An Intuitive Guide To Exponential Functions & e

you might have intended to write “abstruse jargon” instead of “obtuse jargon,” but either debatably works, I think

if you meant abstruse, I (try to) mentally distinguish them using their etymologies (left as an exercise for the reader)

Love this! Wish I knew this 15 years when I returned to calculus, and 23 years ago when I started it!:slight_smile: Thanks Khalid!!

Thanks Chewie and Michael!

Never seen e before (I’m 16) but this explained it pretty clearly in under 10 mins.
Thanks!

Wow Kalid. Thank you very very much. I see you already have hundreds of comments but I’d still just like to add mine saying how good this explanation was.

Wow…wowowowow!

Thank you so much for these articles, really. I have a similar obsession with asking why recursively, even when I finally reach the axiomatic level (“but why did we choose these axioms??!?!” :D).
Unfortunately I started math very late in life (mid-30’s) and my brain is really feeling the pain of not being flexible enough anymore to easily synthesize information in new ways. I sometimes spend 5-10 hours on a single concept just holding my head in my hands until it clicks, only to realize that I’m just scratching the surface of how much intuition I could potentially develop regarding that one concept.

The moment of insight is very rewarding, but my ratio of [energy + time expanded] to [depth of insight] is a little high for my taste.
This is why really appreciate your explanations. Even though I still slog through them, making sure I understand everything fully (whoever said “Math is not a spectator sport” really hit the nail on the head), your insights act as a guiding force through a jungle of confusion. I still have to find the energy to walk in the right direction, but that guidance is priceless.

Thanks again for your insights. They make me (and I’m sure many others) a fundamentally richer human being.

Nicely done…thanks!

As Bob, comment #3, stated, “e” and logs were tools I used to pass exams but now, 40 years after first encountering “e,” I’ve decided to try to understand the concept. Once I become comfortable with “e” and logs, I hope to tackle some of the statistics tests that also fell into the category of “I could apply the rule if I knew the rule applied, but what was really happening here and why this test/rule and not another?” It’s nice to have the time to relearn and to have the internet.

This discussion on ‘e’ has fired up my brain at the age of 68 as nothing has done before. Just one point.As I now see it, the idea of compound interest in bank and the idea of ‘e’ don’t run parallel all the way.At some point they diverge.Am I completely wrong if I say, idea of ‘e’ is a subset of the idea of compound interest? Principle is same. The rate at which an entity changes is proportional to the size of entity at the moment.Frequency of change can be very slow to continuous.
In discussing ‘e’, the second term inside the bracket has to be, .01, .0001, . 000001,.0000001 or some such because we assume that rate is 100%, i.e. the entity is going to double at the end of the period.We take the second term to .0001,.000001 etc in an effort to make the change almost continuous.Even if rate is not 100%,the frequency ,(‘number of chunks’) has to be equal to the rate so that the second term is .1,.01, .00001,.000001 etc. But in compound interest,rate of interest and number of chunks are allowed to be different.A bank can give a rate of , say 9%, and give it in say,27 chunks per year. Formula is then
(1+( (9/100) *1 ) /27 ) )^27
In this case, the idea of ‘e’ doesn’t enter at all.

Just as there is ‘e’ for one dimensional numbers,is there an equivalent constant where an AREA quadruples in a given period? Or even VOLUME becomes 8 times as large in a given period ?

I love this blog, thank you. Bryan from H.K.

Thank you so much for this article! Thank you for explaining mathematical concepts to me as though I am a human being!

“e is the base rate of growth shared by all continually growing processes.”

By reading this one sentence, I felt I finally grasped what e was all about.

Something I am constantly frustrated by is the way that mathematical and technical concepts seem to be written for someone who already understands the concepts - If I already know what the euler equation is all about, why the heck would I be looking it up on Wikipedia!? And often, once I finally understand a concept in spite of the jargon rich, verbose explaination available on the internet, I find I can explain it in a few simple sentences and drawings to almost anyone.

Thanks again for this site, please keep up the good work!

Hi Khalid,

Quick question if you have time. When you’re comparing 50% compounded continuously with 100% compounded continuously, why do you choose the interest subdivision as your comparison basis and not the number of times that the total interest rate is compounded.
What I mean is, why say “with 50% interest we can compound 1% 50 times, while with 100% interest we can compound 1% 100 times”? Why not say “If we break 50% into 50 pieces, we get 1% compounded 50 times or (1+1/100)^50, while if we break 100% also into 50 pieces, we get 2% this time but still compounded 50 times, which gives us (1+1/50)^50”, and compare those two quantities instead?

Why is the comparison that matters the one based a common subdivision of the total interest rates, and not the one based on a common number of subdivisions of the total interest rates?

Thank you, and great writeup by the way!

Hi Kalid,

Quick question if you have time. When you’re comparing 50% compounded continuously with 100% compounded continuously, why do you choose the interest subdivision as your comparison basis and not the number of times that the total interest rate is compounded.
What I mean is, why say “with 50% interest we can compound 1% 50 times, while with 100% interest we can compound 1% 100 times”? Why not say “If we break 50% into 50 pieces, we get 1% compounded 50 times or (1+1/100)^50, while if we break 100% also into 50 pieces, we get 2% this time but still compounded 50 times, which gives us (1+1/50)^50”, and compare those two quantities instead?

Why is the comparison that matters the one based a common subdivision of the total interest rates, and not the one based on a common number of subdivisions of the total interest rates?

Thank you, and great writeup by the way!

Hi Kalid.

Awesome article, as always, but under the “what about different rates section” I don’t understand why we can compare those two formulae for different n values. If they have different n values, surely you can’t compare them?

Great question, it’s a part of the article I’d like to refine more (still not happy with that part entirely).

n represents the quantity we’re going to use to get to a given level of accuracy. Let’s say we want to be accurate to compounded changes of 1%.

When n is “enough” to give us changes of 1% in size, it means:

50% = 50 chunks of 1% interest, compounded together = (1.01) * (1.01) * … [50 times] = (1.01)^50
100% = 100 chunks of 1% interest, compounded together = (1.01) * (1.01) * … [100 times] = (1.01)^100

It’s true that n is different in each case (to reach that level of accuracy) but the two patterns are now compounding changes of the same size, and can be compared. We can see that 50% interest involves half the number of “1% changes” as 100%.

We can pick a tighter precision (compound changes of .1%) and we see the same result: 500 changes of .1% vs 1000 changes of .1%.

Not sure if that helps, my thinking is “No matter what size we pick as the compounding level, the 50% part has half the changes as the other”.

Not bad. I think you lost me at the “But what does it all mean?” section for some reason. Maybe it was at the part to where you stated: “So, if we start with 1.00 and compound continuously at 100% return we get 1e. If we start with 2.00, we get 2e. If we start with 11.79, we get 11.79e.” Wouldn’t it be 1+e or 2+e? I’m also confused about the concept of e being a speed limit. Could you go into further detail? I really want to grasp this insight. I really liked your article about the integration analogy. :smiley:

Kalid, if I could, I’d give you a chocolate box for this awesome explanation. Thank you so much!

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