## Me

Book Reviews

## A Book of Abstract Algebra 2nd ed - Charles Pinter

Pick this math book for your next air-plane trip. Yes seriously, and no I am not insane. The narrative style, historical perspective, and

clarity with which the simple core elements and their meaning are presented is the other end of the spectrum from the standard

theorem-proof-corollary-example-repeat style.

We are continuously assaulted by an enormous amount of interwoven information, whether

sitting in math class, or walking to the car. It is at the core of the human experience to pick out the few bits that are

relevant, discard the rest, and reason with the abstract representations of the important bits we retained. Pinter does

this for us exceedingly well, and so can make the topics clear.

.More than just a nice diversion from the norm, Pinter's style does two things:

- factors out the essence, the meaning, the important concepts, the heart

- makes them understandable

Did you go through math this way: Memorize the patterns of a zillion examples the teacher went through; show

up for the exam; read the questions; try to find all the same components in the same order as one of the examples

you memorized; substitute numbers and try to generate the answer; hope you pass;

with no real idea of what was going on?

Read this book and understand algebra like a mathematician. It's also an enjoyable read. Some excerpts:

"We must make an effort to discard all our preconceived notions of what an algebra is, and look at this new notion of

algebraic structure in its naked simplicity.

structure. There does not need to be any connection with known mathematics. For example consider the set of all colors (pure colors

as well as color combinations), and the operation of mixing any two colors to produce a new color. This may be conceived as an algebraic structure. It obeys certain rules such as the commutative law (mixing red and blue is the same as mixing blue and red).

""When we open a textbook of abstract algebra for the first time and peruse the table of contents, we are struck

by the unfamiliarity of almost every topic we see listed. Algebra is a subject we know well, but here it looks

surprisingly different. What are these differences and how fundamental are they?

First there is a major difference in emphasis. In elementary algebra we learned the basic symbolism and

methodology of algebra; we came to see how problems of the real world can be reduced to sets of equations and how

these equations can be solved to yield numerical answers. This technique for translating complicated problems into symbols

is the basis for all further work in mathematics and the exact sciences, and is one of the triumphs of the human mind."

The book is not entirely an entertaining narrative, it has the breadth and depth of topics to make it useful, but

Pintler's 'filter' brings it out with clarity.

clarity with which the simple core elements and their meaning are presented is the other end of the spectrum from the standard

theorem-proof-corollary-example-repeat style.

We are continuously assaulted by an enormous amount of interwoven information, whether

sitting in math class, or walking to the car. It is at the core of the human experience to pick out the few bits that are

relevant, discard the rest, and reason with the abstract representations of the important bits we retained. Pinter does

this for us exceedingly well, and so can make the topics clear.

.More than just a nice diversion from the norm, Pinter's style does two things:

- factors out the essence, the meaning, the important concepts, the heart

- makes them understandable

Did you go through math this way: Memorize the patterns of a zillion examples the teacher went through; show

up for the exam; read the questions; try to find all the same components in the same order as one of the examples

you memorized; substitute numbers and try to generate the answer; hope you pass;

with no real idea of what was going on?

Read this book and understand algebra like a mathematician. It's also an enjoyable read. Some excerpts:

"We must make an effort to discard all our preconceived notions of what an algebra is, and look at this new notion of

algebraic structure in its naked simplicity.

*Any*set, with a rule (or rules) for combining its elements, is already an algebraicstructure. There does not need to be any connection with known mathematics. For example consider the set of all colors (pure colors

as well as color combinations), and the operation of mixing any two colors to produce a new color. This may be conceived as an algebraic structure. It obeys certain rules such as the commutative law (mixing red and blue is the same as mixing blue and red).

""When we open a textbook of abstract algebra for the first time and peruse the table of contents, we are struck

by the unfamiliarity of almost every topic we see listed. Algebra is a subject we know well, but here it looks

surprisingly different. What are these differences and how fundamental are they?

First there is a major difference in emphasis. In elementary algebra we learned the basic symbolism and

methodology of algebra; we came to see how problems of the real world can be reduced to sets of equations and how

these equations can be solved to yield numerical answers. This technique for translating complicated problems into symbols

is the basis for all further work in mathematics and the exact sciences, and is one of the triumphs of the human mind."

The book is not entirely an entertaining narrative, it has the breadth and depth of topics to make it useful, but

Pintler's 'filter' brings it out with clarity.