ACADEMIA Letters
Mathematics without Boundaries!
David Womack, University of the South Pacific
On Safari in Your Own Backyard1 was the title of an article I published in the Education
Guardian back in 1986. It pointed out that you didn’t need to be part of a team in order
to investigate and carry out significant scientific research. I claimed then that mathematical
investigation could successfully be carried out with nothing more than a pencil and the back
of an envelope. Since then, the rapid output of mathematics in Academia and on the Internet
has been nothing short of phenomenal. Therefore, it is now debateable whether ’back-of-anenvelope’ working is still relevant – even if we do manage to find an envelope! This article is
written in the spirit of what the author has called, Mathematics without Boundaries.
As an amateur mathematician, it is not necessary for you to establish a new principle
nor find a new theorem. For you, there will be no predetermined end-point, destination or
objective. There need be no well-known hypothesis to prove or disprove; it is the mathematical
exploration itself which is paramount. You should not feel obliged to refer to other work in
your topic area - and you probably wouldn’t know the name of your chosen area anyway!
Being a speculative investigation, nothing is off-limits - so no pressure then. Since there
are no boundaries to your inquiry, there is no pressing need to investigate all the ’boundary
conditions’ – for numbers such as negatives, surds, transcendental numbers, complex numbers
and the rest. These can be left to the professionals. Let the good sense of your reader determine
which numbers are valid - and which are not!
1
Womack, David. On Safari in Your Own Backyard. The Education Guardian : Half page feature) 12th August
1986.
Academia Letters, February 2021
©2021 by the author — Open Access — Distributed under CC BY 4.0
Corresponding Author: David Womack, uspdave123@hotmail.co.uk
Citation: Womack, D. (2021). Mathematics without Boundaries!. Academia Letters, Article 305.
https://doi.org/10.20935/AL305.
1
Proofs?
Perhaps in time, you will feel confident enough to ’go public’ with your results. Maybe you
have a periodical in mind? After all, it would be satisfying if all those hours spent ’playing
with numbers’ on the train or just sipping coffee in a café, could result in something more permanent. If you feel the pressing need to justify your results to others, then do so! Attempting
a proof will give you confidence and perhaps convince you that you haven’t made a mistake.
However, note that serious proofs are not for the fainthearted. For example, the final proof of
the historic ’4-colour problem’ consisted of several hundred pages of A4 and took over one
thousand hours of computer time! This example is not typical of course but it makes my point.
At the other extreme is the simple but classic proof given by Euclid around the third century
BC, proving that there are an infinite number of prime numbers.
In his Dictionary of Mathematics (1989)2 , David Nelson writes: ’A proof is a chain of
reasoning using rules of inference, ultimately based on a set of axioms, that leads to a conclusion.’ This sounds straightforward enough(!) but standards of proof change over time and
many theorems which were proved in the past would not meet present standards of rigor. In
fact, mathematicians acknowledge that even today there is no universally recognised proof for
the validity of our primary school arithmetic. Perhaps more realistically then, we should accept that a proof is valid if it convinces a significant number of experts in the field that it makes
sense. However, confirmation of the soundness of any arithmetical ’pudding’ we may have
produced lies entirely in the eating. We should ask, whether the numbers add up according
to our rules, remembering that although we have freedom to choose our rules, they must be
reliable and consistent.
An informal justification should be transparent and make for an easy read. It can however,
rely on the reader’s good sense – for would any person without good sense be interested to read
it anyway? And when you think you have found a ’proof’, invite an amateur mathematician to
disprove it. But don’t attempt to disprove their argument; challenge them on your own ground
– not theirs! It may turn out that you’ve been using a novel principle such as the historical
’adequality’ concept3 .
New numbers?
Why not begin by just experimenting with invented ’new’ numbers. For example, you might
start by creating various ’artificial’ decimal numbers - such as Champenowne’s number. This
2
3
Nelson, David. Dictionary of Mathematics, 1989.
Strogatz, Steven, Infinite Powers, 2019.
Academia Letters, February 2021
©2021 by the author — Open Access — Distributed under CC BY 4.0
Corresponding Author: David Womack, uspdave123@hotmail.co.uk
Citation: Womack, D. (2021). Mathematics without Boundaries!. Academia Letters, Article 305.
https://doi.org/10.20935/AL305.
2
is the concatenation of the counting numbers as in: 0.1234567891011121314 … Notice that
the decimal point ensures the number converges to a finite value (a little more than 0.12…. in
this case).
It’s child’s play to repeat operations on a calculator, so why not try experimenting by
repeatedly taking the square root (or any other root) - taking care to interleave positive terms
to prevent values diminishing to zero.
[Note: Here, the reader might make use of the dot-notation I have used in other articles on
Academia, in which active operations (such as ’divides’ rather than ’divide’) are distinguished
with a dot. Also the bizarre notation for roots and logarithms has been amended (2) ]
Don’t let terms like rational, irrational, transcendental, and the seemingly enigmatic
imaginary numbers, put you off. The current classification of numbers can be intimidating
but many designations are unnecessary for justifying our results. Do we really need to know
the precise definition of ’rational’ or ’transcendental’?
Continued fractions and the danger of decimals.
Perhaps you prefer simple-to-state but historically more challenging areas of mathematics.
You might try (famously!) looking for a pattern in the prime numbers, or tabulating the number of different ways you can partition a number (for example, there are 7 different ways for
the number five, 4+1, 3+2, 3+1+1 etc.). Then there are relatively more difficult topics such as
chaos theory but details of this are not necessary to outline here, since the Internet is replete
with simple explanations.
On the other hand, the reader might like to investigate the fascinating world of continued
fractions - a relatively unexplored topic. A continued fraction is a more “natural” representation of an irrational number than its decimal approximation, for the terms generated are always
(as the name implies), fractions. For example, the famous golden ratio (known as phi - see
Web) is the solution to the equation: y2 − y − 1 = 0.
This results from the constantly changing fraction:- 1) + 1) divides 1) + 1) divides 1) + 1)
…. and so on. There are various ways to determine the value towards which such a continued
fraction is tending; some of which are frowned upon by mathematicians. But if the method
works, then use it.
Inscrutable repetends.
The temptation to use decimal approximations to compare fractions is overwhelming. Decimals free us from extremely tedious calculation unless we use the ’fraction’ key of an elecAcademia Letters, February 2021
©2021 by the author — Open Access — Distributed under CC BY 4.0
Corresponding Author: David Womack, uspdave123@hotmail.co.uk
Citation: Womack, D. (2021). Mathematics without Boundaries!. Academia Letters, Article 305.
https://doi.org/10.20935/AL305.
3
tronic calculator. However, remember that decimals are essentially approximations. For example 333/999 is never going to be the same as 333/1000 – even if we compare 33,333/99,999
with 33,333/100,000. The problem therefore for ’pocket calculator researchers’ is to judge
whether conversion to decimals is always necessary or useful.
Decimals can also lull us into a false sense of security. For example, all non-exact decimal
fractions must have repetends (for example. 1/7 = 0.142857 142857 …) The problem is that
the simple fraction 1/193 produces a decimal of 0.00518134715025906735751295336788 ….
This sequence of digits reveals no repeated section as far as my calculator’s scientific mode
will show. But where would we find such huge fractions, you may ask? The answer is from
the cumulative sums of continued fractions – such as that for pi which gives us the series:4 - 4/3 + 4/5 –4/7 + 4/9 –4/11 + 4/13 -4/15 +4/17 -4/19 …. . The fifth pair of terms gives
us the fraction 8/323. This produces the following decimal according to my Word calculator:
0.02476780185758513931888544891641… Again, there is no observable repetend here!
It’s useful to remember that every irrational number is the result of an infinite continued
fraction and conversely every infinite continued fraction results in an irrational number. But do
we really need to wrestle with the concept of irrational numbers? Since the continued fraction
will never reach infinity, its fractions will never become irrational! The transformation is
hypothetical; the continued fraction is always a fraction wherever and whenever we decide to
examine the series.
Common-sense or rigor (mortis!)
Mathematical rigor in a proof is often insisted upon by publishers. However, this can kill the
beauty of investigation for all but specialists. Rather it can result in a form of rigor mortis
for aspiring mathematicians. Your aim should be to uncover something more than a novel
rearrangement of a known formula; perhaps you might reveal some previously unknown or
surprising property of numbers. In this article, I have encouraged you, the reader, to explore
mathematics without boundaries. If you take up the challenge you will have set off on a
journey, the outcome of which is unknown – which defines a true adventure. And remember,
ultimately at the very least, you can claim that the ideas you present are unique.
Academia Letters, February 2021
©2021 by the author — Open Access — Distributed under CC BY 4.0
Corresponding Author: David Womack, uspdave123@hotmail.co.uk
Citation: Womack, D. (2021). Mathematics without Boundaries!. Academia Letters, Article 305.
https://doi.org/10.20935/AL305.
4