If you have any math discovery or fun problem that you want me to present in a video, please leave it here (with solutions hopefully). I will try my best to make it into a video for you and will credit you in the video.

I wanted to see if I can put a hypothetical situation into an equation. I was thinking it dealt w weighted averages & possibly system of equations. But I wasnt sure about the system of equations part. I think it deals with weighted averages b/c of the consumer price index/basket of goods was something I used to put together a financial map b/f. So this was possibly going to be a continuation of that financial map.

I've also started my research w/ the consumer behavior/economics part. I got an idea from acctng bc of journal entries & bal sheets that making an equation for certain things would be a really cool thing & useful when considering what things to buy while on a pre-existing budget, how much to buy, & maybe when to buy those items. Also I wanted to be able to view in equation form how a budget would be affected. I have a spreadsheet already created for food bill. Which is 1 part of the equation/s I think

Dear Steve, the internet is full of videos on the topic of euler's number, e. However I find that there is something that I don't like about most explanation videos: instead of explaining e, most videos end up explaining exponential functions in general. That's why I would like to present the following explanation, which I think is far more clear:

First consider trying to find the derivative of

f(x) = 10^x

We do this from scratch, starting with the definition of a derivative

f'(x) = 10^(x+h) - 10^(x) / h

= 10^x * (10^h - 1)/h

Note here that the first term 10^x is simply f(x), and that the second term (10^h-1)/h is simply f'(0). Here we are stuck, because we don't have any method to tell what the value of f'(0) is.

WOULDN'T IT BE NICE if f'(0) would just nicely equal to 1? That would solve all our problems for sure! Instead of 10^x, consider a more general form a^x. We can safely claim that there must be some value of "a" where f'(0)=1. To see why this must be true, have a look at https://www.desmos.com/calculator/ukstxtoen1

If you want, you can then go on about methods to find a good approximation for e, since it is just a number on the number line. But I'm just going to leave it here.

Summarizing, we have "invented" Euler's number because this was our only hope at every finding the derivative of an exponential function. No need to convince anyone why it is useful.

I hope you like this approach, let me know what you think if you see this :)

The specific identity that I'm looking forward to you to prove is this:

Founded by American Mathematician Andrew M. Gleason, he used it to construct the regular heptagon.

https://en.wikipedia.org/wiki/File:01-Siebeneck-Tomahawk-Animation.gifHey, do u create custom exercises that deal with consumer behavior?

It's only enough for a tiktok but I found this and left it in a youtube comment

Explanation:

2 = 1+1

2 = 1+(3/3)

2 = 1+(3/2+1)

2 = 1+(3/2+(4/4))

2 = 1+(3/2+(4/3+1))

2 = 1+(3/2+(4/3+(5/5)))

2 = 1+(3/2+(4/3+(5/4+(6/...))))

Dear Steve, the internet is full of videos on the topic of euler's number, e. However I find that there is something that I don't like about most explanation videos: instead of explaining e, most videos end up explaining exponential functions in general. That's why I would like to present the following explanation, which I think is far more clear:

First consider trying to find the derivative of

f(x) = 10^x

We do this from scratch, starting with the definition of a derivative

f'(x) = 10^(x+h) - 10^(x) / h

= 10^x * (10^h - 1)/h

Note here that the first term 10^x is simply f(x), and that the second term (10^h-1)/h is simply f'(0). Here we are stuck, because we don't have any method to tell what the value of f'(0) is.

WOULDN'T IT BE NICE if f'(0) would just nicely equal to 1? That would solve all our problems for sure! Instead of 10^x, consider a more general form a^x. We can safely claim that there must be some value of "a" where f'(0)=1. To see why this must be true, have a look at

https://www.desmos.com/calculator/ukstxtoen1If you want, you can then go on about methods to find a good approximation for e, since it is just a number on the number line. But I'm just going to leave it here.

Summarizing, we have "invented" Euler's number because this was our only hope at every finding the derivative of an exponential function. No need to convince anyone why it is useful.

I hope you like this approach, let me know what you think if you see this :)

Kasper Arfman