Convention for numbering of carbons I'm trying to understand how carbons are numbered in organic molecules (i.e. as in the following image).



(source: Principles of Biochemistry by Nelson and Cox, p. 532)

I've read and heard different methods, among which:


the end of the carbon chain with the highest oxidation state is assigned the lowest number.
the end of the carbon chain with the highest total mass is assigned the lowest number. the number of bonds is used as a kind of multiplier (i.e. O=C-OH corresponds to a total mass of O*2+C+O+H). (source: What is the convention of numbering carbon atoms in organic molecules?)
the end of the carbon chain with the substituent of the highest atomic number is assigned the lowest number (source: Principles of Biochemistry by Nelson and Cox, p. 73).
the carbonyl, acetal or hemiacetal carbon is given the lowest possible number (source: Carbon numbering in carbohydrates)


These methods could come to contradictory conclusions, though. One example I could think of:


Phosphorous would be the highest atomic number constituent, while the remaining rules (carbonyl carbon, total mass and oxidation state) would assign the carbonyl carbon the lowest number.

I assume some are just simplifications. What is the correct method, or is there no single general one?
Your initial example showing derivatives of glucose and fructose might be misleading since the nomenclature of natural product parent compounds does not necessarily follow the usual nomenclature rules of general organic chemistry.

In particular, 


  P-102.2.2 Numbering parent structures
  
  The carbon atoms of a monosaccharide are numbered consecutively in such a
  way that:
  
  (1) a (potential) aldehyde group receives the locant 1 (even if a more senior
  characteristic group is present);
  
  (…)


(This rule is probably easier to understand if you draw the monosaccharides in their acyclic form.)

Generally, however, nomenclature of organic chemistry has different rules for the numbering. The most important simplified criteria for the numbering are:


lower locants for the group that is expressed as suffix
lower locants for multiple bonds
lower locants for prefixes
lower locants for substituents cited first as a prefix in the name


The corresponding actual wording in the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book) reads as follows:


  P-14.4 NUMBERING
  
  When several structural features appear in cyclic and acyclic compounds, low locants are assigned to them in the following decreasing order of seniority:
  
  (…)
  
  (c) principal characteristic groups and free valences (suffixes);
  
  (…)
  
  (e) saturation/unsaturation:
  
  (i) low locants are given to hydro/dehydro prefixes (…) and ‘ene’ and ‘yne’ endings;
  
  (ii) low locants are given first to multiple bonds as a set and then to double bonds (…);
  
  (f) detachable alphabetized prefixes, all considered together in a series of increasing numerical order;
  
  (g) lowest locants for the substituent cited first as a prefix in the name;
  
  (…)


Therefore, the numbering in your own example corresponds to the name 3-phosphinopropanoic acid since the acid group is the principal characteristic group of this compound.