Do American English speakers use note to mean bill (as in ten-dollar bill)? If so, is note a shortening of the word banknote?
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While AmE speakers would understand note if used in context, bill is substantially more common among the general population. Asking someone if they have a twenty-dollar note just sounds odd to my AmE ears.
I believe usage of the term note is more common in professions that deal specifically with currency (e.g. banking, treasury, etc.), and I'll, ahem, note that US currency actually is labeled as a Federal Reserve Note and includes the phrase This note is legal tender. However, in some financial circles, you run the risk of being confused with other types of notes, e.g. a Treasury note (T-Note).
And yes, note is short for banknote.
Ngrams graphs the usages of ten-dollar bill and ten-dollar note:
Bill is much more common than note; you will not hear normal American conversation include the term note in referring to a ten-dollar bill. When ten-dollar note is used in American English, it is usually a numismatic term, a formal term, or else foreign currency.
Note is banknote abbreviated:
Note is rarely—if ever—used as an abbreviation for banknote. The colloquial use of note in American English refers to debt instruments such as promissory notes, deeds of trust, etc. For example, savvy property companies will often try to "purchase the note" on a property going into foreclosure; this allows them to foreclose on the property themselves, avoiding some of the risk of bidding at the auction: even if someone else wins it, instant profit.