Cultural eutrophication activity. Polar regions have less human activity and subsequently less cultural eutrophication. One response to added amounts of nutrients in the aquatic ecosystem is the rapid growth of microscopic algae, also known as an algal bloom. In freshwater systems, the formation of floating algal blooms are commonly nitrogen-fixing cyanobacteria (blue-green algae). This outcome is favoured when nitrogen inputs are reduced and phosphorus inputs are increased. Large amounts of algae reduce the amount of dissolved oxygen available in the water for other organisms, which increases fish mortality rates. These areas affected by algal blooms are known as dead zones, and are
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