Pi is the ratio between circumference and diameter shared by all circles. It is a fundamental ratio inherent in all circles and therefore impacts any calculation of circumference, area, volume, and surface area for circles, spheres, cylinders, and so on. Pi is important and shows all circles are related, not to mention the trigonometric functions derived from circles sin, cos, tan.
Just like every number can be considered a scaled version of 1 the base unit , every circle can be considered a scaled version of the unit circle radius 1 , and every rate of growth can be considered a scaled version of e unit growth, perfectly compounded. So e is not an obscure, seemingly random number. Let's start by looking at a basic system that doubles after an amount of time. For example,.
Splitting in two or doubling is a very common progression. Sure, we can triple or quadruple, but doubling is convenient, so hang with me here. As a general formula:. Clever, eh? So the general formula for x periods of return is:. Our formula assumes growth happens in discrete steps. Our bacteria are waiting, waiting, and then boom , they double at the very last minute. Our interest earnings magically appear at the 1 year mark.
Based on the formula above, growth is punctuated and happens instantly. The green dots suddenly appear. If we zoom in, we see that our bacterial friends split over time:. After 1 unit of time 24 hours in our case , Mr. Green is complete. He then becomes a mature blue cell and can create new green cells of his own. The equation still holds. But money is different. As soon as we earn a penny of interest, that penny can start earning micro-pennies of its own.
So, we earn 50 cents the first 6 months and another 50 cents in the last half of the year:. Sure, our original dollar Mr. Blue earns a dollar over the course of a year. But after 6 months we had a cent piece, ready to go, that we neglected! That 50 cents could have earned money on its own:. Who says we have to wait for 6 months before we start getting interest? Make sense? But see that each dollar creates little helpers, who in turn create helpers, and so on.
Why not take even shorter time periods? How about every month, day, hour, or even nanosecond? Will our returns skyrocket? Our return gets better, but only to a point. Try using different numbers of n in our magic formula to see our total return:. This limit appears to converge, and there are proofs to that effect.
But as you can see, as we take finer time periods the total return stays around 2. The number e 2. But with each tiny step forward you create a little dividend that starts growing on its own. When all is said and done, you end up with e 2. Aside: Be careful about separating the increase from the final result. What can we do here? Ah, something is emerging here. This is pretty interesting.
Because of the magic of exponents, we can avoid having two powers and just multiply rate and time together in a single exponent. It all works out.
Let me explain. This even works for negative and fractional returns, by the way. Examples make everything more fun. Read more about simple, compound and continuous growth.
These examples focus on smooth, continuous growth , not the jumpy growth that happens at yearly intervals. Suppose I have kg of magic crystals. How much will I have after 10 days? Well, since the crystals start growing immediately, we want continuous growth. This can be tricky: notice the difference between the input rate and the total output rate. The net output rate is e 2. In this case we have the input rate how fast one crystal grows and want the total result after compounding how fast the entire group grows because of the baby crystals.
If we have the total growth rate and want the rate of a single crystal, we work backwards and use the natural log.
My bank is generous and gives me the maximum possible compounding. How much will I have after 10 years? How much will I have after 3 years? We go a few months and get to 5kg. Half a year left? We wait a few more months, and get to 2kg.
We get 1 kg, have a full year, get to. As time goes on, we lose material, but our rate of decay slows down. If you want fancier examples, try the Black-Scholes option formula notice e used for exponential decay in value or radioactive decay. This article is just the start — cramming everything into a single page would tire you and me both. Learn Right, Not Rote. Home Articles Popular Calculus. Feedback Contact About Newsletter.
Math books and even my beloved Wikipedia describe e using obtuse jargon: The mathematical constant e is the base of the natural logarithm. Let's see what happens:. In essence you are doing this:. It is a single one. Well, I'll be honest and say looking at it I'd have no idea. It's a good question because it seems to be that this is a violation of "the limit of a product is the product of the limits". If you review the theorems in any texts you'll have, you'll note you can "move inside an exponent" with limits only when the exponent is fixed, for instance.
This is because of the criteria that are necessary to evaluate function composition. We can perform the usual evaluation and substitution provided. This limit has, to coin a phrase, an unremovable composition. So if the limit exists, it must be found in some other fashion.
I think this has been asked before but I am not able to find a link right away so here is the answer. While solving concrete problems in mathematics we usually follow certain rules and do manipulation on mathematical symbols based on these rules.
We don't perform such manipulations based on our whims and fancies. Sadly when dealing with limit most beginners don't realize that there are well defined rules to handle evaluation of limits and we can't just do manipulations disregarding those rules. Next lets argue via intuition. Moreover most people feel that if there is an intuitive explanation for some mathematical result then it is more understandable my personal opinion is contrary to this but let's leave that part.
In an introductory course one may accept that the limit exists without proof and use it as one of the important but unproven formula after all no one bothers to find proof for Heron's formula when one is calculating area of triangles. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.
Create a free Team What is Teams? Learn more. Ask Question. Asked 4 years, 5 months ago. Active 2 years, 3 months ago. Viewed 10k times. Is there a calculus way and intuitive way to understand why this is false? OK 3 3 silver badges 14 14 bronze badges. You have to take the limits "together". An analogy is a tool with two knobs, What each knob will do depends on the setting of other knob. Turning one knob down and then the other will have a different result than turning both knobs at the same time.
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