Polyculture chemicals into the environment such as eutrophication of fresh water or the presence of excess atmospheric nitrogen are greatly reduced in polycultures. Other negative impacts of modern agriculture are similarly reduced. Excessive tillage occurs in most modern agricultural practices, but removes essential microbes and nutrients from the soil that are conserved in polycultures, especially permacultures. Agricultural machinery not only uses fossil fuels and creates waste, but is more costly to farmers. Because polyculture relies on natural systems of crop maintenance these costs can be eliminated as well. Modern agriculture is also dependent on a very limited number of plant species.
Proofs involving the addition of natural numbers induction on "a", and so we have proved the base case "b" = 1. Now, suppose that for all natural numbers "a", we have "a" + "b" = "b" + "a". We must show that for all natural numbers "a", we have "a" + "S"("b") = "S"("b") + "a". We have This completes the induction on "b". Proofs involving the addition of natural numbers This article contains mathematical proofs for some properties of addition of the natural numbers: the additive identity, commutativity, and associativity. These proofs are used in the article Addition of natural numbers. This article will use the