Euler's Identity (Complex Numbers)

How the Fourier Transform Works, Lecture 4 | Euler's Identity (Complex Numbers)

In order to describe the Fourier Transform, we need a language. That language is the language of complex numbers. Complex numbers are a baffling subject but one that it is necessary to master if we are to properly understand how the Fourier Transform works. What is the imaginary number “i” and why it is so useful to us when dealing with the Fourier Transform?

This is the sixth in this series of videos which takes a new and visual look at the maths behind the magic of how the Fourier Transform works.

For a comprehensive and visually intuitive exploration of the Fourier Transform and its workings, I invite you to explore my book series on the Fourier Transform available at:

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Thanks for watching the video How the Fourier Transform Works, Lecture 4 | Euler's Identity (Complex Numbers)

[gk]

What people have to understand is how brilliant these guys were. They had no internet, few if any textbooks. They had to reason things from first principles, so much original. Just stunning

[neilduran]

It is over 20 years since I studied the maths of Euler but this is by far the best explanation I have ever seen. I wish I had seen this video back then. Students of today have it a lot easily than years ago, when you were expected to just get it!

[roger]

For a long time 0 didn't exist, and some people who stupidly claimed that nothing existed had their heads bobbed. Now imagine imaginary numbers. That was like claiming the earth wasn't flat.

[bayestraat]

You should also highlight how euler's identity is nicely shown with multiplication of complex numbers as vectors around a circle plot on the imaginary plane. And how to maintain symmetrical values working out the power spectrum density in FFT.

[JnasBern]

To me what is beautiful is that you have a number with infinite and random digits that is related to exponencial growth/decay, then you raise it to the power of a number that we find impossible to solve and so we call it imaginary, and to another number with infinite and random digits that is related to circles and it's geometry, and then you add a single unit, probably the most basic number that we know, to all of this only to get what we call "nothing"

[petrophilip]

I have seen and used the constant "e" in the study of calculus, complex numbers, infinite series, natural logarithms etc, but no one explained what the number is. This is the simplest explanation I have seen. It takes a special kind of a skill to correctly explain a complex concept in simple terms. Thanks Mark Newman.

[mp]

Now I am waiting for Euler's Supremacy and Euler's Ultimatum...

[isaacrajagopal]

It is not just beautiful "in mathematical terms, " it is just BEAUTIFUL. Period.

[jeanpaulniko]

My first ever comment in 10+ years of watching YouTube. Mark, you nailed it! This video has me feeling ecstatic. You have shown me the connection between sin, cos, i, e, and π as presented in Euler's famous identity. This reveals the deep foundation that underlies all of classical math and ties everything together. Now I have seen the light! Thank you so much.

[audience]

It also links exponents, zero, addition, equality, the identity element under multiplication, and when expanded, trigonometry, division, factorials, and infinite series.

[jacquesjutras]

This gentleman Mark is a very good teacher he is a master.

[nanzhang]

I don't believe Euler named the number after his own name. From what I know Euler was a very modest man, he instead named the number e because it was the next available letter that was not already taken. Listen to the podcast of 'In Our Times' discussing this number.

[markproulx]

“I used to think math was no fun
‘Cause I didn’t know how it was done
But Euler’s my hero
‘Cause I now know that zero
Equals e to the j pi plus 1”

- Paul J. Nihan

[BlackNSB]

That sure as hell is beautiful--especially because, as a student, I didn't understand why this formula was so special. Great video.

[bronzekoala]

This is the best mathematica axplanation I've found so far on YouTube for anything.

[ChdBR]

This is the best explation about Euler's identity!
Thanks.

[martlock]

So beautiful that the simple identity e^(pi*i)+1=0 can link together the most important mathematical concepts (0, 1, i, e, pi) using the most fundamental mathematical operations (equality, addition, multiplication, exponentiation)!

[Soubhik.]

This video is so beautifully made ❤ Absolutely love it.

[Frieza.exe.z]

8:10 "The brilliant thing about mathematicians is that . . . when they are on their way to some wonderful mathematical discovery, they don't let a little thing like "Numbers NOT EXISTING" stop them." Is it safe to say HERO ?

[mingzih]

this explanation is insane, even a grade 9 student can understand.