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Math Help - Finding an exact number

  1. #1
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    Finding an exact number

    Hi,

    Let's say I have the number 1.414213562, now I know that can be expressed as sqr[2]. If I didn't know that, how could I find it out? I just ask as sometimes I get a number and don't know if there is a better/ more accurate way to express it.

    Thanks,

    -Mark.
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  2. #2
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    use a calculator
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  3. #3
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    I mean with a calculator
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  4. #4
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    Hello mark090480
    Quote Originally Posted by mark090480 View Post
    Hi,

    Let's say I have the number 1.414213562, now I know that can be expressed as sqr[2]. If I didn't know that, how could I find it out? I just ask as sometimes I get a number and don't know if there is a better/ more accurate way to express it.

    Thanks,

    -Mark.
    I don't think there's an easy answer to this question.

    If you suspect a number is a simple square/cube root, you can simply square/cube it and see what happens. You need to remember, of course, that 1.414213562 is only an approximation to \sqrt2, so squaring it won't give you an exact answer.

    If it's a recurring decimal - like 0.846153846153... - there's a straightforward way to find its corresponding exact fraction. Using that as an example:
    Let x = 0.846153846153...

    \Rightarrow 1000000x = 846153.846153...


    \Rightarrow 999999x = 846153 (by subtracting the first equation from the second)


    \Rightarrow x = \frac{846153}{999999}

    You've then got to experiment with cancelling the fraction to reduce it to its simplest terms. This one is:
    \frac{11}{13}
    Sometimes, you might just recognise the number; for example 2.718... you may know as the approximate value of e; and, of course, 3.142 which is approximately \pi.

    But if the decimal number you've been given is more complicated than this - for example 0.31783724519578224472575761729617 - then I think there are no set methods for discovering what it is. (Incidentally, that one is approximately \sqrt3-\sqrt2, but I only know that because that's what I started with!)


    Grandad
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  5. #5
    Super Member Bacterius's Avatar
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    Quote Originally Posted by mark090480 View Post
    Hi,

    Let's say I have the number 1.414213562, now I know that can be expressed as sqr[2]. If I didn't know that, how could I find it out? I just ask as sometimes I get a number and don't know if there is a better/ more accurate way to express it.

    Thanks,

    -Mark.
    Hello,
    since a real could potentially be anything, there is no systematical way to find out what it approximates as a function. However, you might want to look at this example :

    0.549306144

    What is this ? Is this a square root ?

    0.549306144^2 = 0,301737240 which is not quite a near-integer or integer.

    Could this possibly be the sine/cosine of some integer angle ?

    \sin^{-1}{(0.549306144)} = 33,319424585, fail.

    \cos^{-1}{(0.549306144)} = 56,680575414, fail.

    Perhaps a logarithm ...

    e^{0.549306144} = 1,732022166

    This reminds me of some square root ... let's see : 1,732022166^2 = 2,999900783540222463303473242058. Yay ! This is the square root of three ! Therefore :

    0.549306144 = \ln{(\sqrt{3})}

    As you can see, by checking a function you wipe out all possibilities for this function, and since there are few "common" functions you can usually try different combinations of them and see what you get (like try square root, then logarithm, then logarithm of square root, then square root of logarithm, ...)

    But since every function is different and that a real can potentially be equal to anything (the logarithm of some enormous integer divided by a constant real added to the sine of one) there is no simple answer to your question
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  6. #6
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    "Recognizing" a number given to many digits is one of the methods of "experimental mathematics". See, for example, this link:

    Recognizing Numerical Constants
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  7. #7
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    Hello awkward
    Quote Originally Posted by awkward View Post
    "Recognizing" a number given to many digits is one of the methods of "experimental mathematics". See, for example, this link:

    Recognizing Numerical Constants
    Wow, this is pretty high-powered stuff! I think I should need quite a little study time to absorb it!

    Grandad
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  8. #8
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    You might also be interested in the on-line Inverse Symbolic Calculator:

    CARMA: Computer-Assisted Research Mathematics and its Applications

    I just tested the "advanced lookup" with 9.7430965594997716285,

    which it was able to recognize as \sqrt{5} \;  \pi + e.
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  9. #9
    Super Member Bacterius's Avatar
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    Quote Originally Posted by awkward View Post
    You might also be interested in the on-line Inverse Symbolic Calculator:

    CARMA: Computer-Assisted Research Mathematics and its Applications

    I just tested the "advanced lookup" with 9.7430965594997716285,

    which it was able to recognize as \sqrt{5} \;  \pi + e.
    It failed to recognize \sqrt{\ln(51) \pi} in advanced mode
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  10. #10
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by awkward View Post
    You might also be interested in the on-line Inverse Symbolic Calculator:

    CARMA: Computer-Assisted Research Mathematics and its Applications

    I just tested the "advanced lookup" with 9.7430965594997716285,

    which it was able to recognize as \sqrt{5} \;  \pi + e.
    It recognized \frac{e^\pi}{\sqrt[e^\pi]{\pi^e}}

    :O!!!
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  11. #11
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    This is extremely clever! I am impressed.

    Grandad
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