> # ictm_ply.mws

Speaker: John B. Cosgrave.

Address: Mathematics Department,
St. Patrick's College,
Drumcondra,
Dublin 9,
IRELAND.

e-mail: [email protected] (College)
[email protected] (Home)

Web: http://www.spd.dcu.ie/johnbcos

Note . The Maple file of this talk (and many other Maple worksheets)
may be accessed at my Web site.

Title of talk :

Using Maple programming to investigate

L- and R-approximations to quadratic irrationalities

Comment on lecture title . Later you may agree that I could just as easily given

my talk the title Using L- and R-approximations to quadratic irrationalities to

introduce Maple programming .

A brief synopsis of my talk .

• Rational numbers, and classic examples of irrational numbers
• Rational approximations to , ...
• Formulating the notions of L- and R-approximations to , ...
• Easily observed relationships between hand computed successive approximations,
• leading to precise formulations and search for proofs
• Surprises in store ...
• When hand computation appears to let one down, what can one do to
  attempt to find some L- and/or R-approximations [this is where Maple
  really makes itself felt ... ]
• Doable projects ... [also some deep, very advanced topics ( en passant )]

A statement of my basic philosophy . Ideally one would like ones students to be

more than passive absorbers of mathematical ideas. One would like ones students

to experience the thrill of making a mathematical discovery. The difficulty is to

identify topics that are neither trivial nor too advanced.

Some motivating questions .

Question(s) 1 .

• What is the value of ? [Students should see , should think of it as being
  - e.g. - the length of the diagonal of the unit square.]
• Is it (perhaps) 1.414213562 (the value given by a 10-digit calculator)?
• Is it (perhaps) the 1000-place decimal given below?:

> evalf(sqrt(2), 10);

> evalf(sqrt(2), 1000);

>

My experience is that I always receive from several students an argument

(perhaps not formulated exactly precisely, but at least in the right direction)

that does not have a terminating decimal value [I assume that the reason

for that immediately occurs to all of my readers. Requiring ones students to write

out of a formal proof of that simple (but non-trivial) result, is an interesting exercise.]

Observation . Every terminating decimal is [trivially] a rational number of a certain kind ,

that is, every terminating decimal is a rational number whose denominator is a power of 10 .

Question(s) (going back to classical times) . Could it be that has a rational value?

That is, could there be integers m and n such that ?

Let's have a look at the following rational number:

> r := 152139002499/107578520350;

>

whose square is:

> r^2;

>

Its decimal value to 10 palces is:

> evalf(r^2);

>

perhaps it is 2?

> evalf(r^2, 20);

> evalf(r^2, 30);

>

O.k., that ' ' isn't 2, but it is pretty close to it, and so ' r ' itself is pretty close to .

There is agreement to at least 20 places:

> evalf(r, 20);

> evalf(sqrt(2), 20);

>

Look at what we get if we lower the numerator of by 1:

> 23146276081390728245000/11573138040695364122500;

>

Definition 1 . Let p and q be natural numbers, then is said to be

• an L-approximation to if ,
  
  equivalently if ,
  
  equivalently if .
  
• an R-approximation to if ,
  
  equivalently if ,
  
  equivalently if .

>

Hand computations to find some L- and R-approximations to :

eventually leads to finding that the numbers and are alternately

L- and R-approximations to .

The start of real progress . Some bright students will invariably call out that they

see some connections [I always try to play the innocent and say something like:

oh yes, like what ? ...].

And what are those connections?

Well they simply jump out at you!!

• The sum of numerator and denominator for a given term of the sequence
  (appear to) produce the denominator of the next rational in the sequence
• The sum of denominators of two consecutive terms of the sequence
  (appear to) produce the numerator of the second of those terms

Question(s) 2 .

• Does that continue to happen (perhaps it is just a fluke on those early ones?)
• Are there any missing L- and R-approximations to ? ['missing' in the sense
  that ... ]

Students can form some more by hand ... , but let's move quickly along to

some simple programs. Let's denote the alternating L- and R-approximations

to by , ... , , ...

A formulation of the apparent connection between the p 's and q 's:

• It appears as though (for , ... )
• It appears as though (for , ... )

Students are easily introduced to the following simple Maple program

(but one has to devote TIME to explaining all the elements involved):

> p[1] := 1:
q[1] := 1:
for k from 2 to 50 do
q[k] := p[k-1] + q[k-1]:
p[k] := q[k-1] + q[k]
od:
seq(p[k]/q[k], k=1..50);

> p[30]/q[30]; # The 'r' from earlier is:

>

Question . But are those rationals alternating L- and R-approximations to ?

Students should do some by hand and calculator before being exposed to:

> seq(p[k]^2 - 2*q[k]^2, k=1..50);

>

Computation on its own is not enough .

Happily a proof of the following elementary theorem is within

the reach of a good second level student (indeed finding a proof

is a really good test, and not an unreasonable one):

Theorem 1 . Let be an L-approximation to ,

then is an R-approximation to ,

and

if is an R-approximation to ,

then is an L-approximation to .

Proof . First, suppose that is an L-approximation to ;

then .

Thus

=

=

= ,

and it follows that is an R-approximation to .

A similar argument [exercise for students] establishes the other half of theorem 1. [ end of proof ]

Comments .

Question (s) .

• What happens when one similarly investigates ?
• Does have L- and R-approximations?
• Are those L- and R-approximations related to each other in
  some similar fashion?
  

First, let's get our definitions [one should first have discussed a proof of the

irrationality of - or at the very least alluded to the probability that

straightened out:

Definition 2 . Let p and q be natural numbers, then is said to be

• an L-approximation to if ,
  
  equivalently if ,
  
  equivalently if .
  
• an R-approximation to if ,
  
  equivalently if ,
  
  equivalently if .

With ones students one could do some hand computations [in fact that

is what I do with mine], along the lines of:

1. When then is not a square, but = ,

and so is an R-approximation to .

2. When then is not a square,

nor is a square.

Thus has no L- or R-approximation with denominator 2.

etc.

Here, students are in for a surprise .

First I will redo to search for L- and R-approximations,

but using a simple Maple program (below).

First, some preliminaries:

> ### WARNING: persistent store makes one-argument readlib obsolete
readlib(issqr);

>

Testing to see if is a square when :

> issqr(2*4^2 - 1);

>

And so it isn't.

Testing to see if is a square when :

> issqr(2*5^2 - 1);

>

And what is it the square of?:

> sqrt(2*5^2 - 1);

>

Now we are ready to systematically look for L- and R-approximations

to . We limit ourselves to testing denominators up to 100:

> for q to 100 do
if issqr(2*q^2 - 1)
then print(L, sqrt(2*q^2 - 1)/q)
elif issqr(2*q^2 + 1)
then print(R, sqrt(2*q^2 + 1)/q)
fi
od;

>

Now let's do the same for :

> for q to 100 do
if issqr(3*q^2 - 1)
then print(L, sqrt(3*q^2-1)/q)
elif issqr(3*q^2 + 1)
then print(R, sqrt(3*q^2+1)/q)
fi
od;

>

So, a surprise!! would appear to have no L-approximations.

A question . Is that true ? [Now that is a really good problem for students.]

An observation and another question . certainly has some R-approximations, but

• how many does it have?
• can one find (and prove) any connection between them?

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