An Intuitive Guide To Exponential Functions & e

Hello Kalid,

Thank you for sharing your intuitions and helping others build their own.

I have a question concerning what seems to be contradictory for me:

We say a %100 growth rate leads to double the original quantity X(%100+%100) = X(1+1) = 2X = X(%200) and that is how we can rewrite it. The %200 is the net result after growth that includes both the original quantity (%100) and the increase/growth part (another %100).

Now if we have a decay rate of %100, which can be re-written as X(%100 - %100) = X(1-1) = 0X = X(%0), which agrees to what you said about the decay of 10kg of radioactive material at a rate of %100 (without compounding of course).

However, the formula for infinitely compounded decay that you have reached near the end of your article: 10 / e^3 or lets take the infinitely compounded decay by one year to be 10 /e ^ 1 = 10 /2.7 which is around 3.7 (less than half the original quantity)

Do not you think that non-compounded decay should be written as X / 2 analogous to non-compounded growth which can be written as 2X. In other words, concerning your example, dividing the 10kg by 2 is slightly larger than 10 / e because as you said infinitely compounded decay rates get smaller and smaller after time.

But X / 2 is not equal to 0X, the one I mentioned above. note X /2 = X(1 - 0.5) which implies a decay of half the original amount only and would result in also half the amount only ( which is contradictory to the wording you chose, i.e. growth of %100 of the original quantity)

Can you please reflect on this apparent contradiction.

Dear Mr.Kalid thanks for your share
and I have a confuse that you said 1.33^3 measure simple interest. I think may be not, 1.33^3 can also means 3 compound interest in one unit time, because tri-compounded Interest’s Trajectory is change, isn’t right?:worried::worried::worried:

I think may be 3*1.33 means 3 periods of 33% simple interest