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This has confused me for so long, after reading the IUPAC rules over and over, I found the parent chain to be from the the CH2 numbered 1 to the one numbered 9, and I considered both CH3's as branches, however my confusion lies in naming the branch in the middle (with the circle around it) since it contains a double bond.

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According to the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book), the traditional name ‘vinyl’ for $\ce{CH2=CH-{}}$ is no longer used for preferred IUPAC names and is only retained for general nomenclature. The preferred name is the systematic name ‘ethenyl’.

Therefore, the name for the compound that is given in the question ‘3,7-dimethyl-5-vinylnona-1,8-diene’ could be found in the literature; however, the preferred IUPAC name in accordance with current recommendations is 5-ethenyl-3,7-dimethylnona-1,8-diene.

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