What is Irrational number: Definition and 43 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. brotherbobby

    Proving that the inverse of a rational number exists

    Problem statement : I cope and paste the problem as it appears in the text below. Attempt : Not being a math student, I try and prove the above statement using an "intuitive" way. Let us have a rational number ##b = \frac{n}{m}##. Multiplying with ##a## from the right, we see ##ab =...
  2. 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...
  3. 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.
  4. 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...
  5. 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...
  6. anemone

    MHB Challenge involving irrational number

    Let $x$ be an irrational number. Show that there are integers $m$ and $n$ such that $\dfrac{1}{2555}<mx+n<\dfrac{1}{2012}$.
  7. 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...
  8. 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...
  9. W

    MHB What Do These Numbers Represent?

    Puzzle: | 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.
  10. 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...
  11. kaliprasad

    MHB Show that sin10∘ is irrational number

    Show that $\sin\,10^\circ$ is irrational
  12. 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.
  13. 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...
  14. 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...
  15. 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 <...
  16. 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...
  17. Ventrella

    A Physics and Integer Computation with Eisenstein Integers

    I realize this question may not have an obvious answer, but I am curious: I am using Gaussian and Eisenstein integer domains for geometry research. The Gaussian integers can be described using pairs of rational integers (referring to the real and imaginary dimensions of the complex plane). And...
  18. 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...
  19. 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...
  20. 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...
  21. 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?
  22. M

    Irrational number to an irrational power

    Homework Statement if a and b are irrational numbers, is a^b necessarily an irrational number ? prove it. The Attempt at a Solution this is an question i got from my first maths(real analysis) class (college) , and have to say, i have only little knowledge about rational number, i would like to...
  23. A

    Irrational power of an irrational number

    What kind of number is sqrt(2)^sqrt(2)? I have noted sqrt(2)^sqrt(2) = 2^(sqrt(2)/2) = 2^(1/sqrt(2)), i.e. a rational number to an irrational power. Now, 1/sqrt(2) is less than 1, but greater than zero. So, given that 2^x is an increasing function, 2^(1/sqrt(2)) is less than 2^1, but...
  24. 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...
  25. 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)."
  26. 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...
  27. 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...
  28. 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...
  29. 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.
  30. 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.
  31. 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
  32. X

    Irrational Number Approximation Error Explained

    Question: Using the fact that \sqrt{2} is irrational, we can actually come up with some interesting facts about other numbers. Consider the number t=1/\sqrt{2}, which is also irrational. Let a and b be positive integers, and a<b. We will prove that any rational approximation a/b of t will...
  33. A

    I wih prove (sqrt(2))^3 is irrational number

    hi well I'm having truple in proving this (sqrt(2))^3 is irrational number!
  34. J

    Is there always at least one irrational number between any two rational numbers?

    and, consequently, infinitely many. I am new to proofs so could you please check if this proof is correct? Let x be an irrational number in the interval In = [an, bn], where an and bn are both rational numbers, in the form p/q. Let z be the distance between x and an, So: x - an = x...
  35. S

    About irrational number

    wat are the answers for these in terms of rational,irrational (irrational number)*(any +ve integer) = ? (any +ve integer) - (irrational number)*(any integer) = ? are the answers also irrational numbers
  36. S

    Analysis - irrational number or positive integer

    Homework Statement Let n and k be positive integers. Show that k^{1/n} is either a positive integer or an irrational number. The Attempt at a Solution I set q = k^{1/n}. Then I set q = \frac{m}{p} . (Where m and p don't have common factors.) Then m^n = k * p^n . So then k is a factor...
  37. L

    Rational and Irrational Number Set proof.

    Hello, here is my problem: how can i prove that if a\in\mathbf{Q} and t\in\mathbf{I}, then a+t\in\mathbf{I} and at\in\mathbf{I}? My original thought was to show that neither a+t or at can be belong to N, Z, or Q, thus they must belong to I. However I'm not certain if that train of thought...
  38. R

    Proof Check (irrational number)

    Hi, can somebody please check my proof(s). I am pretty sure they are right...but i just feel they are too elelmentary. 1) Prove that 12 is irrational -Let x be a rational number -Therefore, x = {a/b: m in Z and n in N} n not 0. -m and n have to be in the LOWEST FORMS...therefore, m and...
  39. B

    Help with gsce irrational number question

    please please help me quick! hi i was practisin a gcse maths paper and need some help with last question; x and y are two positive irrational numbers. x + y is rational and so it x times y. a) by writing the 1/x + 1/y as a single fraction explain why 1/x + 1/y is always rational. b)...
  40. B

    Irrational Number and the Borel Sets

    Took a test in my Analysis class today. One question asked us to prove that the set of Irrational numbers was a Borel Set. After working on the other problems for 90 minutes, I stared blankly at this one for what seemed life a long time. I eventually showed (I think) that the set of Rational...
  41. J

    Non-Negative Integer Binary Concatenation: Is This an Irrational Number?

    I am interested in the following number which is obtained by concatenting the binary representations of the non-negative integers: .011011100101110111... i.e. dot 0 1 10 11 100 101 110 111 ... This is a little bigger than .43 and I assume it irrational since no pattern of bits repeats...
  42. V

    Why Is the Number e Considered Irrational?

    My book does not make sense to me. Here is what it says: I know that: e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + ... + \frac{1}{n!} + \frac{\theta}{n!n}, 0 < \theta < 1 If e is rational then e = \frac{m}{n}; m, n \in Z :confused: And the greatest common factor of m, n is 1...
  43. N

    We know e (exponential) is a irrational number

    we know e (exponential) is a irrational number... how can we prove it??