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Basics of numbers!

  1. Sep 2, 2008 #1
    Numbers are used to tell us about the quantity of something

    Using just numbers and logic how do we prove that say 2+2=4

    we cant use objects here as then the following problem comes up:
    if we have say 3/2+1

    and we take all numbers to be objects..the 3 objects divided by 2 objects is 3/2 +1 object which cant be resolved ...so please clear my doubt


    this is very important as numbers form the basis of math!:biggrin:

    Thanks
     
  2. jcsd
  3. Sep 2, 2008 #2

    HallsofIvy

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    Using Peano's axioms for the natural numbers, (see http://academic.gallaudet.edu/cours...bc4dda7eb1c8525688700561c74/$file/NUMBERS.pdf) every number a has a "successor" s(a) and every number except 1 is the successor of some number. 2 is defined as s(1), 3 is defined as s(2), and 4 is defined as s(3).

    Further addition is defined by
    a+ 1= s(a). If b is not 1, then it is the successor of some number, say c: b= s(c). In that case a+ b is defined as s(a+ c). While that is a "recursive" definition, it can be shown that it defines unique "a+ b" for every a and b.

    Now, 2+ 2= s(2+ 1)= s(s(2))= s(3)= 4.

    If I remember correctly Russel and Whitehead, using concepts simpler than the Peano axioms, required 4 pages to prove 2+ 2= 4!
     
  4. Sep 2, 2008 #3
    i have one thing more to ask

    in physics....we just make use of numbers isnt it...i mean we can do all the operations and mathematical manipulations just because we can use these theorems of numbers isnt it...?

    thanks for all the help
     
    Last edited: Sep 2, 2008
  5. Sep 2, 2008 #4
    Isnt there any simpler way to prove these theorems...i dont really know the set theory and relations..etc
     
  6. Sep 2, 2008 #5

    HallsofIvy

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    No, there is no "royal road" to mathematics! If you are interested in learning the proofs of the fundamental concepts of mathematics, then you will need to learn set theory, etc.
     
  7. Sep 2, 2008 #6
    hehe...but well...people like pythagoras and co. wouldnt have been using these axioms to prove their theorems would they...!

    Is there any primitive proof...which isnt so rigorous but will do for basics in calculus!

    thanks
     
  8. Sep 2, 2008 #7

    CRGreathouse

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    Pythagoras wouldn't have felt the need to prove 2 + 2 = 4. Modern mathematicians did, so we/they invented axiomatic set theory and a plethora of formal logics.

    You can do as the ancients and take 2 + 2 = 4 on faith, or you can learn the basics of set theory (not hard!) to understand these proofs.
     
  9. Sep 2, 2008 #8
    Could you please provide a link..as to where i would get the necessary theory needed to understand Peano's axioms

    and about pythagora's :) dont we DEFINE addition and wouldnt he have thought of multiplication of numbers as different as compared to that of objects...like
    3/2+1

    and we take all numbers to be objects..the 3 objects divided by 2 objects is 3/2 +1 object which cant be resolved further?

    thanks
     
    Last edited: Sep 2, 2008
  10. Sep 2, 2008 #9
    one more thing....dont we define operations like say [tex] \frac{\frac{3}{2} +\frac{1}{3}}{2} [/tex] as being done imagining objects and getting the ans on each operation and then considering only the NUMBER part of the answer and carrying fwd the calculations
     
    Last edited: Sep 2, 2008
  11. Sep 2, 2008 #10
    anant - try to find David Berlinski's (sp??) book "advent of the algorithm." I am reading this now and it goes into some of these questions. And it is written (very) informally, not like a normal mathematical book. In fact, it may be a little too informal in spots.
     
  12. Sep 2, 2008 #11
    Ill surely try to find the book..

    id just lik to ask..is there anything wrong in my defiinition.As this explains the gr8 use of math we can do in physics...as we just take any equation(lik f=ma) and manipulate it according to our needs as anyway operating on two equal numbers will give us equal numbers,no matter what ,as the rules are same for bothe numbers,according to our definition.

    thanks
     
  13. Sep 2, 2008 #12

    CRGreathouse

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    If you mean

    "dont we define operations like say [tex] \frac{\frac{3}{2} +\frac{1}{3}}{2} [/tex] as being done imagining objects and getting the ans on each operation and then considering only the NUMBER part of the answer and carrying fwd the calculations"

    then I can't make sense of it, let alone say that it's correct. If not, which definition?
     
  14. Sep 2, 2008 #13
    the successor function obviously forms the basis of math but we could DEFINE the successor of 9 to be zero in which case we would have modular arithmetic. also we could define more than one successor for a given point. then we no longer have numbers but a graph of interconnected points. what field of math would that be?
     
  15. Sep 2, 2008 #14
    by this i mean...3/2 +1/3 using objects that is say using using an orange...=11/6

    Now,if we take the case of [tex] \frac{ \frac {1}{12}}{3} [/tex]
    one onragen by twelve oranges is 1/12.Now i think of 1/12 as 1/12 oranges again and repeat the algorithm to get my answer...this is OK as a definition of an algorithm isnt it?
    Yes obviously we could do this...but wouldnt make much sense
     
    Last edited: Sep 2, 2008
  16. Sep 2, 2008 #15
    it wouldnt make sense for numbers. it would for a graph. I was just wondering what that field is colled.
     
  17. Sep 2, 2008 #16
    You are not clear enough. I have no idea what you are asking.

    But it does make sense! Modular arithmetic is at the heart of modern algebra.
     
  18. Sep 2, 2008 #17

    CRGreathouse

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    I have no idea what you're trying to say. It doesn't look like a definition to me.

    I don't suppose you think that 1:00 comeing thirteen hours after 12:00 makes much sense, then?
     
  19. Sep 2, 2008 #18
    since the successor function is the basis of math then maybe we shouldnt think in terms of one, two, and three but rather first, second, and third. 2+3 becomes the second after the third. 2*3 becomes the second third. its just semantics but it might make the underlying fundamental idea clearer.

    what does three mean anyway? thirdness?
     
  20. Sep 2, 2008 #19
    division is defined as the inverse of multiplication. multiplication is defined in terms of (multiple) addition(s). addition is defined in terms of (multiple) successor function(s).
     
  21. Sep 2, 2008 #20
    Well this is just a SYSTEM used to measure time..isnt it...?i said it wont make sense to everything like in the case of day to day life objects...but of course we could define anything we like..
    and what i was trying to say is ..that is it that we consider ever number as an object or a number of objects...in our present system of numbers...how else do you prove say [tex] \frac{1}{a} -\frac{1}{b}= \frac{b-a}{ab} [/tex]

    Thanks
     
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