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Morphemes are the units of a language that cannot be broken down into further meanings: to oversimplify, they are words. Morphology is a bit of a complicated topic, as many linguists will argue what does and does not constitute a morpheme. There is a better explanation here.

Many words, to continue our simplification, however, can be broken down into more words or bits of words. A common example is "unsystematically." You can break it down into un / system / atic / al / ly. Each little segment is a full fledged, bonified morpheme, as you shall soon see. In order from left to right, they are, respectively, an adjective prefix, the noun root and an adjective suffix (together forming the stem), another adjective suffix, and, finally, an adverb suffix. However, alone, only the morpheme "system" has any meaning; the rest exist only in the context of other morphemes: they cannot occur "unattached," unlike free morphemes like "system." These are called bound morphemes.

Let us try more examples. Listed are several words broken down into free morphemes and bound morphemes:

boy + ish = boyish
desire + able = desirable
boy + ish + ness = boyishness
desire + able + ity = desirability
The example of desirable and its offshoots is an excellent example of a small problem in morphology and the classification of morphemes: able is a free morpheme AND a bound morpheme. Usually the distinction is clear. (No one goes around just saying "ish" or "ness." Well, no one we take seriously.)

Bound morphemes are further subdivided into derivational morphemes and inflectional morphemes. Those are topics better suited to their own nodes, but, simply, the former changes the classification or meaning of a word, such as changing a noun ("system") to an adjective ("systematic"), and the latter changes the grammatical meaning of a morpheme, such as making a verb past tense ("walk" opposed to "walked").

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