Okay gang, I have my brother in law here, and I’m showing him the SDMB, and told him that I could get an answer for any question he wanted to ask. The question is:
What is the distinction between Algebra and Calculus?
Anyone feel like taking a crack at that?
Basically, algebra is the method of creating and manipulating symbols (variables) that refer to numbers. So if you see an x, it is algebra 
Calculus is a type of math that deals with the limits, derivatives, and integrals of functions.
And you’re probably not going to get too far with calculus if you aren’t using algebra at the same time. Perhaps one could say that calculus is more conceptual, whereas algebra is more practical (as in, a practice)?
Someone will fill in the gaps if I oversimplified or missed the point!
Calculus relies heavily on Algebra and, at points, Trigonometry, so in that sense, they necessarily overlap. It’s the same way that Algebra relies heavily on Arithmetic. As for Algebra being more concrete than Calculus, while this may be true at high school level math, Abstract Algebra is about as abstract as math comes, even more so, I’d say, than Calculus’s pure-math form, Analysis. I think…
Hip shot:
Algebra’s main step away from arithmetic: it teaches us to use symbolic logic (the biggest hump to get over for my friends who bonked their heads on it);
Calculus: relies on algebra a lot, but is mainly concerned with quantifying change and rates of change in sometimes hard to quantify systems.
…ah, I’ll live with that (definitely not a perfect answer).
I generally think of algebra as being finite in nature, and calculus being infinite.
Algebra is the study of operations like addition, subtraction, multiplication, division, and so forth, and the different possible structures you can put on sets with those operations. These are all finite operations: 1+7=8, 5*9=45; you’re only dealing with finitely many elements at a time.
Underlying everything in calculus is the idea of a limit. For example, you can talk about the limit of a sequence, e.g., the limit of the sequence .9, .99, .999, .9999, and so on, limits to the value one (as has been discussed many times on the board). A limit is an infinite operation, in the sense that you’re dealing with infinitely many elements at a time.
I’d say that’s the difference in a nutshell.
Algebra is, or at least at one point was, the study of solving equations. What it’s grown into out of that is something quite different.
The most basic abstract algebra is the study of a set and operations defined on the elements of that set. The one restriction that all algebraic systems I’ve seen have is that the result of applying the operations to elements of the set must also be in the set.
For instance, one of the most basic algebraic structures is a group. A group is a set G with an operation called multiplication that satisfies the following conditions:[list=1][li]If a is in G and b is in G, then ab is in G.[/li]lic = a(bc).[/li][li]There is an element 1 of G such that, for any g in G, 1g = g and g1 = g. 1 is called the identity of G.[/li][li]For any g in G, there is an element g[sup]-1[/sup] such that gg[sup]-1[/sup] = 1 and g[sup]-1[/sup]g = 1. g[sup]-1[/sup] is called the inverse of g.[/list=1]Note that it is not required that ab = ba in general. The simplest example of a group is the integers under addition. The identity is 0, and the inverse of z is -z.[/li]
The idea of a group arose in the study of the solutions of polynomial equations. People had found solutions that would work for any equation of the form ax[sup]4[/sup] + bx[sup]3[/sup] + cx[sup]2[/sup] + dx + e = 0 (where any of a, b, or c may be equal to zero). So they were wondering if solutions could be found for equations involving x[sup]5[/sup] and possibly higher order terms. Evariste Galois developed the theory of groups and showed that 5th-degree equations (those involving x[sup]5[/sup]) had no general solution involving radicals (nth roots).
Calculus, on the other hand, is the study of integrals and derivatives, which are two specific kinds of limits. Basically, it’s a subdiscipline of analysis, which is the study of limits in general.
Er, ultrafilter, I thought the qualities you listed belonged to a field. If they don’t, what does a field have that a group does not?
A field has multiplication and addition. All the elements of a field form a commutative group under addition, and the elements other than the additive identity form a commutative group under multiplication. In addition, the distributive law must hold.
(A commutative group is one in which ab = ba for all a and b in the group.)
I know that this is ultra-simplified, but here goes.
The universe is calculus. Calculus allows many things to change at once. Algebra is a special case of calculus. It is a case where many things don’t change.
example:
If I know the acceleration and initial velocity of an object, I can determine the new velocity of that object at a specific time.
If the acceleration is constant (as it is on earth) I can figure it out with algebra. If the acceleration is not constant (as it can be in many situations) I have to use something other than algebra - calculus works best.
Algebra is what you use when you need to determine the value of a variable.
Calculus is what you use when you need to derive (or prove) the equation that you use the algebra on.
(Or, Algebra is for technicians; Calculus is for scientists.)
What ultrafilter described is Algebra in the most abstract terms, but Algebra existed for a while before this sort of concept was developed, and an introductory Algebra course doesn’t even come close to dealing with things like a group. Anyone who’s taken Algebra I (which should be everyone) should realize this. I’d also like to point out that to solve an equation like ax[sup]4[/sup] + bx[sup]3[/sup] + cx[sup]2[/sup] + dx + e = 0, you need something more robust than a group - specifically, you need a ring, which is defined on two operations, and is a group on one of them. The simplest ring is the integers under addition and multiplication. A field, as ultrafilter said, is a ring which is a group on both of its operations (0 excluded). Basically, a field is something in which you can add, subtract, multiply, and divide. The most familiar field is the real numbers.