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In answer to obywan’s question:
The physicist Richard Feynman called the formula it is derived from “one of the most remarkable, almost astounding, formulas in all of mathematics”.
I’m sure you’ve seen all the symbols in the identity before, but isn’t it weird how they are not connected in any evident, but combine to give such a normal result?
The next part is an excerpt from Surein Aziz’s article the Beauty in Mathematics:
'So, why does this happen? You might think that it is down to some really complex idea — how do we even take a number to the power of i? Well, actually, it isn’t too difficult to see how Euler’s identity comes about - that is one thing that makes the identity so wonderful! But first you have to see Euler’s formula, which leads to his beautiful identity, in full generality:
Doesn’t look quite as nice and neat now, does it? But don’t be put off. To understand how this formula comes about, we need something called Taylor series. These are just a way of expressing functions such as sin (x) or cos (x) as infinite sums. They were discovered by the mathematician Brook Taylor (who was also part of the committee which adjudicated the argument between Isaac Newton and Gottfried Leibniz about who first invented the calculus).
The Taylor series for the function e^ x is:
where n! denotes the product
You can verify this Taylor series using a calculator: choose a number $x$ and see what value the calculator gives you for $e^ x$. Now use the calculator to work out the value of the sum
for as many terms as you like, that is for a number $n$ as high as you like. You will find that the result very nearly equals the result you got for $e^ x$ and the more terms you add to the sum, the closer the two results become. At some point the two results will be the same on your calculator, as their difference becomes too small for the calculator to detect. In “reality”, the two results are equal when you have added an infinite number of terms to your sum.
The Taylor series for the other two functions appearing in Euler’s formula are:
Again you can check this using your calculator, bearing in mind that the angle x is measured in radians, rather than degrees.
Now let’s multiply the variable x in the Taylor series for e^ x by the number i. We get:
But certain powers of i can be simplified – for example, i^2 = -1 by definition, and so i^3 = -i and i^4 = +1, and so on. So we can simplify the above to:
We can gather the terms involving i together to give:
Now notice that these two series are the same as the series for sin (x) and cos (x) from earlier, so we can substitute these in to get:
which is Euler’s formula!
All we have to do now is substitute x = pi. Since sin (pi ) = 0 and cos (pi ) = -1 we get:
So you see, after a sequence of fairly complex mathematics we arrive back where we started — at the (seemingly) simple numbers 1 and 0. That is what I think is so beautiful about this identity: it links very strange numbers with very ordinary and fundamental ones. Seeing why it works feels a bit like treading a little-known path through the mathematical jungle to reach a secret destination somewhere in the thick undergrowth.’
And what about Euler? (from storyofmathematics.com)
Today, Euler is considered one of the greatest mathematicians of all time. His interests covered almost all aspects of mathematics, from geometry to calculus to trigonometry to algebra to number theory, as well as optics, astronomy, cartography, mechanics, weights and measures and even the theory of music.
Much of the notation used by mathematicians today - including e, i, f(x), ∑, and the use of a, b and c as constants and x, y and z as unknowns - was either created, popularised or standardised by Euler. His efforts to standardise these and other symbols (including π and the trigonometric functions) helped to internationalise mathematics and to encourage collaboration on problems.
He even managed to combine several of these together in an amazing feat of mathematical alchemy to produce one of the most beautiful of all mathematical equations, eiπ = -1, sometimes known as Euler’s Identity. This equation combines arithmetic, calculus, trigonometry and complex analysis into what has been called “the most remarkable formula in mathematics”, “uncanny and sublime” and “filled with cosmic beauty”, among other descriptions. Another such discovery, often known simply as Euler’s Formula, is eix = cosx + isinx. In fact, in a recent poll of mathematicians, three of the top five most beautiful formulae of all time were Euler’s. He seemed to have an instinctive ability to demonstrate the deep relationships between trigonometry, exponentials and complex numbers.
Hi! Been following your blog for some time, and enjoyed it much. My interest in math was awaken when i tried parametric design. Rhino and Grasshopper is my tool of choice. Quite often I have to solve problems with the help of geometry, because my knowldege of math is, lets say, limited XD But its not always possible or effective to go that way. So, my question is: how to divide an ellipse into equal parts and with equal distances between division points? :) Sorry for grammar, I'm from Russia. )
Здравствуйте! Рад, что вы наслаждаетесь мой блог!
It is pretty tricky to split an ellipse equally, here are some instructions for a draftsman’s method that may be what you are after?
Alternatively you could find the ellipse perimeter, then use line integration….here are a couple of instruction sets that may help.
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