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Is zero even or odd?

Discussion in 'Electronic Design' started by Gactimus, Dec 20, 2004.

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  1. It does on my digital clock (as does my digital watch) :p
     
  2. Well I basically said that in the part you didn't quote :)
     
  3. Perhaps you could explain what (-x^)2 means then?
     
  4. Gideon

    Gideon Guest

    I remember this as a homework problem in 9th grade algebra class many years
    ago. More recently, my son encounted the same question in 7th grade algebra
    class as an extra credit homework question. He did well but failed to realized
    that one cannot just assume that zero must be even or odd but not both. Most
    other students make the same mistake. The basic proof provided by many of us
    in my algebra class is listed below:

    --------

    Homework question:
    Is zero and even number or an odd number?

    Math definitions:
    Even numbers are numbers that can be written in the form 2n,
    where n is an integer.
    Odd numbers are numbers that can be written in the form 2n + 1,
    where n is an integer.

    First question and answer:
    Is zero an even number?
    That is: Is there an integer n, such that 0 = 2n ?
    Yes. 0=2n => 0/2=n => 0=n (an obvious integer)
    Therefore, 0 is an even number because if can satisfy the definition.

    Second question and answer:
    Is zero an odd number?
    That is: Is there an integer n, such that 0 = 2n +1 ?
    No. 0=2n+1 => -1=2n => -1/2=n which shows that n can not
    be an integer in this case.
    Therefore, 0 is not an odd number because it fails to satisfy the
    definition.

    --------

    Any other math fact about even and odd numbers aren't needed to answer
    the simple question "Is zero even, odd or both?" Of course, other facts about
    even & odd numbers can be used to answer the question if those facts have
    been rigorously proven. For example: A number is even if and only if it is the
    sum of 2 even numbers.


    Note that we cannot automatically assume that the set of even numbers and the
    set of odd numbers are mutually exclusive just from the definitions above. Of
    course, the proof of this fact is rather trivial:
    x = 2n plus x=2m+1 =>
    2n=2m+1 =>
    2n/2 = 2m/2 +1/2 =>
    n=m+1/2, which is impossible since n & m must both be integers

    Based just upon the definitions, we also cannot make the assumption that every
    integer must satisfy at least one of the definitions above and must therefore
    be even or odd.

    I point these out because the even/odd situation is an early introduction to
    number theory and primative proofs for young students. And most students fall
    into the trap of making assumptions which are not supported by the definitions
    alone:
    1) All even or odd numbers must be integers
    2) All integers must be even or odd numbers
    3) A number cannot be both even and odd

    All three statements above are true but must be substantiated via proofs.


    The debate over whether zero is positive or negative is also solved rather
    quickly by reverting to the math definitions:
    Positive numbers are numbers that are greater than zero.
    Negative numbers are numbers that are less than zero.
     
  5. Zero is definitely even. Dividing zero by 2 leaves no "remainder" or
    "fraction".

    However, as a bit of a digression, there are "odd functions" and "even
    functions" - odd ones must have output zero when input is zero. Even
    functions are permitted to have output zero or nonzero when input is
    zero.
    Functions can be odd, even or neither. With an even function, F(x) =
    F(-x). With an odd function, F(-x) = -F(x).

    - Don Klipstein ()
     
  6. No it cant.
    O is specifically excluded from this result as dividing by zero causes
    contradictions.

    Division by zero is *defined* to be undefined. And my usual, "this is
    not debatable" applies to this one. Read it up in any text book.

    The limit:

    L = f(x)/g(x) x->xo, where f(xo)=g(xo)=0

    May or may not exist. If it dose, the limit may be any specific value
    depending on the way the limit is approached.

    In many case the *limit* represents physical reality. The notation 0/0
    is a limit, and as such, is meaningless in mathematics.

    Kevin Aylward

    http://www.anasoft.co.uk
    SuperSpice, a very affordable Mixed-Mode
    Windows Simulator with Schematic Capture,
    Waveform Display, FFT's and Filter Design.
     
  7. No it isnt.

    Kevin Aylward

    http://www.anasoft.co.uk
    SuperSpice, a very affordable Mixed-Mode
    Windows Simulator with Schematic Capture,
    Waveform Display, FFT's and Filter Design.
     
  8. Bondo

    Bondo Guest

    you can say that again! Hilarious!
     
  9. Bondo

    Bondo Guest

    Nice statement. However, this 'proves' once again to me that I
    was never any good at math because I'm nogood at thinking, just
    too lazy! What a shame.
    Bondo
     
  10. Ben Bradley

    Ben Bradley Guest

    In sci.math,
    comp.soft-sys.matlab,
    sci.physics,
    alt.math.undergrad,
    rec.puzzles,
    sci.astro,
    sci.electronics.design and
    comp.lang.perl.misc, on Mon, 20 Dec 2004 21:02:32 +0000, John Woodgate
    Furthermore, 0/0 can GIVE any value. What a versatile expression!
     
  11. Not only usenet: someone once postulated that the only three "values" to
    be used in providing computer resources should be 0, 1 and infinity
    (which he meant to mean "unlimited" (for all practical purposes) (*)).

    (*) My math teacher always said "A sphere's surface is unlimited but not
    infinite", just to highlight the difference between the two.
     
  12. I read in sci.electronics.design that Clifford Heath <>
    How did you manage to move the ^? (;-)
     
  13. Guest

    It is a valid number. And it is even.

    Mati Meron | "When you argue with a fool,
    | chances are he is doing just the same"
     
  14. Guest

    The two are not the same.

    The definition of the ratio a/b is

    a/b = r iff b*r = a

    for the case of n/0 there is no r such that r*0 = n (follows from the
    definition of zero. Therefore n/0 (for non zero n) *does not exist*.

    On the other hand, for 0/0, every r qualifies since for every r, r*0 =
    0 (the definition of zero, again). Therefore, 0/0 is truly undefined,
    in the sense that it is impossible to *uniquely* assign a value to the
    ratio r.

    Mati Meron | "When you argue with a fool,
    | chances are he is doing just the same"
     
  15. So { SET OF ALL INTEGERS } = 0/0 = (0+0)/0 = (2*0)/0 = 2*(0/0)
    = 2* {SET OF ALL INTEGERS } = {SET OF ALL EVEN INTEGERS}?

    Odd.
     
  16. Hogwash. The notation 0/0 is most certainly not a limit, like 4/2 is
    not a limit. And how could you define a limit if there were no
    function values to start with?

    0/0 is clearly, if anything, a constant expression. And it turns out
    that its value is undefined. And limits have nothing to do with that.

    There are "limits of the form 0/0", but this is a shorthand for
    something completely different, and such limits in general _have_ a
    value (depending on just what is taken to the limit here).
     
  17. Fred Bloggs

    Fred Bloggs Guest

    You apparently have stumbled on something else you know damn little
    about. In case you need help with this , you might note that "/" is NOT
    an operator on the integers, it is the "inverse" of a multiplication
    operator. Inverse is a well-defined concept but not necessarily a
    function, it is a set theoretic mapping. E.G. m/n={ q: m=q*n} by
    definition, so that m/n which is actually a set which can be empty, a
    singleton, or infinite. In the case of m/n, it is then m/n = F^-1(m)
    where F(x)= n*x. Your reasoning would lead one to believe /: I x I -> I
    is a function, which it isn't.
     
  18. Peter Wyzl

    Peter Wyzl Guest

    :I know 0 is neither negative or positive but what about odd/even? I think
    : it's even.
    :
    : Odd numbers start at 1 and go every other number 1,3,5,7;1,-1,-3,-5,-7
    : Even starts at 2 and go every other number 2,4,6,8;2,0,-2,-4,-6,-8

    I think it's odd that you even need to ask...
     
  19. Fred Bloggs

    Fred Bloggs Guest

    Wrong- where do you get off saying (2*0)/0= 2*(0/0) ?
     
  20. I should clarify this. This is referring to the notion that different
    f(x) and g(x) will lead to different limits. Usually for the limit to
    have meaning, it must be the same independent of the way a specific f(x)
    and g(x) approaches the limit.
    This was a typo, for which I apologise. It should have been abundantly
    clear from the context that I was saying "is not a limit". Unfortunately
    when I spell checked I inadvertently deleted a word.

    The above wouldn't make logical sense at all otherwise, as I already
    defined "L" a limit, and distinguished it from 0/0. How can a limit be
    physically meaningfull, yet meaningless?
    Which is what I said i.e. "0/0 is meaningless"

    Kevin Aylward

    http://www.anasoft.co.uk
    SuperSpice, a very affordable Mixed-Mode
    Windows Simulator with Schematic Capture,
    Waveform Display, FFT's and Filter Design.
     
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