Wearing an engineer’s hat, I do think e-to-the-pi-j-equals-minus-one really is a beautiful equation. Along with its close cousin e-to-the-pi-over-two-j-equals-j, which is delightful too.
And also useful and practical, because those little equalities help in encoding information into amplitude and phase of a radio carrier signal, and thereby underlie modern telecoms.
Your phone does it all the time: see for example pages 17-19 and 21-22 of this 4G standard:
This is for transmission, not just Fourier analysis in reception. The standard is replete with complex exponentials, because that equality has value as well as beauty. Not just a “trick”. Enjoy!
Name the first Friday after 3/14 Fourier Friday! Doesn't matter what date it falls on, and other dating conventions (ie everywhere not the US) will work with it.
Beauty and simplicity have an intimate relationship. In the end the most comprehensive definitions are analytic tautologies. God has no parts etc...
I think that's the beauty people find captured in Euler's identity. Each part signifies something ostensibly unrelated in its derivation and definition from the point of view of experience, but comprehension makes the relationship seem entirely trivially true almost without really saying anything at all.
Pi was originally defined as the ratio of the circumference to the diameter of a circle. If one (reasonably!) asserts that the radius of a circle is it's defining feature, not it's diameter, then you could also argue that pi 'should' be 6.28.
Euler's formula then becomes e^i(newpi)=0, which seems fairly unremarkable and certainly not worthy of 'most beautiful formula ever'.
Wearing an engineer’s hat, I do think e-to-the-pi-j-equals-minus-one really is a beautiful equation. Along with its close cousin e-to-the-pi-over-two-j-equals-j, which is delightful too.
And also useful and practical, because those little equalities help in encoding information into amplitude and phase of a radio carrier signal, and thereby underlie modern telecoms.
Your phone does it all the time: see for example pages 17-19 and 21-22 of this 4G standard:
https://www.etsi.org/deliver/etsi_ts/136200_136299/136211/08.09.00_60/ts_136211v080900p.pdf
This is for transmission, not just Fourier analysis in reception. The standard is replete with complex exponentials, because that equality has value as well as beauty. Not just a “trick”. Enjoy!
I too am underwhelmed by pi day.
I suggest that we do an Easter like thing:
Name the first Friday after 3/14 Fourier Friday! Doesn't matter what date it falls on, and other dating conventions (ie everywhere not the US) will work with it.
Beauty and simplicity have an intimate relationship. In the end the most comprehensive definitions are analytic tautologies. God has no parts etc...
I think that's the beauty people find captured in Euler's identity. Each part signifies something ostensibly unrelated in its derivation and definition from the point of view of experience, but comprehension makes the relationship seem entirely trivially true almost without really saying anything at all.
Pi was originally defined as the ratio of the circumference to the diameter of a circle. If one (reasonably!) asserts that the radius of a circle is it's defining feature, not it's diameter, then you could also argue that pi 'should' be 6.28.
Euler's formula then becomes e^i(newpi)=0, which seems fairly unremarkable and certainly not worthy of 'most beautiful formula ever'.