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.

View More On Wikipedia.org
Recent contents Top users
  1. J

    Why do irrational numbers appear in quantum physics?

    This is a question I've had for some time, but didn't think to ask whenever I was around someone who might have been able to answer it. If energy and matter are made of quanta, then why is quantum physics coming up with so many irrational results instead of integral ones?
  2. M

    Set of irrational numbers between 9 and 10 are countable

    Homework Statement The set of irrational numbers between 9 and 10 is countable. Homework Equations The Attempt at a Solution My belief is that I can prove by contradiction. first, i must prove by contradiction using diagonalization that the real numbers between 9 and 10 are...
  3. M

    Is \sqrt{2} + \sqrt{3} + \sqrt{5} irrational?

    Homework Statement Is \sqrt{2} + \sqrt{3} + \sqrt{5} rational? Homework Equations If n is an integer and not a square, then \sqrt{n} is irrational For a rational number a and an irrational number b, a + b is irrational a * b is irrational if a is not equal to 0 The Attempt at a Solution...
  4. G

    What Are Some Interesting Properties of Pi?

    So if Pi is an irrational number, and therefore has an infinite line of numbers after the decimal point; my intuition tells me it would take an infinite amount of time to determine its exact value. a) Do calculators and computers somehow know Pi's exact value or is it just an estimate? b)...
  5. M

    Can irrational numbers exist on the numberline?

    This may be an elementary question, but I've been thinking about it a little bit and wondering what other people thought. First, let me say that I'm talking about a number line not as a set but in the more literal sense, like a partitioned line that might exist as the axis of a graph. So...
  6. B

    Rational and irrational numbers proof

    Hey all, I'm new here so I'm a little noobish at the formatting capabilities of PF. Trying my best though! :P Homework Statement Let a, b, c, d \in Q, where \sqrt{b} and \sqrt{d} exist and are irrational. If a + \sqrt{b} = c + \sqrt{d}, prove that a = c and b = d. Homework...
  7. W

    Irrational numbers - incomprehnsible?

    Hi, I am having an issue with irrational numbers and the term irrational. Main Entry: 1ir·ra·tio·nal Pronunciation: \i-?ra-sh(?-)n?l, ?i(r)-\ Function: adjective Etymology: Middle English, from Latin irrationalis, from in- + rationalis rational Date: 14th century : not rational: as a...
  8. G

    Irrational numbers in infinite list of integers

    Is it safe to assume that the absolute value of sin x is greater than zero for all positive integer values of x? I have no real experience in number theory, and I don't know if you can say that there are no irrational numbers in an infinite list of integers.
  9. J

    Irrational Numbers and Real Life I need 6 answers

    Once again my professor asked us to ask 6 people the following question and see how they answer it so if you could respond and give an answer, I would really appreciate it. And if possible can you also tell me a little bit about your mathematics background? We are supposed to write up what...
  10. O

    Question about irrational numbers

    Let p and q be distinct primes. Prove that \sqrt{p/q} is a irrational number.
  11. C

    For every rational number, there exists sum of two irrational numbers

    Homework Statement Prove: For every rational number z, there exists irrational numbers x and y such that x + y = z. Homework Equations by definition, a rational number can be represented by ratio of two integers, p/q. The Attempt at a Solution Is there a way to do this by...
  12. icystrike

    Irrational Numbers: Infinite Numerical Values Explained

    Irrational Numbers are contained by infinite numerical values?
  13. G

    Prove the set of irrational numbers is uncountable.

    Homework Statement Prove the set of irrational numbers is uncountable. Homework Equations The Attempt at a Solution We proved that the set [0,1] is uncountable, but I'm not sure how to do it for the irrational numbers.
  14. S

    Rational and irrational numbers. (semi- )

    Rational and irrational numbers. (semi-urgent) I need to figure this out by tomorrow =/ Homework Statement a. If a is rational and b is irrational, is a+b necessarily irrational? What if a and b are both irrational? b. If a is rational and b is irrational, is ab necessarily irrational...
  15. C

    Irrational Numbers: Proving a Number is Irrational

    What is a proof that a number is irrational. For instance, how do we know the PI goes on forever without a pattern?
  16. C

    Apostol's Analysis 1.11- Irrational Numbers Proof

    Homework Statement Given any real x > 0, prove that there is an irrational number between 0 and x. Homework Equations I'm not sure if the concepts of supremums or upper bounds can used. The Attempt at a Solution Take an irrational number say Pi. We can always choose a number n such...
  17. N

    Irrational numbers and repeating patterns

    Hi, I want to show that an irrational number (let's say pi) can never have an (infinitely) repeating pattern (such as 0.12347 12347 12347 ...). Is it possible to 'proof' (or just make it more acceptable, I don't need a 100% rigorous proof) this easily, without using too much complicated math...
  18. T

    Why Do Irrational Numbers Exist?

    why do irrational numbers exist? I am well familiar with the proof that irrational numbers exist, but why do they?
  19. J

    Converting Equations to Binary & Irrational Numbers

    ... can we convert this equation to binary notation? Also another one, would an irrational number be irrational in any number format?
  20. P

    *Proof* Sum of Rational and Irrational Numbers

    Homework Statement Prove by contradiction: If a and b are rational numbers and b != 0, and r is an irrational number, then a+br is irrational. In addition, I am to use only properties of integers, the definitions of rational and irrational numbers, and algebra. You guys should also know that...
  21. M

    Pigeonhole Principle & irrational numbers

    Homework Statement Let x be an irrational number. Show that the absolute value of the difference between jx and the nearest integer to jx is less than 1/n for some positive integer j not exceeding n. Homework Equations The Attempt at a Solution Ok, I know that it should be solved using...
  22. K

    Is x Irrational if x² Is Irrational?

    Homework Statement Prove that if x^2 is irrational then x must be irrational. Homework Equations The Attempt at a Solution Maybe do proof by contradiction. I'm not really sure where to start.
  23. A

    Definition & Properties of Irrational Numbers

    How do we exactly define irrational numbers.. ive asked this before... but id like to know about any infinite series, if any which is used to define irrational numbers... and how can one prove properties of basic operations for irrational numbers Thanks
  24. K

    Proving Irrationality of 2√2, 2-√2, 17√(1/2)

    I know that √2 is irrational (and I've seen the proof). Now, what is the fastest way to justify that 2√2, 2-√2, 17√(1/2) are irrational? (they definitely "seem" to be irrational numbers to me) Can all/any these follow immediately from the fact that √2 is irrational? Thanks!
  25. H

    Proving Irrationality of Sums and Products of Irrational Numbers

    hi i m hashim i want to solve a qquestion 1.if x is rational & y is irrational proof x+y is irrational? 2. if x not equal to zero...y irrational proof x\y is irrational?? 3.if x,y is irrational ..dose it implise to x+y is irrational or x*y is irrational thanks please hashim
  26. A

    Difference of two irrational numbers

    Im wondering if its possible given x,y irrational, that x-y is rational (other than the case x=y). The reason I am asking this is that I am reading a book on measure theory and they try to construct a non measurable set and they start with an equivalence relation on [0,1} x~y if x-y is rational...
  27. F

    Randomness of digits of irrational numbers.

    How random are the digits of irrational numbers? Can it be said of them (i.e. pi=3.14159...) that given any arbitrarily long string of digits it must occur at some point in any irrational number? And would anyone know of anywhere I could find out more on this topic?
  28. Pythagorean

    The Might of Occam's Blade Stops Irrational Numbers

    I have discovered the mighty blade of Occam, I shall destroy all who advocate the existence of irratonal numbers. Spiders are your gods.
  29. Pythagorean

    Irrational Numbers Don't Exist: Worship Spiders

    don't exist! and spiders are your new gods, worship them...
  30. C

    Proof that q^2 is divisible by rm^2

    http://www.artofproblemsolving.com/Forum/weblog.php?w=564 Could someone help me with 2b? Thanks
  31. S

    Are all irrational numbers rational?

    Since pie is the ratio of the circumference of the circle to its diameter, isn't it possible that there exist a fraction for all nonrepeating going on forever decimal values?
  32. W

    Can Pi be Used as a Random Number Generator for Proving Normality?

    Just wondering, if you group decimal places of an irrational number, let's say into sequences of groups of 10, for example, if k is irrational 4.4252352352,3546262626,224332 (I made that up) they you group (.4252352352) (3546262626) (and so on) then my question is that the probability...
  33. M

    Is √5 + √3 Irrational by Contradiction?

    we can prove that √5 is irrational through contradiction and same applies for √3. but can we prove that √5 + √3 is irrational by contradiction?
  34. S

    Explore Irrational Numbers: Find Out How Close You Can Get!

    Just curious about a thing I've been thinking of: It's true that that there are numbers that aren't rational... let's say x is such a number. Now we take two integers, a and b where a is the integer if x is rounded up, and b is the integer if x is rounded down. Forming their arithmetic...
  35. T

    Sme question about irrational numbers

    Some question about irrational numbers Our teacher showed us Cantor's second diagonal proof. He said that by this proof we can show that there are more irrational numbers than rational numbers. He also said that the cardinality of natural numbers or rational numbers has a magnitude...
  36. C

    Proving the Irrationality of √3 and Other Non-Perfect Square Roots

    Hello all I encountered a few questions on irrational numbers. 1. Prove that \sqrt{3} is irrational [/tex]. So let l = \sqrt{3} . Then if l were a rational number and equal to \frac{p}{q} where p, q are integers different from zero then we have p^{2} = 3q^{2} . We can assume that...
  37. L

    Rational and Irrational Numbers

    I need to show that a rational + irrational number is irrational. I am trying to do a proof by contradiction. So far I have: Suppose a rational, b irrational. Then a = p/q for p, q in Z. Then a + b = p/q + b = (p + qb) / q But I don't know where to go from here because I still have a...
  38. C

    Schwarz Inequality and Irrational Numbers

    -------------------------------------------------------------------------------- Hello everyone. I have 2 questions. 1. Prove that the cube root (3) + sqrt (2) is irrational. My Solution Assume l is an irrational number of the form p/q where p and q are integers not equal to 0...
  39. C

    Schwarz Inequality and Irrational Numbers

    Hello everyone. I have 2 questions. 1. Prove that the cube root (3) + sqrt (2) is irrational. My Solution Assume l is an irrational number of the form p/q where p and q are integers not equal to 0. Then p^6 / q^6 = [(cube root(3) + sqrt (2))]^6 I concluded that it must be in the...
  40. W

    Irrational numbers vs. Transcendental numbers

    It would seem that an irrational number would have to be a transcendental number. If a transcendental number is a number which goes on infinitely and never repeats, then all irrational numbers would have to be transcendental, because if they repeated then you could find a fraction doing the...
  41. N

    Understanding Irrational Numbers: Is it Possible to Exact Measure?

    Okay, I was thinking about irrational numbers, and I came to this conclusion: It is impossible exactly measure an irrational number.I am probably wrong, and that's why I posted this thread to check the validity of that statement. Here is my proof: If you wanted to cut a piece of paper...
  42. J

    Irrational Numbers: Expressible as Infinite Summations?

    I apologise if this belongs in another place, but: Can all irrational numbers be expressed as infinite summations, ie like Pi and e? I'm looking for: provable, disprovable, or neither. This is essential to something else I am working on. sincerely, jeffceth
Back
Top