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
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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.
?
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? ???
If you want to work with me see AI opportunities we are recruiting ?
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Image source - my euler’s identity tshirt - I really have this t shirt
My job was a general worker. The time I was coaching. I was helping teachers to learn from them. That's why I say I was coaching start.
3 个月by comparing the serie expansions of the exponential and trigonometric expressions. The formula was first published in 1748 in his foundational work Introductio in analysin infinitorum
MMM Enabler at UHM Uncovering Hidden Meanings
3 个月I am looking forward to read your new book! Congratulations! Silvino Santos
Data Scientist | M.Sc. Physics & M.Sc. Engineering
3 个月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 | 1:1 @ topmate.io/MLwhiz
3 个月One of the best identities in Math.