Oxymorons of Mathematics
You may be aware of the well-known mathematical paradox surrounding the tale of Achilles, an ancient Greek. He challenged the tortoise to a race and graciously gave him a head start. The crux of the paradox is that Achilles never has a chance of winning because the tortoise will always be fractionally ahead during all stages of the race.
In this article I will not presume to offer scenarios as illustrious as the oxymoron surrounding Achilles who, although swift of foot, cannot manage to keep pace with the laboriously slow and lethargic turtle.
Rather, below are two offerings that will challenge your mathematical nous.
9.9999... = 10
Let’s begin with the decimal 9.99999…. which consists of an infinite number of the digit 9 after the decimal point. Represent this decimal by N.
N = 9.99999…. (1)
Multiplying both sides of the equation by 10 gives
10N = 99.9999… (2)
Now subtract equation (1) from equation (2).
10N – N = 99.9999… - 9.99999….
9N = 90
Thus N = 90 ÷ 9 = 10
However, it was stated initially that N = 9.99999…..
Thus we conclude that 9.999999…. = 10
This contradiction can also be obtained using the properties of a geometric series.
Write the decimal as the sum of an infinite number of terms.
9.999999… = 9 + 0.9 + 0.09 + 0.009 + 0.0009 + 0.00009 + ….
The infinite sum of the geometric sequence
a + ar + ar2 + ar3 + ar4 + …. is a/(1 – r)
Our geometric sequence is 0.9 + 0.09 + 0.009 + 0.0009 + 0.00009 +….
Hence a = 0.9 and r = 0.1
Therefore a/(1 – r) = 0.9/(1 – 0.1) = 1
So 9.99999…. = 9 + 1 = 10
Have you sorted out why we are getting 9.9999…. = 10?
A = B, B = 0
Suppose we have two numbers, A and B such that A = B.
Multiply both sides of the equation by B.
Then A × B = B × B or AB = B2
Now subtract A2 from both sides of the equation.
AB - A2 = B2 - A2
AB - A2 can be factorised to obtain A(B – A).
B2 - A2 can be factorised using the difference of two squares method to obtain (B + A)(B – A).
Thus we have
A(B – A) = (B + A)(B – A)
(B – A) is common to both sides in the equation, so it can be cancelled.
A = B + A
Now subtract quantity A from both sides to get
A – A = B + A – A
0 = B
But we started by assuming A = B, so what’s happened to A?
Inside is outside
Another well known area of mathematics that deserves the appellation of oxymoron is topology. We’ve all looked at our reflection in fun house mirrors. Our distorted, grotesque image -reminiscent of Edvard Munch’s painting “The Scream”- is an example of rubber sheet geometry.
Using this idea, can we consider a coffee cup to also be a doughnut? And can you remove your jumper without first removing your overcoat?
In a branch of mathematics known as topology, the answer to both questions is ‘yes’.
A coffee cup and a doughnut are topologically invariant. Topologically speaking, there is no difference between a coffee cup and a doughnut because one shape can be transformed by stretching and without tearing to obtain the other shape.
(Admittedly, I would still prefer to eat a doughnut rather than ingest the coffee cup!)
If you’re a fan of the TV character Mr Bean, you may recall the episode when he put on his swim trunks over his trousers and then removed his trousers without moving his bathing suit.
You might rightly ask if my ramblings on topology, coffee cups and Mr Bean’s bathing attire have any significance. As a mathematics teacher, I observe students in the same predicament as Mr Bean. They struggle at first but, by adhering to the principle of topological invariance, they persevere to demonstrate that seemingly impossible mathematical challenges can be resolved. Via judicious “kneading”, of the problem, they transform it to a manageable form whose solution is within their ken.
Topology can also be used to appreciate that there is no ‘inside’ or ‘outside’ in the Mobius strip shown below.
Mentally trace the dotted line to convince yourself of this truth.
Also of interest is to see what happens when you cut along the dotted line. Try it.
First construct the Mobius strip by giving a length of a paper a half twist and gluing together its ends.
You will then appreciate the humour in the joke:
Why did the chicken cross the Möbius strip?
To get to the same side!
The explanations to appreciate the oxymoronic investigations described above depend on the level of your mathematical training and proficiency.
But take heart that a common sense approach together with a clear explanation can rival the efforts of the most academic of mathematicians who inundate with a barrage of symbols and mathematical theorems.
Please leave comments as to how you might approach the problems.