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Consider the following organic compounds:

(1) 1-methylcyclohexa-1,4-diene; (2) 1-methylcyclohexa-1,3-diene; (3) 1,4-dimethylcyclohexa-1,4-diene

Compound $(1)$ has two double bonds, which I'll label $C1$ and $C4$ such that the methyl group is attached to $C1$ and thereby assign lowest numbers to all functional groups and substituents. Officially its name is 1-methylcyclohexa-1,4-diene.

Generally we try to give just enough information to be able to reconstruct the original compound, but not too much. With that in mind, I feel that compound $(1)$ should be named simply 1-methylcyclohexa-4-diene:

  • Since it ends in -diene, it's understood that there are two double bonds.
  • Since it ends in -4-diene, it's understood that one of them begins at $C4$.
    • If the second double bond were located after $C4$, the numbering is backwards; instead of, say, -cyclohexa-4,6-diene, it should've been named -cyclohexa-1,3-diene.
    • If the second double bond were located at $C2$ or $C3$, then priority was given to the methyl substituent over the alkene functional groups. Rather than calling it 1-methylcyclohexa-2,4-diene, it should've been named 5-methylcyclohexa-1,3-diene.
    • Therefore, the only option left is for the second double bond to be at $C1$, and therefore specifying 1-methylcyclohexa-1,4-diene is extraneous.

Following a similar line of thought, compound $(2)$ ought to be named just 1-methylcyclohexa-3-diene, rather than its proper name of 1-methylcyclohexa-1,3-diene.

Taking this logic to an extreme, the third compound should be named 1,4-dimethylcyclohexadiene, instead of its proper name of 1,4-dimethylcyclohexa-1,4-diene, because the same logic used above to determine that the second double bond is located at $C1$ should be sufficient to determine the locations of both double bonds in this case, where they both have substituents!

Why in all of these cases, and others like them, do we specify both locants of the alkenes, when one (for compounds $(1)$ and $(2)$) or none (for compound $(3)$) is sufficient?

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The question is based on a false premise:

Generally we try to give just enough information to be able to reconstruct the original compound, but not too much.

This isn’t true. The prime aim of IUPAC nomenclature is clarity and the construction of unambiguous names.

Three logicians walk into a bar
The bartender asks: “Do you all want a drink?”
The first logician says: “I don’t know.”
The second logician says: “I don’t know.”
The third logician says: “Yes.”

IUPAC nomenclature doesn’t work like that. Generally, the complete information about a structure is explicitly given by the name and does not rely on any implied information or hidden logic involving other structures.

The fact that “1-methylcyclohexa-2,4-diene” would actually be called 5-methylcyclohexa-1,3-diene does not change the preferred name for 1-methylcyclohexa-1,4-diene.

Nevertheless, there are a few exceptional cases where locants are omitted, but only when there is no ambiguity. For preferred IUPAC names, the nomenclature rules explicitly stipulate when locants are omitted. In most cases, if any locants are essential for defining the structure, then all locants must be cited for the structure.

Also note that your third example is clearly wrong even when following your unorthodox logic. Your name “1,4-dimethylcyclohexadiene” could mean 1,4-dimethylcyclohexa-1,4-diene as well as 1,4-dimethylcyclohexa-1,3-diene or even 1,4-dimethylcyclohexa-1,2-diene, which are all correct IUPAC names.

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I second @Loong's statement about IUPAC, the system based the nomenclature on rule that avoid any two person from coming to a different structure derived from the same name. Generally the functional group is your starting point based on priority and goes sequentially with priorities until you complete the name. Exceptions are still there for some basic compounds. In your example, all nomenclature based on alkene group plus methyl group, this is your number 1, the direction of numbering depends on the direction (location) of the first alkene.

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