# Is everything an expression?

Is everything that you can write in math (that makes mathematical sense) an expression? If not, what would be examples of non-expressions? And would all expressions be composed of expressions themselves?

Also, are operators (like the differential operator) by themselves expressions?

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What do you mean by "expression"? – Harry Stern Jan 16 '11 at 19:03
@Harry whatever mathematicians like to use it for. – wrongusername Jan 16 '11 at 19:06
@wrongusername: the term does not, to my knowledge, have a fixed meaning in mathematics. – Qiaochu Yuan Jan 16 '11 at 19:20
Look up en.wikipedia.org/wiki/Well-formed_formula for an example of defining what an "expression" is in one part of mathematics. – Yuval Filmus Jan 16 '11 at 20:04
– Simon Jan 17 '11 at 0:13

According to Merriam-Webster's Collegiate Dictionary, in mathematics, an expression is "a mathematical or logical symbol or a meaningful combination of symbols". Thus $a$, $+$, $a+b$, and $a+b=c$ are expressions, while $+\!=$ and $==$ are not. Whether an arbitrary fragment of a meaningful expression, such as $=c$, is necessarily still an expression is dubious. I am inclined to think not, but would agree with the statement that all expressions are themselves composed of expressions, down to the (atomic) level of single symbols.