The Number e
Some facts about it
The remarkable number now commonly referred to as e, has a value of
2.7182818…
It is an irrational number, which means it can not be defined as the division of any two real integers. It is a non-recurring decimal which goes on for ever.
The Scottish mathematician John Napier (1550-1617) is credited with inventing logarithms, which by means of tables, can be of great assistance when multiplying or dividing large numbers. Log tables were commonly used prior to the invention of the calculator and computer.
For these purposes, logarithms to base 10 (the "common log", log_{10} or just log) would generally be used. However, logarithms to base e (the "natural" log, or ln) were found to have special unique properties. In fact, Napier and others, are believed to have discovered natural logarithms, even before the actual number e, their base, was explicitely known of, or referred to.
e is the infinite sum
∞ ∑ n=0 |
1 ——— n! |
which is
1 —— 0! |
+ |
1 —— 1! |
+ |
1 —— 2! |
+ |
1 —— 3! |
+… |
The Swiss mathematician Jakob Bernoulli (1654-1705) discovered the constant e, when studying a problem concerning compound interest.
If money is earning interest at a rate of 100% per annum, and the interest itself also starts earning interest as soon as it is put on, then this is called compound interest. Since the interest is itself earning interest, the frequency with which interest is added makes a difference to the total amount of interest earned.
If interest is added at the rate of 100% once a year, then at the end of the year, the total amount of money will have doubled.
If 50% interest is added, twice a year, then the total amount of money after half a year, when interest is first added, will be the original amount multiplied by 1.5.
Then at the end of the year, when interest has again been added, the total amount of money will be 1.5 times the original amount, multiplied by 1.5 again, which is 1.5 squared, or 2.25, times the original amount.
If 25% interest is added, four times a year, then the total amount of money after the complete year has passed, will be the original amount multiplied by 1.25, multiplied by 1.25 again, then 1.25, then 1.25 yet again. Thus there will be the original amount multipled by 1.25 raised to the fourth power, which is 2.44141.
As one increases the frequency with which interest is added, however, the total amount at the end of the year approaches a limit, which turns out to be the number e.
More generally then, e is the limit of
^{ }_{ }(1+ |
1 —— n |
)^{n}_{ } |
as n approaches infinity.
e^{x} is often referred to as
exp(x).
log_{e}x is often written
ln(x).
The function exp is its own derivative, so that
d —— dx |
exp(x) = exp(x) |
The derivative of ln is the reciprocal, thus
d —— dx |
ln(x) = |
1 —— x |
For positive real numbers x, ^{x}√x reaches a maximum when x is equal to e
For positive real numbers x, x^{x}_{ } reaches a minimum when x is |
1 —— e |
For any non-negative real number x (provided x is not e), e^{x}_{ } is greater than x^{e}_{ }
Last updated 27 December 2010