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help with abstract math
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Last
edited 2/10/2009 3:14:00 PM
One characteristic of people that are good
at math is that when they are faced with a new concept, they immediately start thinking
of examples.
When you learn about a new concept
START THINKING OF EXAMPLES

Suppose you just learned the definition of prime number. (A positive integer n is
prime if its only positive divisors are
2 is prime,
Hold on! If n is prime it has to be odd, because
otherwise
So if n is odd it
must be prime, because it is not divisible by
And so
on…
It is vital to
generate example of any new concept (if you can!) because that is the fastest road to
understanding the concept.
There are many different kinds of examples, discussed in Varieties of Mathematical Prose, section 2.3.1. In another sense all examples are the same they either fit the definition or they don’t (more about this under definitions.)
The example above is a baby example of what you might go through when you first learn about primes. Abstractmath.org uses baby examples a lot. Their value is that you can concentrate on what is being illustrated without getting stuck in unfamiliar math. Some students sneer at baby examples. “I already knew that!” But that example does not exist to teach you about primes. It exists to teach you how, when you come across a definition new to you, you use examples to understand it.
Do not diss baby examples.
Suppose you need to know the largest integer n for which . One way to do it is to calculate: 4! = 24, 5! = 120, so the answer is n = 4.
When I gave a problem that came down to this calculation in my discrete
math class, most students solved it correctly, but several wrote apologies on
their paper for doing it by trial and error. Apologies are not necessary:
Trial and error is a method.
In the case of this problem it is probably the easiest
way to do it. Even if you need to know
the largest integer for which n! < 4,000,000 it makes sense to do it using trial
and error with a calculator or a program such as Mathematica or Maple.
Guessing at the
answer to a problem and then using a theorem to prove it is correct is
legitimate. Some students don't believe
this!
You need to find . You remember that you just did that integral
yesterday! Wasn’t the answer ? Let’s see, the derivative of would be ,
which is . So the Fundamental Theorem of Calculus
says that the answer is correct!
This is a perfectly respectable
way to do math.
It is good math behavior
to guess at an answer and then prove it is correct
You need not use a specific method (for
example substitution or integration by parts) to get an integral. Any way of coming up with the answer is OK as long as you
can check it out (in
this case by using differentiation). Note.
Of course, if you do have a method,
you may be better off than you are if you can only guess. The integral is
easy to do by substitution: Let and . Then
Method. A teacher might insist that you solve an integral problem by using
substitution. Of course, if you practice
doing five integrals by substitution you will get better at it. That didn’t stop me from being irritated by
such practices when I was a student. Return.