Inverse Element

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For the set of integers $Z$ , an arithmetic operation $\ast$ is defined as $a \ast b = a+b+2ab$ . How many elements of $Z$ have an inverse element for $\ast$ ? (The identity element is a value $e$ such that for all values $n$ , $e \ast n = n$ and $n \ast e = n .$ The inverse element of $n$ is a value $m$ such that $n \ast m = e$ and $m \ast n = e .$ )

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1 solution

Patrick Corn
Mar 11, 2014

Clearly $0$ is the identity element. Now, if $a + b + 2ab = 0$ , then $4ab + 2a + 2b = 0$ , so $4ab + 2a + 2b + 1 = 1$ , so $(2a+1)(2b+1) = 1$ , so $2a + 1 = \pm 1$ . This leads to the two solutions $a = 0$ and $a = -1$ ; each is its own inverse.

Inverse Element

1 solution

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