# Is there a constant that is not a constant?

1. Apr 19, 2015

### Shahid Manzar

I was wondering why is there a constant that isn't really a constant in mathematics?

I am talking about "e", as in exponent e which is the base of the natural logarithm. By definition e = lim (1 + 1/n)^n as n approaches infinity, but doesn't this make e a non-constant since infinity always changes? In fact we know that value of e varies between 2.70 to 2.80 (P. 248 Calculus 5th edition, Stewart)
Also if value of e varies then how does a calculator pick a given value for e to solve natural log problems and what is the logic for choosing that particular value for e.

Thank you for the help!

2. Apr 19, 2015

### Staff: Mentor

There isn't.
A constant always has the same value.
Infinity is not "always changing." The fact that e is defined as a limit has nothing to do with its value.
No, you are mistaken. e doesn't vary. Its value is somewhere between those two numbers, but that doesn't mean that its value is changing.
A calculator uses an approximation of e. Different calculators might use different approximations, but that doesn't mean the e's value is changing.

3. Apr 19, 2015

### SteamKing

Staff Emeritus
According to this article:

http://en.wikipedia.org/wiki/E_(mathematical_constant)

the value of e has been calculated to 1 trillion (1012) decimal places.

Unlike similar calculations for the value of π, the calculation of the number e gets far less attention in the press.

4. Apr 19, 2015

### Mentallic

Just because the value of $S_n=(1+1/n)^n$ as n increases keeps changing, doesn't mean that it won't converge to a value at infinity.

$$S_1=2$$$$S_2=2.25$$$$S_3\approx 2.37$$$$S_4\approx 2.44$$
$$S_{10}\approx 2.59$$
$$S_{100}\approx 2.705$$
$$S_{1000}\approx 2.717$$
$$S_{10^6}\approx 2.71828$$

Notice as n gets larger, the value of Sn changes much more slowly. As n approaches infinity, Sn will approach a certain constant value which is irrational and we give this constant the symbol $e\approx 2.7182818284590...$

This is equivalent to, for example,

$$\lim_{n\to \infty}\frac{n+1}{n}=1$$
Even though we have

$$S_n=\frac{n+1}{n}$$

$$S_1=2$$$$S_2=1.5$$$$S_3=1.33$$$$S_{10}=1.1$$$$S_{100}=1.01$$

We will find that $S_{\infty}=1$

5. Apr 19, 2015

### rootone

A constant is a value which is what it is.
The degree to which we can be accurate about it can be improved, but this does change the value.
A simple example , take the number two.
There is is no such thing as more or less 'two', it has an exact numerical value.
Two cats sleeping on my couch are not approximately two cats, they are exactly and precisely two cats,
not a range of possible cat quantities somewhere between 1.9 and 2.1 cats.

Last edited: Apr 19, 2015
6. Apr 20, 2015

### statdad

Two cats on a couch equals two more cats than is needed.

7. Apr 23, 2015

### Stephen Tashi

Some of the words involved in the mathematical definition of "limit" suggest a process that takes place in time or takes place in steps. However, if you examine the precise wording of the definition of "limit" (the so-called "epsilon-delta" definition), there is no mention of something being done in steps or progressing as time passes. Illustrating limits by showing a computation done in steps is merely an intuitive way to present them. It's rather surprising how the mathematical definition of limit expresses an idea that is intuitively associated with doing a calculation in steps without mentioning anything about steps or a process taking place in time.

8. Apr 23, 2015

### symbolipoint

The base of the natural logarithm, e, DOES NOT VARY. e is a constant; e is a constant value. When it is written with digits, the value shown is an approximation only, because e is an irrational number.

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