"algebra deals with equality but analysis deals with inequality and error terms" - sorry, but that is not correct: error terms originate from limes-considerations, but you cannot say that analysis deals only with these terms or inequalities.
For example the one of the most fundamental equations of analysis is not an inequaltity: the definition of the differential quotient.
Also analysis does not deal only with individual objects. As a counter-example: you can have a set consisting of only one individual object, but it is still algebra or a series with an infinite number of terms and it is still analysis...
Furthermore your definition is unfortunately not very rigid in a mathematical sense, you can look up better definitions in wikipedia (just as an example):
http://en.wikipedia.org/wiki/Mathematical_analysis
http://en.wikipedia.org/wiki/Algebra
Algebra deals with sets of elements and their connections while analysis deals with limits (mostly "0" or "infinity"), but both are equal in a mathematical sense.
Objective of algebra is to solve the problem and analysis is the process to obtained broader understanding.
@Rahul: what is that supposed to mean?
The solution of a problem is mainly dependent on the problem itself - some are easier to solve with algebra, some are easier to solve with analysis.
Take quantum mechanics as a practical application of maths: you can either solve QM problems in the Heisenberg-picture (so with algebra) or in the Schrödinger-picture (which is analysis).
The simplest approach is: Algebra = sets and their connections, Analysis = limits of series.
In algebra we take collection of objects as sets, group, field. in these we emphasize on structure of the collection of objects. in analysis primarily studies of individual object. algebra deals with equality but analysis deals with inequality and error terms. analysis deals with details while algebra takes a broader view. Both are important to have in mathematics, and any interesting problem would most likely contain a mix of the two. e.g
Homotopical Algebra with Applications to Mathematical Physics
Kac-Moody Algebras, Vertex Algebras, Quantum Groups, and Applications
"algebra deals with equality but analysis deals with inequality and error terms" - sorry, but that is not correct: error terms originate from limes-considerations, but you cannot say that analysis deals only with these terms or inequalities.
For example the one of the most fundamental equations of analysis is not an inequaltity: the definition of the differential quotient.
Also analysis does not deal only with individual objects. As a counter-example: you can have a set consisting of only one individual object, but it is still algebra or a series with an infinite number of terms and it is still analysis...
Furthermore your definition is unfortunately not very rigid in a mathematical sense, you can look up better definitions in wikipedia (just as an example):
http://en.wikipedia.org/wiki/Mathematical_analysis
http://en.wikipedia.org/wiki/Algebra
Analysis is all about continuous processes, which, as a practical matter, means everything is based on the concept of limit.
Algebra is the study of algebraic structures. An algebraic structure is a non-empty set together with some binary operations defined on it and a set of axioms. Analysis is the study of both algebraic and geometric structures using the process of limits.
Well, large parts of modern algebra deal with limits as well, so I wouldn't see a neat opposition between algebra and analysis. Much work has been done to investigate the algebraic structures that occur in analysis, and much work has been done to make analytical tools available in algebra (algebraic geometry). So, even if I do not see an opposition, I still use the words 'algebraic structure' (groups, rings, algebras) and 'analytical tools' (derivative, limit, etc.); both are needed in most of mathematics.
Just an example: Continuity is an analytical concept. The set S of continuous real-valued functions on a topological space X is an R-algebra. If the space in question X is compact and A is a sub-algebra o S that satisfies the conditions of the Stone-Weierstrass theorem, this algebra is dense in S with respect to uniform convergence. The maximal ideals of S corresponds to the points of X... etc. This is analysis with a strong flavour of algebra. But, if however the space X is an algebraic variety, you may as well say, that we deal with algebra in an analytical language.
Basically, we may accept algebra as discrete mathematics, but analysis is not discrete mathematics anyway.
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