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  "body": "\n## Summary\n\nThe Grothendieck group of a commutative [[monoid|0N6R902FG1]] is the [[abelian group|082GHNPJ53]] obtained by adding formal differences, in the universal way.\nIt turns the natural numbers into the integers.\nApplied to isomorphism classes of finitely generated [[projective modules|BFFA03GE3V]] under direct sum, it gives the group $K_0$ of a ring, the start of [[algebraic K-theory|5GVFZWMWC8]].\nThe same idea applies to vector bundles and to abelian categories.\n\n## Explanations\n\n### L1 human\n\nSome structures can be added but not subtracted, such as natural numbers, or vector spaces under direct sum.\nThe Grothendieck group adds negatives in the most economical way.\nIts elements are formal differences $a - b$, with $a - b = c - d$ when $a + d$ and $c + b$ become equal after adding a common term.\nFrom the natural numbers one gets the integers.\nFrom finite-dimensional vector spaces over a field one also gets the integers, through the dimension.\n",
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