What is the golden ratio doing here?!

And,

Then the solutions are:

Source: https://www.wolframalpha.com/input/?i=xy%3Di%2C+x%5E2%2By%5E2%3D1

Just messing around with stuff. I basically combined a circle with a complex hyperbola. Maybe you can get a nice connection with and

Looking at different functions with Taylor series/maclaurin series in polynomial vector space gives a pretty nice analogy

If you type have the full ma

If you look at the polynomial vector space and just ignore the coefficients and only look at if there is an entry or not, than you find a nice summary of these functions

As you can see cos and sine are orthogonal.

Cool new way to see zero divided by zero!

So we learned in school that if you ask what 0/0 is you have to say: undefined or you are not allowed to do that! That’s illegal!. Officially that is correct but it is also kind of boring. Like if you are allowed to take the root of negative numbers, surely you divide by zero?

I’m reading a really charming book called Shape which I highly recommend. When I read books I always have the feeling that you get to know the author. It’s like climbing in the auther’s head. Sometimes I cannot continue the book because I can’t stand the author because certain views or so. Anyway, this author is really nice it seems through his book.

But back to dividing by zero. So if you talk about it in a specific example, i.e. you add information, you could define it! For example if we talk about the ratio of conversion rate between fahrenheit en celsius, you have to divide both numbers. E.g. 98/37 = around 2.4. That is the conversion rate. But this should always be the same, so 0/0 is 2.4 in this case. He also uses the example of area to perimeter for a square. For a square with length 1 it is 1 / 4 = 0.25. If you make the square smaller the ratio will go to zero, so that would mean that an appropriate answer in this case for 0/0 would be 0!

Why is Euclid’s book called ‘The Elements’?

I wonder why the second most read book in history is called ‘the Elements’? It is about math and mostly geometry. What does this have to do with Elements? It cannot refer to elements in chemistry because those were obviously not known yet.

Another definition is given:

A component or constituent of a whole or one of the parts into which a whole may be resolved by analysis.

So it could be that the axioms and propositions Euclid gives in his book are like the components of math and in this way it is called the ‘Elements’.

Another reason could be because it is named after Plato’s Elements or Plato’s Platonic Solids. These are 3D geometrical objects where all the surface and angles between the surface are all equal. 5 solids satisfy these criteria:

Fire

Earth

Air

Water

‘Heavens/Aether’

 

Plato was amazed by these and thought that they represented the elements as regarded as important that time: fire, earth, water, air and the fifth one he corresponded to the ‘heavens’. Aristotle later corresponed to ‘quintessence/aether’.

Euclid proves propositions about these in the last and most difficult book of the Elements. I wonder that this was the whole goal, and that is why it is called the elements. To show these beautiful symmetric objects and the importance they thought these had in physical world.

Ironic

Obviously the world isn’t build up with these objects even though how nice they look. But what make them so nice is their symmetries. And in a certain sense, symmetry and group theory (studies of symmetries) are really important, maybe the most important thing using elementary particle physics and physics as a whole. Physicist danced around this principle for ages, but Noether appreciated this fully.

 

 

 

Russell’s Paradox

 

Mechanical calculator dividing by zero

Image of infinities by Gamow

From the book: one, two, three …. infinity:

Best but rarely used reason why complex numbers should exist

Closure

The set R under exponentiating is only closed if complex number are allowed to exist. ‘Closed’ means that algebra can be done in that set with that operation. It allows to solve the equation Ax=B. Algebra also means solving.

If not the set isn’t closed under a commonly used operation. It feels like building a home but with an open roof.