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Posers and Puzzles

Posers and Puzzles

  1. 17 Dec '03 16:35
    mathmatics is based on a number of self evidant truths, they can not be proven, but they are held true because it is conveint.

    what happens if you ignore one? take X=X for example. if we disregard that rule, and instead say 1=2, and still hold all other truths, then it follows that for given numbers X and Y, where both X and Y are real numbers, X=Y. in other words, all real numbers are the same

    what about unreal numbers, look at complex numbers: Ai+B
    A and B are real numbers, thus they are all the same, i always represents (-1)^(1/2). thus all complex numbers are the same. but are real numbers the same as imaginary numbers?

    where can you pepole take this?
  2. Standard member TheMaster37
    Kupikupopo!
    17 Dec '03 17:24
    Originally posted by fearlessleader
    mathmatics is based on a number of self evidant truths, they can not be proven, but they are held true because it is conveint.

    what happens if you ignore one? take X=X for example. if we disregard that rule, and instead say 1=2, and still hold all other truths, then it follows that for given numbers X and Y, where both X and Y are real numbers, X= ...[text shortened]... e same. but are real numbers the same as imaginary numbers?

    where can you pepole take this?
    '=' is a relation defined with the following properties;

    1) X=X
    2) If X=Y then Y=X
    3) If X=Y and Y=Z then X=Z

    If you throw out the first property, you can't use '=' anymore in the way you usually use it, not even to say '1=2'

    Further more; The number '2' is defined as 1+1, the number following 1. Therefore you can't define '1=2' anymore, because then you'd cancel the definition of '2'. And with that you'd cancel the definition of all positive integers, intergers, fractions, reals and next complex numbers. You'd only have '1', and maybe '0'. Per definition those two are different as well, so defining '1=0' would leave you with just one of the two, then there would be no other numbers.

    Ton
  3. 17 Dec '03 20:49
    Originally posted by TheMaster37
    '=' is a relation defined with the following properties;

    1) X=X
    2) If X=Y then Y=X
    3) If X=Y and Y=Z then X=Z

    If you throw out the first property, you can't use '=' anymore in the way you usually use it, not even to say '1=2'

    Further more; The number '2' is defined as 1+1, the number following 1. Therefore you can't define '1=2' an ...[text shortened]... g '1=0' would leave you with just one of the two, then there would be no other numbers.

    Ton
    i can see that all real numbers would be de-difined, but they wouldn't sease to exsist. and i do not see how it would effect unreal numbers, like i or infinity or 1/0.
  4. Standard member Fiathahel
    Artist in Drawing
    18 Dec '03 11:26
    Originally posted by fearlessleader
    i can see that all real numbers would be de-difined, but they wouldn't sease to exsist. and i do not see how it would effect unreal numbers, like i or infinity or 1/0.
    the problem is: if you define 1=2, and keep the definition of + the same, then 0=1-1=2-1=1. If 0=1 then by induction also 2=3, 3=4, 4=5,... . The only number that can differ from 0 are the reals minus the integers, with of course 0.5 = 1.5 = 2.5. But keep in mind that there is no element a anymore with a*b=b*a=b. So 0.4/0.2 isn't defined anymore. Your first idea might be 0.4/0.2=2=0, but 0*0.2 =/= 0.4.

    Infinity can still be defined, cause you can't apply standard rules on it. i doesn't exsist anymore, cause there is no -1.
  5. 20 Dec '03 21:18 / 2 edits
    Originally posted by Fiathahel
    the problem is: if you define 1=2, and keep the definition of + the same, then 0=1-1=2-1=1. If 0=1 then by induction also 2=3, 3=4, 4=5,... . The only number that can differ from 0 are the reals minus the integers, with of course 0.5 = 1. ...[text shortened]... dard rules on it. i doesn't exsist anymore, cause there is no -1.
    no, all real numbebers will be equal, as all ratinal numbers can be defined by two intigers divided by eachother, and all iratinals are equal to a ratinal plus another iratinal. it seems to me that i would still be seperate from the reals, as their wouls still be negatives, thay would just be the same as positives. what i would be is unclear to me, I think it would be the same exept that it would be the same as all the other complex numbers.

    also, as i understand it, in calc, 2infinity is greater than infinity, but 2 has lost its definition, thus all the products of infinity are the same, efectivly screwing calc.
  6. 21 Dec '03 11:37
    You can get the same kind of effect in normal maths by doing arithmetic on multiples of infinity.

    1*inf = 2*inf = 3*inf

  7. Standard member TheMaster37
    Kupikupopo!
    21 Dec '03 12:20
    Originally posted by fearlessleader
    no, all real numbebers will be equal, as all ratinal numbers can be defined by two intigers divided by eachother, and all iratinals are equal to a ratinal plus another iratinal. it seems to me that i would still be seperate from the reals, as their wouls still be negatives, thay would just be the same as positives. what i would be ...[text shortened]... lost its definition, thus all the products of infinity are the same, efectivly screwing calc.
    i = sqrt(-1) = sqrt (1) = 1
  8. Donation Acolyte
    Now With Added BA
    21 Dec '03 13:25 / 2 edits
    I can see why 1=2 implies that all rational numbers are equal, but what about irrational numbers? Wouldn't R just become R/Q (if that makes sense to anyone)?

    EDIT: Stratch that, all algebraic numbers would be equal. But what about the transcendentals? Now that would be a freaky set: R divided by the field of all real algebraic numbers...
  9. Standard member TheMaster37
    Kupikupopo!
    21 Dec '03 14:31
    Originally posted by Acolyte
    I can see why 1=2 implies that all rational numbers are equal, but what about irrational numbers? Wouldn't R just become R/Q (if that makes sense to anyone)?

    EDIT: Stratch that, all algebraic numbers would be equal. But what about the transcendentals? Now that would be a freaky set: R divided by the field of all real algebraic numbers...
    Hmm, let's see where tis reasoning strands:

    All reals can be approximated by rationals. Eg, for Pi we have 3 le Pi le 4. Since 3=4=1, we have that 1=3=Pi=4. You can do tihs with more accurate bounds as well, since all rationals are equal as well.
  10. 21 Dec '03 22:51
    Originally posted by TheMaster37
    i = sqrt(-1) = sqrt (1) = 1
    yes, but i dont know

    maby your right, and i would cese to exsist.
  11. 21 Dec '03 22:53
    Originally posted by Acolyte
    I can see why 1=2 implies that all rational numbers are equal, but what about irrational numbers? Wouldn't R just become R/Q (if that makes sense to anyone)?

    EDIT: Stratch that, all algebraic numbers would be equal. But what about the transcendentals? Now that would be a freaky set: R divided by the field of all real algebraic numbers...
    algebraic vs. transcendentals?

    explain.
  12. Standard member Fiathahel
    Artist in Drawing
    22 Dec '03 12:12
    Before we continu this thread it could be useful to give a definition of R we want to use. Cause, for example, you cannot use that it can be writen as an endless chain of numbers: lik Pi =/= 3.14...... Cause that is a limit with elements in Q, and therefor must equal 1.
  13. 26 Dec '03 21:13
    Originally posted by Fiathahel
    Before we continu this thread it could be useful to give a definition of R we want to use. Cause, for example, you cannot use that it can be writen as an endless chain of numbers: lik Pi =/= 3.14...... Cause that is a limit with elements in Q, and therefor must equal 1.
    god damn my mathmatical inferiority!

    can someone explaing in more drawn out terms what these pepole are talking about?:'(
  14. Standard member TheMaster37
    Kupikupopo!
    27 Dec '03 22:29
    Originally posted by fearlessleader
    god damn my mathmatical inferiority!

    can someone explaing in more drawn out terms what these pepole are talking about?:'(
    Ok, suppse you have a row of numbers, an endless row. And all those numbers are elements of Q (the rational numbers, numbers wich can be written as a fraction). Suppose the numbers in that row are coming very close to Pi the further you go down the row. For example;

    3, 31/10, 314/100, 3141/1000, 31415/10000, 314159/100000, ...

    We say that the row converges, and has limit Pi.

    Now you can probabely see that all reals (everything you can make without i) can be approximated by fractions. That's what Fiathahel was talking about, in more complicated words.