Mathematics is an activity of investigation and exploration.
Informally, both calculi and an algebras are tools which consist
of sets of symbols and systems of rules (usually called axioms)
for manipulating those symbols. Calculi tend to be
specified/defined/explored/used to answer questions of
"calculation" or reckoning, in some very general sense. Calculi
tend to be used to investigate properties of objects (i.e "What
is the area under the curve?") Algebras tend to be
specified/defined/explored/used to answer questions about how
different "things" are related, in some very general sense.
Algebras tend to be used to study the relationship between
objects. (i.e. "Is this equation 'the same' as that equation?")
I think it is safe to say that the term "algebra" today, carries
a bit more meaning to most mathematicians than the general teram
"calculus". The Calculus (as taught in high-school or
undergraduate university), also known as "infinitesimal
calculus", is a calculus focused on limits, functions, derivativ