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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/ 
Associated Topics:
College Modern Algebra
High School About Math
High School Basic Algebra

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