# Algebra and Analysis [closed]

I'm a math major at university and my tutor told me that for most people it's best to focus on either algebra or analysis, however I have trouble understanding the difference between them.

What are the actual differences between algebra and analysis, and how do I tell which is for me?

• "more about discrete and analysis more about continuous" is not really a true distinction. Studying either one would benefit you a lot. You don't have to decide between the two. Commented Aug 10, 2015 at 18:42
• It depends what your interests are and where you think you might want to go. If you've only done one year of university, there's no reason to already focus on one. Some people change their minds about their focus in graduate school.
– Moya
Commented Aug 10, 2015 at 18:43
• There's a lot of overlap: analysts frequently use some algebra and algebraists frequently use some analysis. Any mathematician needs to know the basics of both. Commented Aug 10, 2015 at 18:48
• You can do analysis using algebra - it's called smooth infinitesimal analysis - read A Primer of Infinitesimal Analysis by John L Bell.
– user117644
Commented Aug 10, 2015 at 20:27

Algebra is about combining things together with operations while analysis focuses more on studying the closeness or "connectedness" between points.

Some of your confusion might stem from the fact that algebra and analysis can often work together. If you take courses in abstract algebra and real analysis, you will study each topic on its own, mostly divorced from the other.

To expand on algebra a little...

High school algebra involves addition and multiplication of real numbers and the properties of these operations. For example, if $ab=0$, then you know either $a=0$, $b=0$, or both.

In later algebra courses you would learn about more interesting algebraic structures that have different rules. If you have studied linear algebra you would know that, unlike real number multiplication, matrix multiplication doesn't commute in general, $AB \neq BA$, and knowing $AB=0$ doesn't imply that $A$ or $B$ is the zero matrix. You could also study algebraic rules for complex numbers, polynomials, and many other structures. (Quaternions are another example where multiplication doesn't commute.)

In an abstract algebra class you will forget about "what it is" you are studying and focus your study on the operations used to combine objects. In a way you are studying how things behave rather than what they are. This leads to the idea of groups, rings and fields among other structures which let you recognize common algebraic patterns throughout all areas of math instead of focusing on concrete specifics.

I have less experience in analysis, but I can say a few things...

You would already be familiar with some of the ideas of analysis from calculus. Many calculus classes don't give a rigorous definition of a limit and simply assume it works based on intuition (indeed, this is how calculus was developed historically).

A study of analysis would revisit limits and give a rigorous definition what it means for a series to "converge" and what it means for a function to be "continuous". From here one can develop rigorous definitions for derivatives and integrals.

Later these ideas will be generalized to work in $n$-dimension space, for functions of complex variables, and any spaces in which the idea of "continuity" of functions makes sense. (These spaces go by the name of topological spaces).

After studying them separately, analysis and algebra tend to show up together all over the place. A simple example might be the product rule from calculus, which is an algebraic rule that results from studying analytical properties of functions:

$$d(fg) = df g + fdg$$

Another example is the inner product, which grew out of the dot product from linear algebra but can also be applied to pairs of continuous functions.

• well to be exact, i would say the study of 'closedness' or 'connectedness' belongs to topology, not analysis. Commented Aug 13, 2015 at 17:50
• @Mathcho I'm honestly not sure how to define "analysis" in general. Much of analysis, including limits and series, do generalize to topology. My description sort of leaves out measure theory, but in a way this also deals with how "near" certain sets are to other sets. Keep in mind I say "closeness" and not "closedness". Commented Aug 14, 2015 at 12:08
• Oh i'm an undergraduate and i'm not too much into analysis either. if you put in that way, i understand what you're saying here. Commented Aug 14, 2015 at 12:25
• And according to wikipedia, ...Modern mathematical analysis is the study of infinite processes... And ...algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics... The sources are 1)en.wikipedia.org/wiki/Analysis#Mathematics 2) en.wikipedia.org/wiki/Algebra Commented Oct 28, 2016 at 3:21

Abstract algebra is largely (but not only) about sets with operations and their properties.

Mathematical analysis is largely (but not only) more about topology, measure, and how you can apply topology and measure to functions, namely integration and differentiation.

Contrary to your comment about continuity and discreteness, there are many interesting algebraic structures with topologies that you can study. For example, there are rings of continuous functions from a topological space into itself.

Similarly, analysis is no stranger to discreteness. For example, summable sequences are nothing more than integrable functions from $\Bbb N\to \Bbb R$ where the measure is the counting measure.

Functional analysis (at the very least) is a complex marriage of the two disciplines. There is no reason to draw a line between the two disciplines as you described.

You need to know both. Honestly at higher levels of math things aren't broken down into analysis or algebra. There's functional analysis. Algebraic topology etc.

Both are important to know.

The real question however, is how do you eat your corn?

• I can't really recall how I ate my corn the last time I did it. I think I hold it horizontally and then go around, but I'm not sure, could be that I eat horizontal strips to. I guess I need to find out.
– Lara
Commented Aug 10, 2015 at 19:09

At an elementary level, algebra has shorter proofs; and sometimes a formula says all, whereas in geometry or topology a picture can say a lot. At a higher level, these areas begin to merge. For example, algebraic number theory very often uses analytic tools (see the "analytic class number formula", to give an example), and this holds for several other areas of modern mathematics.

Typically, a math major takes after calculus: linear algebra, differential equations, elementary real analysis, abstract algebra and possibly complex analysis. You really don't specialize in algebra or analysis until you get to grad school in mathematics.

• You need a broad base in mathematics if you plan on taking the GRE subject test to go to grad school in mathematics. Commented Aug 10, 2015 at 21:17