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Math Help - Existence of a Multiplicative Identity

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    Existence of a Multiplicative Identity

    According to Michael Spivak's book on calculus, in his initial discussion of the basic properties of numbers, he talks about 1 not being equal to 0 in regards to the existence of a multiplicative identity property. He does attempt a brief explanation as to why this needs to be stated, however, his meaning is still very unclear to me. Can someone please clarify this for me?

    He is a link to the book in google books. It is P6 on pp. 5-6.

    Calculus - Google Books
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    Quote Originally Posted by ldv1452 View Post
    According to Michael Spivak's book on calculus, in his initial discussion of the basic properties of numbers, he talks about 1 not being equal to 0 in regards to the existence of a multiplicative identity property. He does attempt a brief explanation as to why this needs to be stated, however, his meaning is still very unclear to me. Can someone please clarify this for me?

    He is a link to the book in google books. It is P6 on pp. 5-6.

    Calculus - Google Books
    If F is a field with multiplicative identity ( 1_F) equal to the additive identity ( 0_F) you have a pretty interesting field. Want to know why?

    Let x\in F then x=x\cdot 1_F=x\cdot 0_F=x\cdot \left(0_F+0_F\right) =x\cdot 0_F+x\cdot 0_F=x\cdot 1_F+x\cdot 1_F=x+x\implies x=0_F=1_F
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    Thank you for your quick response. I am not very familiar with fields so this may simply be beyond my abilities at this time, unless there is a more intuitive way to understand this idea. Basically though, is "one is not equal to zero" an axiom that must be stated in order for a working proof of the multiplicative identity property?
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    Quote Originally Posted by ldv1452 View Post
    Thank you for your quick response. I am not very familiar with fields so this may simply be beyond my abilities at this time, unless there is a more intuitive way to understand this idea. Basically though, is "one is not equal to zero" an axiom that must be stated in order for a working proof of the multiplicative identity property?
    The idea is that if the multiplicative identity is equal to the additive identity your space can only have one element.
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    Why does this need to be stated with the property? Why not state "1 is not equal to 2" as well?
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    Quote Originally Posted by ldv1452 View Post
    Why does this need to be stated with the property? Why not state "1 is not equal to 2" as well?
    Because "2" is meaningless. It is kind of confusing to think of them as 0,1 which is why I wrote them with subscripts. A field need not have "numbers" in them in the traditional sense. Really, all " 0" means is that anything "added" to it doesn't change. I could have equally called it e, \coprod, or \overset{\text{. .}}{\smile}.
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    I'm definitely going to have to ponder this a bit more, but I'm starting to get the line of thinking. This is very interesting. So would it be correct to think of "x" as an element that has both additive and multiplicative identity properties? And that the only way such an element could exist is if were the only element in such a field? And therefore could not exist in a field with multiple elements, such as the field of real numbers?
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    Quote Originally Posted by ldv1452 View Post
    I'm definitely going to have to ponder this a bit more, but I'm starting to get the line of thinking. This is very interesting. So would it be correct to think of "x" as an element that has both additive and multiplicative identity properties? And that the only way such an element could exist is if were the only element in such a field? And therefore could not exist in a field with multiple elements, such as the field of real numbers?
    Yes. If one has a field that has a multiplicative and additive identity that conincide then it can only have one element. So, yes in particular it couldn't be the real numbers.
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    The concept, as well as Spivak's comments, make a lot more sense now. I'm going to have to educate myself in some abstract algebra basics. Thanks for all the help.
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