Could someone tell me what is significant about the constant e ( apprx. 2.71828 )?

Know a little bit of calculus?

dy/dx = k*y is a very very common logistic equation in nature. it describes population growth, radioactive decay, and zillions of other physical situations.

the solution to that equation is an exponential growth (or decay). the base of the exponent depends on the value of k in the logistic equation, in some complicated way.

it turns out that rather than figure out what that complicated base of the exponent is as a function of k, it is easier to just write down the corresponding solution for the k=1 case, i.e. the solution to dy/dx = y. it s not too easy to find the solution to that equation, but once you do, you re done. you can now write the solution very easily to any equation of the original more general form, in terms of the solution that you found of the special case.

the solution of the special case is y = e^x.

then it is very easy to find the solution of the more general case in terms of the special case solution: y = e^kx.

so e came about out of some calculus considerations, trying to find the solution to dy/dx=y. to calculate this number, well there are several ways. the most usual introductory one that i have seen is lim_x->0 (1+1/x)^x

e is the base of natural logarithms. It is also called Euler's number (hence e).

The function y=e^x is the only one for which the gradient at any value of x is equal to the y value of the curve at that point. Another way of saying that is the function f(x) = e^x is its own derivative (and it is the only function which is its own derivative).

e crops up all over the place. Many physical functions turn out to be either natural exponentials or natural logarithms.

Originally posted by LionHearted
Could someone tell me what is significant about the constant e ( apprx. 2.71828 )?


http://mathworld.wolfram.com/e.html

e is one cool number. Never described why before, so I’ll probably think of better ways of putting it after posting this.

If you do any statistical applications, you’ll soon discover the power of e. Example: Check out this chart (http://finance.yahoo.com/q?s=MSFT&d=c&k=c1&a=v&p=s&t=my&l=off&z=l&q=l) of Microsoft’s stock price history. Looks like the company had exponential growth up until 2000, right? Wrong. Check out the log version (http://finance.yahoo.com/q?s=MSFT&d=c&k=c1&a=v&p=s&t=my&l=on&z=l&q=l) of the chart. The natural logarithm that is, where the natural logarithm is the logarithm to the base e. This chart reveals that Microsoft’s growth was actually more linear up to 2000. In other words the growth rate year after year was about the same percentage. Note the scale of the y-axis.

Stock prices that rise the same percentage every day make for a misleading exponentially growing chart, unless you look at the log version of the chart to more easily see the truth. Yahoo Help puts it this way: “In a logarithmic scale, the distance between each unit of distance reflects an equal percentage change. Logarithmic scales usually allow for more meaningful comparisons over longer periods of time, whereas linear charts are preferable for shorter time frames.” The same applies to population growth charts.

Here’s another example. Take the percentages 50% and 200%. These are equivalent to 1/2 and its reciprocal, 2/1. But that may not be obvious looking at 50% and 200%. The natural log of 50% and 200% is -0.693 and 0.693 respectively. Much easier to see a reciprocal. Likewise with log(14.29%) and log(700%), which is log(1/7) and log(7), which is -1.946 and 1.946.

I don't understand this:
If you do any statistical applications, you’ll soon discover the power of e. Example: Check out this chart of Microsoft’s stock price history. Looks like the company had exponential growth up until 2000, right? Wrong. Check out the log version of the chart. The natural logarithm that is, where the natural logarithm is the logarithm to the base e. This chart reveals that Microsoft’s growth was actually more linear up to 2000. In other words the growth rate year after year was about the same percentage. Note the scale of the y-axis.


If the chart of the LOG VERSION is linear then the growth itself IS exponential.

If the "growth rate was the same percentage" then the growth
IS exponential.

You’re right. I misstated that. Where I said “growth” I should have said “growth rate.” Let me restate: Looking at the linear version of the chart, you might think that the company had exponentially better performance each year up to 2000, if performance is gauged by percentage growth of stock price. But the performance was actually about constant as more easily seen in the log version of the chart. In the log version, you can simply compare the slope of last year’s data to this year’s to determine whether performance improved or suffered.