Why natural constant “e” is called “natural”
Most of us won’t feel strange to the symbol “e” in math. It is an irrational number whose value is approximately equal to 2.71828182845904523536…Meanwhile, if “e” is the base of the logarithm, then it will be called “natural logarithm”. Besides these two common characters, have you ever thought why we use the word “natural” to describe it?
Looks like the variation between natural food and proceed food, natural constant is a kind of number which is discovered rather than created by the human. But how we find the existence of “e”?
“e” in interest
The concept of interest was already commonplace in Middle Eastern civilizations as early as 5000 BC. In terms of interest, we always consider which way we store the money can get the highest profit. Now let us make a very simple example. We consider different interest rate and payment way.
Assume the interest rate of a bank is 100% and the interest is paid once a year. I we store $1 in the bank; we will get $2 at the end of year.
If the ratio doesn’t change, but payment way will change to every half year. Then the interest rate will change to 50% in half year. So when we get $0.5 in the middle of the one year, we will store $1.5 into bank and get the $1.5*50%=$0.75 in next half year. So at the end of this year, we will get $2.25 in total.
If the interest is payed each four months. Store the money in the same way as before, we will get $2.37 at the end of year.
So what will happen if the bank gives back the interest every day? Such capitalization of the interest will give you approximately $2.71456748202 in the end of year. What about pay interest every second? There is 31536000 second in a year, and you will get $2.7182817813 at last.
Now this number is close to the value of “e”. Actually this is the celling of the profit from interest. In the case of situation where annual interest rate is 100%, no matter how the payment method or the store approach changes, the sum of money won’t exceed the value of “e”.
“e” in the calculous
As we all know, the derivative of ex is itself. Also, if we display the ex in rectangular coordinate system, it is shown as the graph below.
The interesting thing is, if we put it in the polar coordinate system, the follow figure will be showed.
This is called “Logarithmic spiral”.
Many photography is based on this beautiful figure:
Also, this pattern can be easily found in natural world.
“e” in the nature
Another interesting thing is many moving trail in real world can be changed from straight to logarithmic spiral. Let’s take the moth as an example. The phenomenon that moth often flies into the flame or light misleads many people. The prevalent explanation is moth has phototaxis. However, the real reason is that moth depends on moon light for navigation at night. Due to moon light is parallel, moth can fly straight if it keeps the fixed angle between moving trail and light.
However, with the advent of lights and flame, the light in the natural area is not parallel any more. Instead, it is disordered cause the light from the flame and electronic lights are radioactive. Shown as the below graph.
Hence, if the moth keeps flying in the same way, the moving trail will be changed to logarithmic spiral. That’s why we always see that moth flies rotationally into the flame and lights.
In the end:
Naming “e” with “natural constant” is the appreciation from the scientists. In 2004, when Google wanted to be listed on Nasdaq, two founders Larry Page and Sergey Brin who are obsessed with the beauty of “e” decided to set the total financing as $2718281828. Similar to the value of “e”.
“e” is not merely a symbol, the beauty of it deserves more explorations.