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Irrational numbers and number systems.

  • 07-12-2006 8:26pm
    #1
    nobodythere
    Closed Accounts Posts: 2,349 ✭✭✭


    I've been learning about fractions in binary numbers and how for example some numbers canot be represented in different bases - for example certain decimal numbers could not be converted to binary because after the decimal point (errrrr) in a binary number, each number would represent 2^(-1), 2^(-2), 2^(-3) and so on. Similarly in the decimal system numbers after the point represent 10^(-1) etc etc.

    Is it therefore conceivable that an irrational number like pi or e that we normally consider irrational could be expressed rationally in another base?


Comments

  • #2
    Son Goku
    Closed Accounts Posts: 1,475 ✭✭✭Son Goku


    Is it therefore conceivable that an irrational number like pi or e that we normally consider irrational could be expressed rationally in another base?
    A lot of numbers, which are infinitely repeating in a base aren't in other bases. However other numbers such as pi or e can never be expressed as a finite decimal in any base.

    This is because a decimal in any base is just a sum of a finite number of powers of that base.
    c1 X n^(-1) + c2 X n^(-2) + c3 X n^(-3) +................ as you said

    Where n is the base and c are some numbers that multiply the base.

    In some bases a fraction will be infinite, in others it won't.

    However you can prove that there exists no base in which the sum is finite for numbers like pi or e, it will always be infinite.

    This is related to why 0.999999....... = 1.


  • #3
    Farouk.Bulsara
    Closed Accounts Posts: 50 ✭✭Farouk.Bulsara


    grasshopa wrote:
    I've been learning about fractions in binary numbers and how for example some numbers canot be represented in different bases - for example certain decimal numbers could not be converted to binary because after the decimal point (errrrr) in a binary number, each number would represent 2^(-1), 2^(-2), 2^(-3) and so on. Similarly in the decimal system numbers after the point represent 10^(-1) etc etc.

    Is it therefore conceivable that an irrational number like pi or e that we normally consider irrational could be expressed rationally in another base?

    A fraction is by definition a rational number - it is the ratio of two integers. All fractions can be converted to binary - however, some have non-terminating binary expansions. Just like 1/3 has a non-terminating decimal expansion 0.333333333 etc. In base 3, this would just be 0.1, so no problem there.

    Note however that numbers like e and pi are irrational - they are *not* the ratio of two integers. The major difference in representation (in any base) is that they have non-terminating AND non-repeating expansions. If the pattern of digits *did* repeat, then it is easy to show that the number is in fact a rational number (i.e. a fraction). So irrational numbers can never be represented with a finite or repeating expansion of digits.

    Farouk


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