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Category Archives: Mathematics

The answer is 47

Blog Off Posted on 19th April 2013 by Simon Jensen21st April 2017  

I have been working with a mathematical proof for a whole day … just to finally give up. The technical details are quite abstract, so I’ve created a PDF file with all the information you need.

I have “discovered” a new arithmetic function which has a funny property. I just got the sequence for this function approved on OEIS (sequence A224914). But no one has proved the conjecture, so far. It’s quite thrilling!

Are you a math student or an expert on number theory? In that case, you might want to take a look at this:

The answer is 47 (open PDF in a new window)

Can you prove this new conjecture?

Posted in Mathematics | Leave a reply

Definitions and axioms – and inconsistency

Blog Off Posted on 10th June 2009 by Simon Jensen21st March 2016  

Previously, I studied mathematics for a couple of years, and thus walked the first few steps along this beautiful road. In the mathematical conventions, I discovered, from time to time, a lack of consistency which seem to have passed by unnoticed, or maybe not?

Here’s an example:

Sometimes, when we want to make a new definition by the use several axioms, a strange kind of “semi-inconsistency” arises.

Let us assume we want to define a new concept, C, by using a description, D (which can refer to one or more previous definitions), together with a requirement that D satisfies some axioms (let us name them A1, A2 and A3). Such a definition would often be stated like this:

Definition: C is a D that satisfies the following axioms:
i) A1
ii) A2
iii) A3

By this we mean that all the axioms (in this case three of them) must be satisfied by D. But then we could as well formulate the definition in this way:

Definition: C is a D that satisfies the following logical statement: A1 ⋀ A2 ⋀ A3

which can be written in several different ways (since the logical operator ⋀ is commutative).
An example could be:

Definition: C is a D that satisfies the following statement: A3 ⋀ A1 ⋀ A2

and thus we see that the original definition could as well be written this way:

Definition: C is a D that satisfies the following axioms:
i) A3
ii) A1
iii) A2

In short, if we want to define something by using a list of axioms (strictly speaking an unordered list, even though it is often numbered), then the order of those axioms should not have any influence on the definition. This means we need to express all the included axioms in such a way that they are independent of one another. But this is not always the case with some of the fundamental mathematical definitions, see the example below.

In several books discussing fundamental abstract algebra, the following definition of a group is given:
Normal definition of group
In this definition Axiom ii) must be presented before Axiom iii), because otherwise the symbol e (the neutral element) has not been described when it is used in Axiom iii).

You could, in principle, imagine very complicated definitions that contain a list with hundreds of axioms. Then, imagine that these hundreds of axioms depend on each other in a complex way.

I suggest a new convention for this kind of definitions:
In all definitions that use a list of axioms, we should always state the listed axioms in such a way that they do not depend on each other.

As an example, I rewrite the group definition mentioned above:
New definition of group
This new definition is better, since the three group axioms now “have the same status” (i.e. they are mutually independent, and commutative) – and the order in which we list the axioms is unimportant here. We would have to define the term neutral element separately, but that is no problem (actually, it is better). What is your opinion here?

Posted in Mathematics | Tagged axioms, definitions, mathematics | Leave a reply

Jensen’s Paradox

Blog Off Posted on 4th June 2009 by Simon Jensen19th March 2016 15

Everyone who is interested in mathematics, physics or common philosophy knows the most famous of Zeno’s paradoxes, namely the one with Achilles and the tortoise. Many similar paradoxes exist, and when you understand the first one it is easy to understand them all.

But one day when I was playing with my thoughts, I found a “paradox” that actually requires some extra thinking. We can call it Jensen’s Paradox (don’t mix it up with Jensen’s inequality or The Jensen’s Inequality “Paradox”, a paper on economics by Susan Woodward).

Here it comes:

Consider the figure below which contains the points A, B and C:
The points A, B and C
If we want to move ourselves from the point A to the point C, we can choose the following path: First we go in a horizontal line to the point B and then we continue in a vertical line upwards to the point C. If the distance is 1 between A and B, and the distance is 1 between B and C, then the total distance (let us call it S) will be 1+1, which is 2, when we choose to go this way.

Now, consider the next figure:
Horizontal half the way to B, and then vertical half the way to C
Above, we move ourselves horizontally half the way to B, and then vertically half the way to C. We repeat these two steps once and arrive in the point C. The distance S becomes 2 again.

Let us develop this procedure a bit further:
We have split every previous step into two new steps
We have now split every previous “step” from the figure above into two new steps. The distance S is still 2.

We could continue in this way for some time. Below, we have 8 steps:
The distance S is still 2

And now 16 steps:
The distance S is still 2

And 32 steps:
The distance S is still 2

One more time. Here, we have 64 steps:
The distance S is still 2
The distance between A and C have now been split into 128 small distances (64 horizontal and 64 vertical), and it is easy to see that the complete distance, S, is still 2. With a little mathematics, this can be written in the following way:
Sum of 64 steps
Let us now imagine that we split the distance into many horizontal and vertical parts (let us assume we then have n steps). Then it looks like this:
Sum of many steps
The sum of all the parts (the small vertical and horizontal distances) will always be 2, i.e. the total distance is always 2, even if we split the distance into an infinitely large number of steps:
Sum of infitely many steps
We conclude that the distance below equals 2 (since the limit above is 2):
Infinitely many steps
But according to Pythagoras’ theorem, the hypotenuse of the triangle below is equal to the square root of 2:
Pythagoras theorem used here
So, here is the “paradox”: We have now seen that the distance S between the point A and the point C is always 2, and we do know, simultaneously, that it is always the square root of 2 (according to Pythagoras). Hope you like it!

Posted in Mathematics | 15 Replies

A small exercise

Blog Off Posted on 4th June 2009 by Simon Jensen10th January 2015 2

When calculations are performed, people often use several arithmetic and algebraic rules without thinking about how these rules really work. The following example illustrates this fact:


So: 1 equals -1.
Apparently at least one of the equal signs above must be wrong. But which one? Or which ones?

I have noticed that many high school teachers find this problem difficult to solve. And that says more about the teachers than the problem!

Posted in Mathematics | 2 Replies
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