer is our measurements are limited to the instruments that we use, is it not the same for other...
  16. 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 >...
  17. R

    Maxwell's distribution of velocities of molecules

    The expression relating the mean number of molecules with velocities in the range v and v + dv and position r and r + dr is given by where n = N/V is the number density of molecules. My question is: Since LHS is an integer, how do we ascertain the RHS is an integer, since it involves pi and an...
  18. 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...
  19. 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...
  20. 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.
  21. 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:
  22. Pejeu

    Irrational numbers could they be more

    consistently thought of as actually emergent functions that take the desired accuracy as input? As them being numbers would imply the apparently paradoxical concept that infinite complexity can exist in a finite volume of space.
  23. B

    MHB Irrational numbers forming dense subset

    Hello. I have some problems with proving this. It is difficult for me. Please help me.:confused: "For arbitrary irrational number a>0, let A={n+ma|n,m are integer.} Show that set A is dense in R(real number)
  24. C

    Irrational Numbers: Is It Possible?

    Is it possible to have an infinite string of the same number in the middle of an irrational number? For example could I have 1.2232355555555.....3434343232211 Where their was an infinite block of 5's. Then I was trying to think of ways to prove or disprove this. It does seem like it might...
  25. paulmdrdo1

    MHB Sums and Products of Rational and Irrational Numbers

    Explain why the sum, the difference, and the product of the rational numbers are rational numbers. Is the product of the irrational numbers necessarily irrational? What about the sum? Combining Rational Numbers with Irrational Numbers In general, what can you say about the sum of a rational...
  26. B

    How do irrational numbers play a role in physics?

    Hi, I have some theories about physical facts derived from the size of powers in physics, compared to the first fraction of an irrational number. I do not know if this is redundant with present day science, but I am curious about it. Regards, Justin
  27. J

    Differentproof there are more irrational numbers than rational numbers

    you can list and match up all rational numbers with irrational numbers this way.. lets say i have an irrational number 'c'. Rational->Irrational r1->cr1 r2->cr2 . . . rn->crn There exists an irrational number that is not on this matching, (not equal to any of the crx's) this...
  28. L

    A naive question about irrational numbers

    I've been thinking about this recently and couldn't find the answer to my question (even though I assume it's a really simple one, so forgive me if it's too trivial). Let's say we have two rods of length 1 meter and we put them at right angles to each other. Then we cut a third rod just long...
  29. C

    I solving a proof dealing with the set of irrational numbers.

    Homework Statement Let x,y,t be in the set of all real numbers (R) such that x<y and t>0. Prove that there exists a K in the set of irrational numbers (R\Q) such that x<(K/t)<y Homework Equations if x,y are in R and x<y then there exists an r in Q such that x<=r<y The Attempt at a...
  30. D

    Clarifications on the least upper bound property and the irrational numbers

    Hello everyone. I desperately need clarifications on the least upper bound property (as the title suggests). Here's the main question: Why doesn't the set of rational numbers ℚ satisfy the least upper bound property? Every textbook/website answer I have found uses this example: Let...
  31. P

    How to find 'self locating digits' in irrational numbers

    Let us take the most mainstream irrational out there, (Pi). Now write (Pi) as: 3. 14159265... Let us number the decimals of Pi. 0 gets paired with 1 1 gets paired with 4 2 gets paired with 1 . . . 6 gets paired with 6 Thus 6 is a self locating digit. My question is then...
  32. K

    Can Rational Numbers Approximate Irrational Numbers Arbitrarily Closely?

    Prove the theorem comprising that an irrational number β can be described to any limit of accuracy with the help of rational. Attempt- Taking the β to be greater than zero and is expressed with an accuracy of 1/n For any arbitrary value of β, it falls between two consecutive integers which...
  33. G

    Solving Irrational Numbers: Exploring My Reasoning

    Can anyone explain what is wrong with my reasoning? Suppose x = \frac{p}{q} and let x = \sqrt 2 + \sqrt 3 . Also, let a,b,c \in {\Bbb Z} and assume a < xc < b. If I show that xc must be an integer, and I know there does not exist c such that \sqrt 2 c, or \sqrt 3 c is an integer. Then...
  34. 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...
  35. 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 ...
  36. 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...
  37. 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...
  38. 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...
  39. 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...
  40. 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...
  41. 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: .
  42. 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?
  43. 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...
  44. 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?
  45. 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}...
  46. 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...
  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. 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
  49. 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...
  50. 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...