Editing Addition (section)

=== Ordering ===
[[File:XPlusOne.svg|right|thumb|[[Log-log plot]] of {{nowrap|1={{mvar|x}} + 1}} and {{nowrap|1=max ({{mvar|x}}, 1)}} from {{mvar|x}} = 0.001 to 1000<ref>Compare {{harvtxt|Viro|2001}}, p. 2, Figure 1.</ref>]]
The maximum operation <math> \max(a,b) </math> is a binary operation similar to addition. In fact, if two nonnegative numbers <math> a </math> and <math> b </math> are of different [[orders of magnitude]], their sum is approximately equal to their maximum. This approximation is extremely useful in the applications of mathematics, for example, in truncating [[Taylor series]]. However, it presents a perpetual difficulty in [[numerical analysis]], essentially since "max" is not invertible. If <math> b </math> is much greater than <math> a </math>, then a straightforward calculation of <math> (a + b) - b </math> can accumulate an unacceptable [[round-off error]], perhaps even returning zero. See also ''[[Loss of significance]]''.

The approximation becomes exact in a kind of infinite limit; if either <math> a </math> or <math> b </math> is an infinite [[cardinal number]], their cardinal sum is exactly equal to the greater of the two.<ref>Enderton calls this statement the "Absorption Law of Cardinal Arithmetic"; it depends on the comparability of cardinals and therefore on the [[Axiom of Choice]].</ref> Accordingly, there is no subtraction operation for infinite cardinals.{{sfnp|Enderton|1977|p=[http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA164 164]}}

Maximization is commutative and associative, like addition. Furthermore, since addition preserves the ordering of real numbers, addition distributes over "max" in the same way that multiplication distributes over addition:
<math display="block"> a + \max(b,c) = \max(a+b,a+c).</math>
For these reasons, in [[tropical geometry]] one replaces multiplication with addition and addition with maximization. In this context, addition is called "tropical multiplication", maximization is called "tropical addition", and the tropical "additive identity" is [[extended real number line|negative infinity]].{{sfnp|Mikhalkin|2006|p=1}} Some authors prefer to replace addition with minimization; then the additive identity is positive infinity.{{sfnp|Akian|Bapat|Gaubert|2005|p=4}}

Tying these observations together, tropical addition is approximately related to regular addition through the [[logarithm]]:
<math display="block">\log(a+b) \approx \max(\log a, \log b),</math>
which becomes more accurate as the base of the logarithm increases.{{sfnp|Mikhalkin|2006|p=2}} The approximation can be made exact by extracting a constant <math> h </math>, named by analogy with the [[Planck constant]] from [[quantum mechanics]],{{sfnp|Litvinov|Maslov|Sobolevskii|1999|p=3}} and taking the "[[classical limit]]" as <math> h </math> tends to zero:
<math display="block">\max(a,b) = \lim_{h\to 0}h\log(e^{a/h}+e^{b/h}).</math>
In this sense, the maximum operation is a ''dequantized'' version of addition.{{sfnp|Viro|2001|p=4}}