The Algebra that High School and College Share in Common
Date: 12/05/2016 at 19:22:49 From: James Subject: High School Algebra and Abstract Algebra How is the algebra you learn in high school connected to abstract algebra? How are the topics of a high school algebra course that come out of a textbook titled "Algebra" related to what one finds in an abstract algebra textbook? Simple topics such as exponents, graphing functions, rational functions, exponential functions, logarithmic functions, factoring, lines, etc., do not seem at all related to the very general and abstract topics from an abstract algebra course. In fact, when one looks up the word algebra, the definition fits the high school algebra course but not the abstract algebra one. It's obvious to me how real analysis is related to calculus. Abstract algebra and high school algebra, by contrast, could not seem any more different -- this, despite sharing a keyword in common!
Date: 12/06/2016 at 13:14:25 From: Doctor Vogler Subject: Re: High School Algebra and Abstract Algebra Hi James, Thanks for writing to Ask Dr. Math. The word "algebra" in mathematics most properly refers to the "rational" operations (addition, subtraction, multiplication, division) and their use in solving equations. When you do this with real numbers, what this means is solving polynomials. While trig functions (like sin and cos) and other functions (like exp and log) are sometimes taught in high school algebra classes, these functions are not *technically* algebraic functions. Square roots, on the other hand, *are* algebraic functions, because they come from solving polynomial equations. The term "algebraic number," in particular, means a number (real or complex) that is the root of a polynomial with integer coefficients. Since many more people have experience with high school algebra, this is the definition usually found in a (non-mathematical) dictionary. In the case of abstract algebra, the main concept is applying operations to "abstract" numbers, which might not even be "numbers" in the traditional sense. They might represent ideas -- like geometric rotations, or functions, etc. But we still study how these behave under operations, and we often still call those addition or multiplication, and we still invert them to get subtraction and division. So the common feature here is how these structures relate to operations. This is in contrast to, for example, notions of "continuity" (i.e., continuous functions), or limits, or approximations, all of which are ideas not related to operations and therefore are not properly "algebraic" ideas. I hope that explanation makes sense. If you have any questions about this or need more help, please write back, and I will try to offer further suggestions. - Doctor Vogler, The Math Forum at NCTM http://mathforum.org/dr.math/
Date: 12/06/2016 at 14:10:50 From: James Subject: High School Algebra and Abstract Algebra Why does an algebraic number have to be the roots of a polynomial with *integer* coefficients? Why can't the polynomial have rational coefficients or irrational ones? Similarly, why are polynomials limited to only positive integer exponents? Another point of confusion: if algebra is applying the operations of addition, subtraction, division, and multiplication on objects, then how can you apply those operations to the abstract objects studied in abstract algebra? For instance, how can you multiply things that aren't numbers? I know you can do that with, like, say, matrices and vectors; but now that I think about it, operations on matrices and vectors aren't the same as those operations on numbers. So how is it decided which way to define those four operations on other objects? And lastly, if things like solving trig or logarithmic equations aren't algebra, then what would they be?
Date: 12/06/2016 at 19:34:40 From: Doctor Vogler Subject: Re: High School Algebra and Abstract Algebra Hi James, Allow me to address your questions in turn. > Why does an algebraic number have to be the roots of a polynomial with > *integer* coefficients? Why can't the polynomial have rational > coefficients or irrational ones? If you change "integer" to "rational," then you get the same thing. For example, take any polynomial with rational coefficients. Multiply the whole thing by the product of all of the denominators (or by the least common multiple), and you get a new polynomial with integer coefficients -- but it still has the same roots. So "integer" or "rational" coefficients give you the same numbers; and either one may be used in the definition of "algebraic number." Now, if you allow *irrational* coefficients, you just get all numbers -- and this is not a useful concept. > Similarly, why are polynomials limited to only positive > integer exponents? Similarly to the rational-coefficients question: If you have a "polynomial" but some of the exponents are *negative* integers, then this "polynomial" can be written as a polynomial with only non-negative integer exponents divided by a power of x. For example, consider x^2 + x + 2 - 3x^-1 + 5x^-3 The roots of this "polynomial" with negative exponents are the same as the roots of the numerator of this polynomial: (x^5 + x^4 + 2x^3 - 3x^2 + 5)/x^3 So you still get the same set of numbers. > Another point of confusion: if algebra is applying the operations of > addition, subtraction, division, and multiplication on objects, then how > can you apply those operations to the abstract objects studied in > abstract algebra? For instance, how can you multiply things that aren't > numbers? I know you can do that with, like, say, matrices and vectors; > but now that I think about it, operations on matrices and vectors aren't > the same as those operations on numbers. So how is it decided which way > to define those four operations on other objects? That is precisely what abstract algebra studies. Those operations can be defined as you wish; and the way you define them determines what properties they have; and the properties they have determine whether the thing you have defined is a group, or a ring, or a field, or a module, or a vector space, or a monoid, and so on. These are the things you study in abstract algebra. In order to have a group, for example, you need one operation which is required to have a couple of properties (e.g., associative). That operation we sometimes might call addition (especially if it's a commutative operation), and other times we might call multiplication. But that operation might be matrix multiplication if you're studying a group of matrices, or function composition if you're studying a group of functions (or permutations), and so on. In rings, you have two operations, which are normally called addition and multiplication (and they have to satisfy certain properties for this to be a ring), even if those operations are something like "AND" and "OR" in a Boolean algebra, or something else. And so on. > And lastly, if things like solving trig or logarithmic equations aren't > algebra, then what would they be? Trigonometry. Calculus. Analysis. Solving exponential equations. - Doctor Vogler, The Math Forum at NCTM http://mathforum.org/dr.math/
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