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PATTERN RECOGNITION
Contents
The product
rule for derivatives
The substitution rule for integration
Some
bothersome types of pattern recognition
An expression as an instance of substitution
Two different substitutions give the same expression
Technical problems in carrying out pattern matching
Recognizing
an instance of substitution
When you do
math, you must recognize abstract patterns that occur in
¨ Symbolic expressions
¨ Geometric figures
¨
Your own mental representations of mathematical objects.
This happens in
high school algebra and in calculus, not just in the higher levels of abstract
math.
Easy examples
At most
The definition of "at most" says that "x is at most y" means .
To understand this definition requires recognizing the pattern " x is at most y"
no matter what occurs in place of x and y. For example,
" sin x is at most 1"
means that .
The
product rule for derivatives
The product rule for differentiable functions f and g tells you that the derivative of f(x)g(x) is .
You recognize that the expression
fits the pattern f(x)g(x) with
and
. Therefore you know that the derivative of
is
. More about the product rule below.
The
quadratic formula
The quadratic formula for the solutions of an equation of
the form is usually given as
If you are asked
for the roots of ,
you recognize that the polynomial on the left fits
the pattern
with 3
for a, -2 for b and -1 for c. Then substituting those
values in the quadratic formula gives you roots equal to -1/3 and 1.
Difficulties with the quadratic
formula
A little problem
The quadratic
formula is easy to use but it can still cause pattern recognition
problems. Suppose you are asked
to find the solutions of . Of course you can do this by simple algebra
but pretend that the first thing you thought of was using the quadratic formula
and:
you got upset
because you have
to apply it to
and has only two terms
and has three
terms…
(help!)
Do
Not Be Anguished:
Write
as
(so a =
3, b = 0 and c = -7).
Then put
those values into the quadratic formula and you get .
This is an example of the following useful principle:
Write zero cleverly.
I suspect that most people reading this would not have had the
problem with that I just described. But before you get all insulted, remember:
The thing about really easy examples is that they give you the point
without getting
you lost in some complicated stuff you don’t understand very well.
A fiendisher problem
Even college students may have trouble with
the following problem (I know because I have tried it on them): What are the solutions of the equation ? The answer
is wrong: the correct answer is
When you remember a pattern with particular letters
in it
and an example has some of the same letters in it,
BE CAREFUL!
The substitution rule for integration
The chain rule says that
the derivative of a function of the form is
. From this you get the substitution rule
for finding indefinite integrals:
Example
To find ,
you recognize that you can take
and
in the formula, getting
. Note that in the way I wrote the integral
the functions occur in the opposite order from the pattern. That kind of thing happens a lot.
The integral
Another example
is given here.
Some bothersome types of pattern recognition
Dependence on conventions
Definition: A quadratic
polynomial in x is
an expression of the form .
Examples
¨
is a quadratic polynomial: You have to
recognize that it fits the pattern in the definition by writing it as
¨
So is :
You have to recognize that it fits the definition by writing it as
.
¨
The expression is not a
quadratic polynomial in x. You have to be aware that saying it is
a “quadratic polynomial in x” binds
the x: You can’t substitute for it and
call the result a quadratic polynomial in x.
Some authors would just say, “A quadratic
polynomial is an expression of the form ” leaving you to deduce from conventions on variables that it is a polynomial in x instead of in a (for example). Tsk.
Note also that I have deliberately not mentioned what sorts of numbers a, b, c and x are. The definition makes sense for them to be any elements of a ring. The authors may assume that you know they are using the real or complex field.
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An expression as an instance of substitution
One particular type of pattern recognition that comes up all
the time in math is recognizing that
a given expression is an instance
of a substitution into a known expression.
Example
Students are sometimes baffled when a proof uses the fact
that for positive integers n. This requires the recognition of the patterns x + x
= 2x and
. Similarly
.
Example
The assertion
" "
(1)
has as a special case
" "
(2)
which involves the substitutions and
. If you
see (1) in a text and the author
blithely says it is never negative, that is because it is of the form “
”. The
fact that there are minus signs in (2) and that x and y play different roles in (1) and in (2) are red herrings. See ratchet effect and variable clash.
Note that when you make these substitutions you have to
insert appropriate parentheses (more here). After
you make the substitution, the expression of course can be simplified a whole
bunch, to
“
”
A common cause of error in doing this (a mistake I make sometimes) is to try to substitute and simplify at the same time. If the situation is complicated, it is best to substitute as literally as possible and then simplify. See substitution.
Two different substitutions give the same expression
Some proofs involve recognizing that a symbolic expression or figure fits a pattern in two different ways. This is illustrated by the next two examples. I have seen students flummoxed by Example ID, and Example ISO is a proof that is supposed to have flummoxed medieval geometry students.
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This proof is over two millenia old and is called the pons asinorum (bridge of donkeys). It became famous as the first theorem in |
Example ID
Definition: In a set with an associative binary operation and an identity element e, an element y is the inverse of an element x if
(1)
In
this situation, it is easy to see that x has
only one inverse.
Theorem: .
Proof.
I am given that is
the inverse of x. By definition,
this means that
(2)
To
prove the theorem, I must show that x is the
inverse of , which requires that
(3)
But (2) and (3) are equivalent!
If you
don’t understand this proof, struggle with it,
and if
necessarily sleep on it and then struggle again. It is worth it.
Example ISO
This sort of double substitution occurs in geometry, too.
Theorem: If a triangle has two equal angles, then
it has two equal sides
Proof: In the figure, assume . Then triangle ABC is congruent to triangle ACB
since the sides BC and CB are equal (they are the
same line segment!) and the adjoining angles are equal by hypothesis.
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The point is that although triangles ABC and ACB are the same triangle and sides BC and CB are the same line segment, the proof involves recognizing them as geometric
figures in two different ways.
Technical problems in carrying out pattern matching
¨ You can easily get confused if the patterns involve a switch in the order of the variables. Example.
¨
In matching a
pattern you may have to insert or remove parentheses. For example, if you substitute for a,
2y
for b and 4 for c in the expression
,
you get
. If you did the substitution literally without editing the expression
so that it had the correct meaning you would get
,
which is not the result of substituting
for a, 2y for
b and 4 for c in the expression
. (In fact it is not the result of performing any substitution in
. Can you prove that?)
More about pattern matching
¨ Textbooks in abstract math generally give proofs without telling you what the pattern of proof is. See Patterns of Proof for examples.
¨ Another example of pattern matching is discussed under ratchet effect.
Recognizing an instance of substitution
Integration by Parts
The rule for integration by parts says that
Suppose you need
to find . (In abstractmath.org, log means
).
Then we recognize this integral as having the pattern for the left side
of the parts formula with
and
. Therefore
How did I think
to recognize as
?? Well, to tell the truth because some nerdy guy
(perhaps I should say some other
nerdy guy) clued me in when I was taking freshman calculus.
This is an
example of another really useful principle:
Write 1 cleverly.
Explicit descriptions
An explicit description
is a description not
requiring pattern recognition to understand.
Example
This is an explicit
description:
(EA) “The derivative of the square
of a function is
A form of this rule that does require pattern recognition is:
(PRA) "The derivative of is
.”
The point is that that in (EA) the rule is stated in a way
that you
don't have to decode patterns to understand what the rule says.
Of course, applying
(EA) requires pattern recognition: you must recognize that you have
the square of a function and you must recognition what the function is.
(This is not necessarily obvious to beginners: consider 
the functions ,
,
and
. They are all squares of functions, namely
,
and sin x
respectively.)
Most definitions and theorems in mathematics do require pattern recognition and many would be difficult or impossible to formulate clearly without it. Note that assertion (EA) is considerably harder to read the assertion (PRA). It is also considerably harder to write it so that it is correct and unambiguous!
Terminology
The terminology explicit description,
meaning one not requiring pattern recognition, is peculiar to this
website. “Not requiring pattern
recognition” is not an entirely crisp definition.
Example
You could give the product rule for
derivatives as “The derivative of the
product of two functions is the derivative of the first function times the second
function plus the first function times the derivative of the second
function.” Is this crisp? When you see an expression you have to match
as the “first function” and
as the “second function”. On one hand, it seems to me that the very meaning of the phrase “first function” requires
to be the first
function. On the other hand, why
couldn’t the “first function” be x and the
“second function”
? (If you take
it that way, though, you still get the right answer.)
