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Last edited 2/10/2009 3:14:00 PM

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Think of examples. 1

Trial and error 2

Guessing. 2

Think of examples

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. 

Example

Suppose you just learned the definition of prime number.  (A positive integer n is prime if its only positive divisors are 1 and n.)  You might have thoughts like this go through your head:

2 is prime, 3 is prime, 4 is not prime because 2 is a divisor, 5 is prime, 6 is not prime because 2 is a divisor, 7 is prime, 8 is not prime because 2 is a divisor…

Hold on!  If n is prime it has to be odd, because otherwise 2 is a divisor!  Wait: 2 is a prime anyway even if it is even.  So the only even prime is 2. 

So if n is odd it must be prime, because it is not divisible by 2.  Hold on!  It might be divisible by something else.  That’s right, 9 is odd but it is not a prime! 

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.

Kinds of examples

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.

Trial and error

Example

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:

 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

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!

Example

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. 

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

Notes

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.

References

[Selden & Selden]