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Last edited 2/12/2016 8:08:00 AM
MORE USEFUL BEHAVIORS
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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.
When you learn about a new concept
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.
Do not diss baby examples.
Trial and error
Example
Suppose you need to know the largest integer n for which n! < 109 . 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
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
ò sin x cos x dx .
You remember that you just did that integral yesterday! Wasn’t
1 2
1
1
sin x + C ? Let’s see, the derivative of sin2 x would be 2 sin x cos x , which is
2
2
2
1
sin x cos x . So the Fundamental Theorem of Calculus says that the answer sin2 x is correct!
2
This is a perfectly respectable way to do math.
the answer
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
ò sin x cos x dx
is easy to do by substitution: Let u = sin x and du = cos x dx . Then
ò sin x cos x dx = ò u du =
1
2
u2 + C =
1
sin2 x + C
2
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]