Acid growth protons (H ions) to be pumped out of the cell into the cell wall. As a result, the cell wall solution becomes more acidic. This activates expansin activity, causing the wall to become more extensible and to undergo wall stress relaxation, which enables the cell to take up water and to expand. Acid growth Acid growth refers to the ability of plant cells and plant cell walls to elongate or expand quickly at low (acidic) pH. This form of growth does not involve an increase in cell number. During acid growth, plant cells enlarge rapidly because the cell walls are
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