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Math Help - Multiplicative Identity Proof

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    Multiplicative Identity Proof

    Hey all, I can't seem to figure out the following proof:

    Let x be an element of the Integers. If x*x = x then x=0 or x=1.

    Any help would be appreciated!
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by jstarks44444 View Post
    Hey all, I can't seem to figure out the following proof:

    Let x be an element of the Integers. If x*x = x then x=0 or x=1.

    Any help would be appreciated!
    Well it follows from the axioms of the integers that

    \begin{aligned} x\cdot x = x &\implies x\cdot x - x = 0\quad\text{\phantom{x.}(additive inverse)}\\ &\implies x\cdot(x-1)=0\quad\text{(distributive law)}\end{aligned}

    Since the integers form an integral domain (not sure how much abstract algebra you know), there are no zero divisors. So either x=0 or x-1=0.

    Does this make sense?
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