Intuition for i to the power i | Lockdown math ep. 9

Hi Grant thank you for the series, they are soooo good. You know what? As an 8th grader, your videos really open my mind and thank you for doing all that : )

Thanks Simón! It's been fun.

3blue1brown

If you start with df/dx = f and f(0)=1, you get that the only solution is f(x) = exp(x). In fact, if you start with df/dx = f, the only solutions you get are C*exp(x) for some constant C. This is usually shown in an intro class on linear ordinary differential equations. See https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem for more info on the theorem that exp(x) is the unique solution. You cannot derive f(a+b)=f(a)f(b) from df/dx = f alone. Consider f(x)=2exp(x) and plug in some values. If you add the constraint that f(0)=1, then yes, you can derive f(a+b)=f(a)f(b) since as stated above f(x)=exp(x) is the only solution. If you have f(a+b)=f(a)f(b), you also cannot deduce that df/dx=f. Consider f(x)=1 everywhere. This satisfies the first property, but df/dx = 0, which is not equal to f. I don't know if there are more interesting counterexamples.

Aman Karunakaran

In the ep.7, you showed us if the exponential property f(a+b)=f(a)f(b) and the derivative of f exists, the coefficient of the derivative is ln(A), thus if A = e, then df/dx=f if f=exp(x). I wonder if we are given two properties: f(0) = 1, df/dx=f, what do we get? I got exp() and here is simplified version (not exp(x), but exp(1)). https://youtu.be/Pxg-2F-nMZk But I have a question: can we derive the exponential property f(a+b)=f(a)f(b) from df/dx=f? Or df/dx=f exists from f(a+b)=f(a)f(b)? I would like to know the relationship between the two properties: df/dx=f and f(a+b)=f(a)f(b). Or these are independent. From f(a+b)=f(a)f(b), you show f(0)=1 this time. So I feel this is a missing ring for me. (This is the most time-consuming question I ever had if I includes the above video.)

Hitoshi Yamauchi

Hey Grant, I just wanted to thank you for the effort you are putting into this series, so sad it's gonna be over

This special case can then be extended in high school math to solving equations of the form $a+bj = f(x+yj)$.

Reginald Carey

it also helps in the transition from a number line to the Cartesian plane. Except students would think x-i coordinate system instead of x-y.

Reginald Carey

in that mindset sqrt(-1) might not seem such a leap of faith to young learners.

Reginald Carey

Grant, just thought of a great way to introduce complex numbers very early. $a+bj = f(a)$ instead of $b=f(a)$ when ever someone talks about a function over a real (or complex) number.

Reginald Carey

Potentially! It's on the list.

3blue1brown

"keep loving math" could become Grant's equivalent of "Fly safe" by Scott Manley.

William Smith

Can you do a video on the derivation of the gamma function? How does factorials extend in to negative numbers? Complex numbers?

PseudsPie

I followed about half of that at one sitting but will be going over it again. Your passion is very infectious.

Simon Allen

SECOND LAST IN THE SERIES!? NOOOOOOO!

Edward N