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According to Hückel's rule a compound is aromatic if it has conjugation throughout the compound and the number of conjugated electrons is $4n+2$. This compound does not satisfy both:

1,2-dihydronaphtalene

The benzene ring itself is aromatic, though. Is the compound considered aromatic because of this?

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    $\begingroup$ Hückel's rules cannot be applied to this compound, it is not a monocycle. $\endgroup$ – Martin - マーチン Jun 5 '17 at 4:28
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Hückel's rules are a very concise set of very strict rules. Most aromatic compounds do not comply with these.

Hückel (4n + 2) rule
Monocyclic planar (or almost planar) systems of trigonally (or sometimes digonally) hybridized atoms that contain (4n + 2) π-electrons (where n is a non-negative integer) will exhibit aromatic character. The rule is generally limited to n = 0 – 5. This rule is derived from the Hückel MO calculation on planar monocyclic conjugated hydrocarbons (CH)m where m is an integer equal to or greater than 3 according to which (4n + 2) π-electrons are contained in a closed-shell system. Examples of systems that obey the Hückel rule include:

  • cyclopropenyl cation (m = 3, n = 0)
  • cyclopentadienyl anion (m = 5, n = 1)
  • benzene (m = 6, n = 1)

Systems containing 4n π-electrons (such as cyclobutadiene and the cyclopentadienyl cation) are 'antiaromatic'.
From IUPAC Gold Book, DOI: 10.1351/goldbook.H02867

Based on this it is actually quite difficult to assess aromaticity to more complex molecules. Aromatic molecules that do not comply with Hückel's rules are for example naphthalene, thiophene, or pyridine, among many more. A more suitable definition, which also makes quite clear how fuzzy the whole subject is:

aromatic

  1. In the traditional sense, 'having a chemistry typified by benzene'.
  2. A cyclically conjugated molecular entity with a stability (due to delocalization) significantly greater than that of a hypothetical localized structure (e.g. Kekulé structure) is said to possess aromatic character. If the structure is of higher energy (less stable) than such a hypothetical classical structure, the molecular entity is 'antiaromatic'. The most widely used method for determining aromaticity is the observation of diatropicity in the 1HNMR spectrum.
    See also: Hückel (4n + 2) rule, Möbius aromaticity
  3. The terms aromatic and antiaromatic have been extended to describe the stabilization or destabilization of transition states of pericyclic reactions The hypothetical reference structure is here less clearly defined, and use of the term is based on application of the Hückel (4n + 2) rule and on consideration of the topology of orbital overlap in the transition state. Reactions of molecules in the ground state involving antiaromatic transition states proceed, if at all, much less easily than those involving aromatic transition states.

From IUPAC Gold Book, DOI: 10.1351/goldbook.A00441

While we tend to say a molecule is aromatic, that is hardly ever the case. Think about substituted benzene molecules; while the phenyl ring retains aromatic character, the side chains are not necessarily aromatic. When describing the molecule it is better to refrain from absolutes.

Within this framework it is possible to identify molecular entities that posses aromatic character and refer to these. Strictly speaking the answer that 1,2-dihydronaphthalene is aromatic is therefore incorrect. Better is to say that because of the phenyl subunit, 1,2-dihydronaphthalene possesses some aromatic character. In this particular case you could even argue that because of the additional double bond, the main reactivity of this molecule is more similar to alkenes than it is to aromatic compounds.
Aromatic character is sometimes like a unicorn: it's hard to describe in unambiguous terms, but when you see it, you'd probably know what it is.

There is another definition of relevance in the IUPAC Gold Book, which actually gives a clearer, more systematic aproach for determining aromatic properties, and can be seen as an extension of the above:

aromaticity
The concept of spatial and electronic structure of cyclic molecular systems displaying the effects of cyclic electron delocalization which provide for their enhanced thermodynamic stability (relative to acyclic structural analogues) and tendency to retain the structural type in the course of chemical transformations. A quantitative assessment of the degree of aromaticity is given by the value of the resonance energy. It may also be evaluated by the energies of relevant isodesmic and homodesmotic reactions. Along with energetic criteria of aromaticity, important and complementary are also a structural criterion (the lesser the alternation of bond lengths in the rings, the greater is the aromaticity of the molecule) and a magnetic criterion (existence of the diamagnetic ring current induced in a conjugated cyclic molecule by an external magnetic field and manifested by an exaltation and anisotropy of magnetic susceptibility). Although originally introduced for characterization of peculiar properties of cyclic conjugated hydrocarbons and their ions, the concept of aromaticity has been extended to their homoderivatives (see homoaromaticity), conjugated heterocyclic compounds (heteroaromaticity), saturated cyclic compounds (σ-aromaticity) as well as to three-dimensional organic and organometallic compounds (three-dimensional aromaticity). A common feature of the electronic structure inherent in all aromatic molecules is the close nature of their valence electron shells, i.e., double electron occupation of all bonding MOs with all antibonding and delocalized nonbonding MOs unfilled. The notion of aromaticity is applied also to transition states.
From IUPAC Gold Book, DOI: 10.1351/goldbook.A00442

One of the key elements is the thermodynamic stability and the tendency of retaining the structural type during the course of a reaction. This certainly reflects the benzene-like aspect of the earlier definition.
The last part comes directly from the original (4n + 2) criterion, but significantly weakens it, as it is not reduced to the very symmetry restricted number Hückel derived from the hydrocarbon cycles.
For simple compounds like thiophene it is easy to construct an approximate MO diagram (if you for further simplification ignore the σ-sub-structure). And the same can be done more complex molecules like 1,2-dihydronaphthalene.
In general I am not a big fan of oversimplifying complicated situations; questions that are asking for absolute statements are always oversimplified. In case of aromaticity it is tempting to reduce it to a simple electron-count, which therefore still prevails in textbooks although horribly incomplete; but it is also dangerous, because it neglects a lot of the underlying chemistry.

Aromaticity is a beast; fuzzy definitions, or criteria, the ongoing debate among scientists only show that you need to be open minded and accept that many things are not as easy as it seems at first glance.
For the time being you can help yourself by identifying molecular entities, or moieties that are derived from (simpler) aromatic compounds and check whether a conjugated π-system was destroyed while creating them. As one of the key features you can be almost certain, that aromaticity is retained in all but the most extreme cases.

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  • $\begingroup$ By your logic, would benzaldehyde be non-aromatic because it shows reactions of aldehydes? The non-benzenoid ring can be broken into two and they can be treated as two substituents, right? $\endgroup$ – Yashas Jun 8 '17 at 13:07
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    $\begingroup$ @Yashas Yes and no; benzaldehyde is still aromatic, at least the phenyl ring. It's also an aldehyde. It's just not that simple, and it would be wrong regarding the one without the other. Certain reactions can target one functionality, while others target the second. $\endgroup$ – Martin - マーチン Jun 8 '17 at 13:27

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