The alpha carbon in organic chemistry refers to the first carbon that attaches to a functional group (the carbon is attached at the first, or alpha, position). By extension, the second carbon is the beta carbon, and so on.
This nomenclature can also be applied to the hydrogen atoms attached to the carbons. A hydrogen attached to an alpha carbon is called an alpha-hydrogen (α-hydrogen), a hydrogen on the beta-carbon is a beta-hydrogen, and so on.
This naming standard is sometimes considered to be not in compliance with IUPAC nomenclature (which encourages that carbons be identified by number, not by Greek letter), but it nonetheless remains very popular, in particular because it is useful in identifying the relative location of carbons to other functional groups (a carbonyl in the example on the right).
Organic molecules with more than one functional group can be a source of confusion. Generally the functional group responsible for the name or type of the molecule is the "reference" group for purposes of carbon naming. For example, the molecules nitrostyrene and phenethylamine are very similar; the former can even be reduced into the latter. However, nitrostyrene's α-carbon is adjacent to the styrene group; in phenethylamine this same carbon is the β-carbon, as phenethylamine (being an amine rather than a styrene) counts its atoms from the opposite "end" of the molecule.
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“Imagination is a valuable asset in business and she has a sister, Understanding, who also serves. Together they make a splendid team and business problems dissolve and the impossible is accomplished by their ministrations.... Imagination concerning the worlds wants and the individuals needs should be the Alpha and Omega of self-education.”
—Alice Foote MacDougall (18671945)