Irrational numbers

  1. adjacent

    adjacent 1,533
    Gold Member

    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:
     
  2. jcsd
  3. Not quite right. An irrational number is any real number which cannot be expressed as the ratio of two integers. 3.62566 = 362566 / 100000 and so it is rational. Any decimal which terminates is therefore rational.
     
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  4. adjacent

    adjacent 1,533
    Gold Member

    Oh, thank you for that.
    :shy:
    I think I was relying on my stupid calculator too much
     
  5. Mentallic

    Mentallic 3,684
    Homework Helper

    But also, any decimal that has a repeating pattern is rational too.

    [itex]0.333... = 0.\overline{3} = 1/3[/itex] (repeats every digit)

    [itex]0.142857142857... = 0.\overline{142857} = 1/7[/itex] (repeats every 6 digits)

    Generally, if we have a number between 0 and 1 which repeats after n digits, then it's of the form

    [tex]0.\overline{a_1a_2a_3...a_n}[/tex]

    So if we let

    [tex]x=0.\overline{a_1a_2a_3...a_n}[/tex]

    then

    [tex]10^nx = a_1a_2a_3...a_n.\overline{a_1a_2a_3...a_n} = a_1a_2a_3...a_n + 0.\overline{a_1a_2a_3...a_n}[/tex]

    and so

    [tex]10^nx-x=a_1a_2a_3...a_n[/tex]

    Solving for x,

    [tex]x(10^n-1)=a_1a_2a_3...a_n[/tex]

    [tex]x=\frac{a_1a_2a_3...a_n}{10^n-1}[/tex]

    which is a ratio of 2 integers, hence x is a rational number.

    You can use this procedure to easily find, for example, what 0.5454545... is in fractions.

    Let [tex]x=0.\overline{54}[/tex]

    [tex]100x=54.\overline{54}[/tex]

    [tex]100x-x = 54.\overline{54} - 0.\overline{54} = 54[/tex]

    [tex]x=\frac{54}{99}[/tex]
     
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  6. micromass

    micromass 18,543
    Staff Emeritus
    Science Advisor

    There are uncountably many irrational numbers. An irrational number itself cannot be uncountable, I have no idea what that would mean.
     
  7. adjacent

    adjacent 1,533
    Gold Member

    Count ##\pi##. I am sure you can't
     
  8. Yes, I can: "Pi!"
    That wasn't so difficult.

    If you meant the digits of pi, these are still not uncountable, even if there is an infinite number of them.
     
  9. adjacent

    adjacent 1,533
    Gold Member

    So funny. But I asked you to count it :p and ##\pi !## returns a math error
    You will have to do that forever then.
     
    Last edited: Jun 13, 2014
  10. Mentallic

    Mentallic 3,684
    Homework Helper

    Can you count 2.5 to show us what you mean? :confused:

    Because your calculator's factorial function is only defined for non-negative integers. Wolfram alpha gives 7.188... (an irrational).


    There are a countably infinite amount of digits in any irrational or rational that doesn't terminate. The number of irrational numbers however is uncountably infinite.
     
  11. adjacent

    adjacent 1,533
    Gold Member

    Two point five
    That's why my calculator is stupid. (Sometimes I forget to set to degrees and get the wrong answer in tests)

    That is so strange. There is a joke:
    Hahah, if you can count a set of things infinite in number it means you do that for eternity?
     
  12. Nugatory

    Staff: Mentor

    A countably infinite set is a set whose members can be put into one-to-one correspondence with the set of integers. That's clearly the case for the digits of ∏; just map the n'th digit to the integer n. On the other hand, the set of real numbers is not countably infinite; it can be proved that no such mapping can be constructed.

    If you want an amusing diversion... You might try googling for "Hilbert's hotel"
     
  13. Mentallic

    Mentallic 3,684
    Homework Helper

    Ok then count wasn't the word you were looking for. Reciting the digits is what you want.

    It's not stupid, it just wasn't programmed to solve anything and everything that Mathematicians have ever discovered. The calculator wouldn't have enough memory for that anyway. If you need to find the factorial of a non-integer, then you're likely at a level where a calculator just isn't going to be enough to do everything for you. That's what computers and math programs like matlab were made for. Do you need to work with larger numbers than 10100? Then use a computer :smile:

    Well then that's not the calculator's fault, is it :tongue:
    Unless you're doing a test in physics or some other science that works solely in degrees, keep to radians and then can convert when needed (which should be rarely). Also, you want to work towards having an intuition about the magnitude of the numbers you would be expecting. If you are calculating something in radians but your calculator is unfortunately set to degrees, then you should be able to spot the fault when you get a value that is 2 orders of magnitude less than expected.


    Countably infinite means that you would technically be counting for an eternity, yes. The difference however is that you can't find a way to count all of the irrational numbers (if you could count for eternity).
     
  14. D H

    Staff: Mentor

    There is one and only one number that qualifies as ##\pi##. So the answer to your question is one.

    The integers are a subset of the rationals, which in turn are a subset of the algebraic numbers, which in turn are a subset of the computable numbers. Even though the computable numbers are a superset of the integers, both sets have the same cardinality. They are countable sets.

    Note that ##\pi## and ##e## are computable numbers.

    The reals are not countable. Most ("almost all" is the technical term) or the reals cannot be computed.
     
  15. micromass

    micromass 18,543
    Staff Emeritus
    Science Advisor

    Count ##1/3##, you can't do that either. It also has an infinite number of digits.
     
  16. Nugatory

    Staff: Mentor

    And spend some time understanding the limitations of the algorithms your computer is using to perform arithmetic.... There have to be some limitations somewhere, because the computer can only manipulate a finite number of finite-length bit strings.

    An entire generation has been raised in blissful ignorance of what life was like before we discovered fire, toolmaking, and IEEE floating-point arithmetic. (Not that IEEE arithmetic is perfect - far from it - but it is good enough that we can generally survive despite our complacency).
     
  17. There is some terminology confusion on adjacent's part. Just because a set is infinite doesn't mean it is uncountable. That I am a human and have a short lifespan is irrelevant to mathematical truth. Being countable or uncountable has nothing to do with what humans can do in their lifetimes, and everything to do with an existence of a one-to-one correspondence with the naturals. Think about it, if being an infinite set meant you were uncountable, then the naturals would be uncountable, but that is the very set upon which counting is based!


    As for the π issue, unfortunately, people associate numbers so much with how they are written, rather than how they are defined, that the concept of a nonterminating, nonrepeating decimal number like π somehow becomes this mysterious entity that we can "never know exactly" in their minds.

    Irrationals, rationals, etc are not defined by what they look like or how they can be written. This is merely a result of the mathematical definition. Numbers are not symbols on paper. Symbols on paper are a means of communicating the idea of a number. I actually had to convince someone that 4/2 was an integer recently.

    An irrational number is not a number that does not terminate or repeat. Numbers do not terminate or repeat, such a statement is nonsense. An irrational number is a number that is not equal to any ratio of integers. As a result, when writing this number in decimal form, that notation will not allow for repetition or termination, but that is not a definition.
     
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  18. adjacent

    adjacent 1,533
    Gold Member

    Ok everyone, thanks for helping me. I think I get it now :smile:
     
  19. Mentallic

    Mentallic 3,684
    Homework Helper

    Maybe they should start with coding in C without using imported packages, run into a few overflow and precision issues and a lack of functions, then they'll begin to appreciate calculators again.
     
  20. stevendaryl

    stevendaryl 2,800
    Science Advisor

    In mathematics, there are different "sizes" of infinity. There are infinitely many integers: 0, 1, 2, ... It never stops, so it's infinite. There are also infinitely many real numbers. But there is a sense in which there are more real numbers than there are integers. The sense is this:

    Two sets [itex]A[/itex] and [itex]B[/itex] are said to be "the same size" (technically, the same cardinality) if you can set up a correspondence between the two sets, so that every element of [itex]A[/itex] is matched with exactly one element of [itex]B[/itex], and vice-verse (technically, a one-to-one mapping). For example, the sets

    [itex]A = \{ cat, dog, pig \}[/itex]
    [itex]B = \{ red, yellow, blue\}[/itex]

    are the same size because they can be put into correspondence many different ways, but here's one: [itex]cat \leftrightarrow red,\ dog \leftrightarrow yellow,\ pig \leftrightarrow blue[/itex]

    Infinite sets can be put into a one-to-one correspondence, also. For example, the set [itex]A = [/itex] the positive integers and the set [itex]B = [/itex] all integers:

    [itex]1 \leftrightarrow 0[/itex]
    [itex]2 \leftrightarrow -1[/itex]
    [itex]3 \leftrightarrow +1[/itex]
    [itex]4 \leftrightarrow -2[/itex]
    [itex]5 \leftrightarrow +2[/itex]
    etc.

    You can also set up a one-to-one correspondence between the integers and the rationals. That's a little harder to describe, but it can be done.

    Any set that can be put into a one-to-one correspondence with the positive integers is called a "countable" set.

    Some sets are not countable. The easiest example is the set of reals. There is no way to set up a one-to-one correspondence between the positive integers and the reals.
     
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  21. adjacent

    adjacent 1,533
    Gold Member

    I see, thank you so much :smile:
     
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