However did Euler come up with the Euler’s identity?
Background
On friday, we finished the #AI and #cybersecurity course.?
That was the last of the teaching at the #universityofoxford? before the next major course starting in January on? see? my course on AI at the University of Oxford? ???
Since I have nothing to do on the friday evening! I have gone back to my favourite pastime of asking #chatgpt abstract maths related questions such as mathematical concepts beyond space and time??
So, this time, the question is:?
However did Euler come up with the Euler’s identity?
Euler’s Identity
I think Leonhard Euler was the greatest mathematician ever for formulating Euler’s Identity?
Euler's identity is a remarkable mathematical equation that elegantly combines five fundamental mathematical constants into a single equation:
Where
Euler's identity is celebrated for its beauty because it links:
The question is - however did Euler come up with something like this??
Wikipedia and other sources do not explain exactly how euler came up with the euler’s identity??
Hence, my fri evening abstract qs to chatGPT
I am of course also interested in using these ideas in my forthcoming book on the mathematical foundations of data science.?
How did Euler come up with Euler’s identity
?
He derived its Taylor series expansion:? This gave a way to express the exponential function as a series of? real numbers??
Then, a genius moment, when Euler noticed that the Taylor series the exponential function matched the Taylor series expansions for cosx and sinx c?
Now, considering the special case of x =? π
For me, one final piece of the puzzle was: a recap of Taylor’s series.??
Taylor series is a mathematical tool that represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. It is widely used in calculus, mathematical analysis, and numerical approximations.
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also
Euler was studying the exponential function , which was already well-known as the solution to the differential equation means
?So, there you have it :)?
PS I can write latex in word but not sure how in linkedin. the images are from chatGPT
If you want to study with me see? my course on AI at the University of Oxford? ???
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Image source - my euler’s identity tshirt - I really have this t shirt
MMM Enabler at UHM Uncovering Hidden Meanings
5 天前I am looking forward to read your new book! Congratulations! Silvino Santos
Managing Partner at zConatus
5 天前Well, math is just a language and as such it admits hyperboles and all kinds of figurative language. It took more than 50 years for this, so called identity, to be accepted (though I like the more adequate term: imposed). For instance: there are many identities that equals to 1. The trigonometric identity that was used constrains the answer to a single cone of possibilities in an infinite range. Therefore I say it IS imposed. Therefore I say it IS a constraint where should be none. We can also argue that < i > is another methaphor ... invented and very badly used for centuries... Math: the ancient art of torturing symbols until they fit wherever you intended.
Data Scientist | M.Sc. Physics & M.Sc. Engineering
5 天前I remember deducing the identity when I learned about the Taylor series expansion of e^x. I basically followed the same steps (substituting for ix just out of curiosity) and I initially thought this was a mistake; it didn't make sense to me that e^iπ was equal to -1. However I asked my maths teacher, and she indeed confirmed that was called Euler's identity and my mind was blown away
Staff ML Engineer | Meta, Roku, Walmart | Sometimes known as MLWhiz | Making AI Simple Again(MASA)
5 天前One of the best identities in Math.