Water pollution Oxygen-depleting substances may be natural materials such as plant matter (e.g. leaves and grass) as well as man-made chemicals. Other natural and anthropogenic substances may cause turbidity (cloudiness) which blocks light and disrupts plant growth, and clogs the gills of some fish species. Alteration of water's physical chemistry includes acidity (change in pH), electrical conductivity, temperature, and eutrophication. Eutrophication is an increase in the concentration of chemical nutrients in an ecosystem to an extent that increases the primary productivity of the ecosystem. Depending on the degree of eutrophication, subsequent negative environmental effects such as anoxia (oxygen depletion) and severe reductions
Proofs involving the addition of natural numbers 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 Peano axioms for the definitions of addition of the natural numbers, and the successor function S(a). In particular: For the proof of commutativity, it is useful to define another natural number closely related to the successor function, namely "1". We define 1 to be the successor of 0, in other words, Note that for all natural numbers