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Is (nearly) all mathematics addition?

  1. May 16, 2013 #1
    So far after observing the math I've been learning for the past 10 years of my school career it seems to me like everything can be simplified into addition. Ill show by example;

    Addition: obviously
    Subtraction: Addition of negative numbers
    Multiplication: Repeated Addition
    Division: 1/Multiplication
    Exponents: Repeated Multiplication which in turn is repeated addition

    Nth roots: Any root can be found using the equation;
    (X_g+(a/(x_g)^n-1)/2=x_g2

    Where x_g is an initial guess
    n is the root you are looking for
    A is the number you are rooting
    and x_g2 is substituted back into the formula until x_gt (t being in place of some integer) equals u [where u =a/(x_g1)^n-1]
    So now
    Nthrt(a)=(X_gt+u)/2 when u=x_gt

    So now by my definitions above; Although tedious; any real root can be found by addition!

    Now since e^x=
    lim n-->ifiniti (1+x/n)^n
    so
    ln(x)=lim n-->ifiniti n(x^(1/n)+1)
    So using above definitions and the change of base rule, the log of any real base to any real number can be found by addition as well!

    The trigonometric functions can be expanded into a taylor series and found by adding as well.

    Is there a function that cannot be broken down into addition?! Thanks!
     
  2. jcsd
  3. May 16, 2013 #2

    phinds

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    Gold Member

    You are basically talking about arithmetic, which yes, is pretty much as you describe. This is a small part of math.
     
  4. May 16, 2013 #3
    Logarithmic Functions aren't arithmetic
     
  5. May 16, 2013 #4
    Using first principles to find derivatives can be broken down into addition and algebra!
     
  6. May 16, 2013 #5

    WannabeNewton

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    Science Advisor

    Break down the definition of compactness of topological spaces down to something that is purely addition.
     
  7. May 16, 2013 #6
    Although i see your point and i knew there were holes in my "theory" before i posted, try to be a little less vague next time you try to disprove someone. I dont know if you mean sequentially compact, if you mean a Metric space. And besides if its say a function Space you could most likely simplify the functions, see if Function set a is a subset of The function space etc. so on a whole youre right i cant simplify an entire area of mathematics to addition, but if i were you i would have just pointed out integration or multiplication of negative numbers.
    Thanks for the reply though! :)
     
  8. May 16, 2013 #7

    Mark44

    Staff: Mentor

    How would you evaluate ## e^{i \pi}## using only addition?
     
  9. May 16, 2013 #8
    I agree with the posters against this hypothesis. Multiplication into addition is only valid for integers. How then, would you turn (2.5)*(5) into addition´╝č
     
  10. May 16, 2013 #9

    Mark44

    Staff: Mentor

    This one's actually pretty easy.

    Write it as 25 * 5, keeping in mind that you moved the decimal point one place to the right.
    Code (Text):

     25
     25
     25
     25
    +25
    ---
    125
    Now move the decimal point one place to the left to get 12.5.
     
  11. May 16, 2013 #10
    How would you justify all of your steps (namely all the extra factors of 10) with addition?
     
  12. May 16, 2013 #11
    2.5+2.5+2.5+2.5+2.5
     
  13. May 16, 2013 #12
    e^ipi=cos(pi)+isin(pi)
    Cosine and sine can be evaluated using a taylor series expantion which can be simplified to addition (but would be very tedius)
    sin(pi)=0 so i*0 which means you add i no times
     
  14. May 16, 2013 #13

    Mark44

    Staff: Mentor

    Not only tedious, but how do you take into account that there are an infinite number of terms, each of which involves calculating (-1)n##\frac{\pi^{2n}}{(2n!)}##?

    Let me know when you're done...
     
  15. May 16, 2013 #14

    WannabeNewton

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    Science Advisor

    It isn't a whole area of mathematics; it's just a single definition. If this were a metric space it wouldn't even matter because open cover compactness and sequential compactness are equivalent for metric spaces. If this were a general topological space, like I asked, then go ahead and work with either one. You're "hypothesis" will still fail. If your hypothesis fails for an extremely important concept in mathematics, then this thread is quite pointless.
     
  16. May 16, 2013 #15

    phinds

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    Gold Member

    I second the motion
     
  17. May 16, 2013 #16
    Youre right, nevermind i should just stop being curious about mathematics, or asking questions to people with greater mathematical knowledge than I.
     
  18. May 16, 2013 #17

    phinds

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    Gold Member

    How about these? Do you reckon they reduce to addition?

    geometry
    topology
    abstract algebra
    prepositional logic
    Fourier analysis
    Laplace transforms
    differential equations
    set theory
    game theory

    and those are just ones that come to mind and I don't know all that much about math even at the undergraduate level. I'm sure there are lots I'm leaving off that really have little or nothing to do with addition. Your hypothesis is seriously flawed.
     
  19. May 16, 2013 #18
    My post was literally asking to be proven wrong! I never wanted it to seem as though i was saying math was redundant. I think math is a beautiful thing, and besides i asked "Is nearly all math addition" I never declared "All Math can be broken down to addition"
     
  20. May 16, 2013 #19

    MathematicalPhysicist

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    Basically, yes you can break everything to addition besides also primitive notions in set theory such as membership predicate. The compactness is basically saying that a set that is covered by some union of open sets can be covered by finite number of them. If you know addition and 'being a member in a set' you basically understand the definition.

    The fact that these are basica notions don't say that the other notions aren't as important. Just like in physics you have space and time as the basic given information, you still can define velocity,momentum.
     
  21. May 16, 2013 #20
    What about the function

    f ( x ) = 0 when x is irrational
    1/n when x = m/n is rational

    ?
     
  22. May 17, 2013 #21
    It's really possibel to translate everything back to integers. You don't even need addition, you can define addition with the successor function.

    Your number x would be an infinite sequence of rational numbers. The sequence would be rational if it converged to a rational number.
    A rational number is a pair of integers (p,q) such that gcd(p,q) = 1.
    (you might need a triple of integers if you want negative numbers as well).
     
  23. May 17, 2013 #22
    Perhaps you mean the math on some very polite structures, where Fields play a good role. A really narrow domain! Most of mathematics lies behind...
     
  24. May 17, 2013 #23
    Maybe the question should be asked... what math (in principle) might not be possibly implemented using a computer?

    They seem to be fundamentally based on addition...

    Analog computers clearly are based on addition (of angular velocities).
    Digital computers are based on the half-adder (the addition of 2 bits).
    Two half-adders make a full adder (arithmetic sum of 3 input bits).
    The binary adder-subtractor adds (arithmetic sum of 2 binary numbers of any length) and subtracts (using 2's and 1's complements)
    Shift registers
    Logic gates
    Onward and upward, etc...
     
  25. May 17, 2013 #24

    jbriggs444

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    Science Advisor

    Computers could equally well be considered to be based on NAND or NOR gates. Or flip-flops. A fellow named Turing had a model as well. Nothing special about addition.

    Not all analog computers are based on rotating gears and mechanical integrators. Nor am I sure that a mechanical integrator running a knife-edge driven wheel at a distance of 1.5 cm from the center of a drive wheel running at 1.5 rotations per second counts as an "addition" of 1.5 times 1.5 yielding 2.25.

    [Edit: fixed the discrepancy between driven and drive wheels in an integrator]

    Mathematics can contemplate that which physical computers cannot calculate.
     
    Last edited: May 17, 2013
  26. May 17, 2013 #25
    That really gets to the point of the question, and why I stipulated "in principle", because if by mathematics you mean the contemplations of a mathematician's physical brain, then what prevents one from imagining a computer sufficiently sophisticated (in principle) simulating a mathematician's brain contemplating these things?
     
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