A recommendation on how to read
textbooks:
ON READING BOOKS
Serge Lang, "Basic Mathematics"
In writing the book, the whole subject has to
be
organized in a totally ordered way, along lines and pages, which is not
the way
our brain works naturally. But it is unavoidable that some topics have
to be
placed before others, even though our brain would like to perceive them
simultaneously. This simultaneity cannot be achieved in writing, which
thus
gives a distortion of the subject. It is clear, however, that I cannot
substitute for you in perceiving various sections of this book
together. You
must do that yourself. The book can only help you, and must be
organized so
that any theorem or definition which you need can be easily found.
Another
way of reading this book is to start at the
beginning, and then skip what you find obvious or skip what you find
boring,
while going ahead to further sections which appeal to you more. If you
meet
some term you don't understand, or if you need some previous theorem to
push
through the logical development of that section, you can look back to
the
proper reference, which now becomes more appealing to you because you
need it
for something which you already find appealing.
Finally,
you may want to skim through the book
rapidly from beginning to end, looking just at the statements of
theorems, or
at the discussions between theorems, to get an overall impression of
the whole
subject. Then you can go back to cover the material more
systematically.
Any of
these ways is quite valid, and which one you
follow depends on your taste. When you take a course, the material will
usually
be covered in the same order as the book, because that is the safest
way to
keep going logically. Don't let that prevent you from experimenting
with other
ways.