There is a very good book about math that everyone should read: "e: The story of a number". It is thin, easily digestible, and fun. The contents of this page are based on parts of this book.
The story of the number e = 2.718281828... intermingles with the history of calculus. The efforts of calculating the area of the deceptively simple curve y=1÷x guided Newton and Leibniz to the study of infinitesimal processes, and then stumbling upon the "e" number.
Many other scientists had already approximated the value of e, even the Babylonians in financial calculations. I have myself stumbled upon it in an Excel financial simulation, but neither me nor the Babylonians recognized the importance of the number at the time, since we didn't know calculus :)
The e number has many, many facets. The one I intend to show is related to something relatively simple: the power function, that we learn at primary school like this:
104 = 10 × 10 × 10 × 10
Power is, at first sight, just a compact form of expressing repeated multiplication. When the exponent (in the example, 4) is a positive integer, this definition is good enough. The primary school also teaches some properties involving powers of the same base, all easily proven:
104 × 102 = 10(4+2) = 106
104 ÷ 102 = 10(4-2) = 102
(104)2 = 10(4×2) = 108
Now, would be acceptable a zero or negative exponent? The initial definition cannot handle that. But the properties of powers of the same base can be used to accept these exponents:
103 ÷ 103 = 10(3-3) = 100 = 1
(since 1000 ÷ 1000 = 1, therefore 100 must be 1)
102 ÷ 104 = 10(-2) = 1 ÷ 102 = 0,01
(since 10 ÷ 1.000 = 0,01, therefore 10(-2) = 1 ÷ 102)
Thanks to these, we can express the reciprocal function y=1÷x in the more elegant form y=x-1.
Now, what about a non-integer exponent? Let's begin with the case 1÷x. Which result could come out from
Tinkering a bit, we could say that the result should be "a third" of 8, but a "multiplicative third", such that
8(1÷3) × 8(1÷3) × 8(1÷3) = 8
But 2 × 2 × 2 = 8, so the "multiplicative third" is just the cubic (third) root of 8. An exponent of 1÷2 would be equivalent to square root, 1÷4 would be the fourth root, and so on.
This conjecture seems to be true because it works for all previous cases. Example:
[10(1÷2)]4 = 10(4÷2) = 102 = 100
If we take the square root of 10 and elevate the result to the 4th power, we get 100. Therefore, expressing roots as reciprocal exponents (1÷x) looks indeed correct.
If the exponent is a rational number (i.e. the ratio of two integers x÷y), just consider that
b(x÷y) = b powered to x, then take the y-th root.
There are some tests that we should make to assert that all these extensions to the basic power function are indeed sound. The power function must be "smooth" to be valid, that is, there should be a small response to small changes of the exponent (or the base), without sudden jumps or kinks.
The power function looks smooth, and monotonic, for integer exponents:
22 = 4
23 = 8
24 = 16
By the way, a monotonic function grows in only one direction. The power function is clearly monotonic: when the base is greater than 1, the result grows as we grow the exponent, with no exceptions. When the base lies between 0 and 1 (exclusive), the result shrinks as we increase the exponent, with no exceptions.
Since the power function is monotonic, we can use the "sandwitch theorem":
If x1 <= x2 <= x3, then
f(x2) must lie between f(x1) and f(x3)
if the function is smooth
In practice: If x1=1, x2=2 e x3=3, we have 21=2 e 23=8, then the result of 22 must lie between 2 and 8 (and indeed it does: the result is 4).
We can only accept negative and non-integer exponents as "valid" if they pass the sandwitch test. Testing with some exponents between 2 and 3:
22 = 4
23 = 8
22,5 = 25÷2 = 5,6568...
22,25 = 29÷4 = 4,7568...
22,125 = 217÷8 = 4,3620...
22,0625 = 233÷16 = 4,1771...
22,03125 = 265÷32 = 4,0875...
At least in empirical tests, the power satisfies the sandwitch theorem. Of course, there are rigorous proofs which guarantee the power function is smooth. We can relax about this, and go straight to the "big trouble": irrational exponents.
A smooth function should not have too many "holes", that is, be undefined for certain values. For example, y=1÷x is undefined just for x=0 but it is smooth around this singularity.
On the other hand, y=x! is useless for real numbers because factorial is undefined for non-integer and negative values of x. The singularities are the accepted values of x (integers), with abysses around them.
Our current definition of power function cannot cope with irrational exponents, Consider these cases:
2√2 = ?
10π = ?
Both √2 and π are irrational numbers, that is, they are cannot be expressed as a ratio of two integers (not exactly, at least).
If we cannot express π as a ratio, 10π does not exist, because we cannot separate the exponent into its "power" and "root" components. This is a hole. Worse: there are infinitely many irrational numbers, they are even more abundant than rational ones, therefore the power function might be shock full of holes, making it un-smooth.
One way around this hole is using a rational approximation that "converts" a irrational exponent into a rational one, like this:
10π might not exist, but 10(31415926÷10000000) exists and it is a good approximation.
This technique seems to "solve" the problem, but two issues remain. The "small" issue: we would have to handle huge numbers, with millions of digits. The big issue is we haven't proven that a power with irrational exponent really exists.
There are other problems. When the base is negative, a fractional exponent a÷b takes the b-th root of the base. For example, (-1)1÷2 is the square root of -1, whose result is not a real number. The result can only be real when b is odd and/or a is even.
But, if the exponent is irrational, we simply can't determine whether a or b are even or odd, so we don't know whether the result is real or complex. This fact invalidates the approximation technique.
To fill these holes, we must redefine the power function. The teachings from the primary school will suddently look useless. I hope you can handle that :)
Consider the reciprocal function y = 1÷x = x-1. It looks simple. Many mathematicians tried to find the area beneath this curve, and didn't succeed. This problem was finally solved when calculus was invented by Newton and Leibniz.
The reciprocal function is smooth and has no holes for any values of x whatsoever, rational or irrational, save for x=0.
Before calculus, one thing was "known" (actually suspected, without proof): the area grows logarithmically as x grows. Due to this, the function of this area was baptized "ln".
ln x is the area beneath the curve x-1, starting from x=1, going forwared or backward. The start is at x=1 because x=0 is a hole. To calculate ln 3, we integrate the area from x=1 up to x=3. To calculate ln 0.5, we integrate from x=1 to x=0.5 going backwards, and taking the result as negative.
With calculus, it was possible to prove that all properties of logarithms (those ones taught at high school) also hold for the function ln x. Then ln must be, for all intents and purposes, a logarithmic function. It is a case of "duck typing" in math: if the function squawks like a duck and walks like a duck, it is a duck. Due to the many special properties of ln, it is nicknamed "natural logarithm".
If the reciprocal function exists for all x>0, ln x also exists for all x>0.
Recalling the definition of logarithm:
log 1000 = 3, because 103 = 1000, and log x is the base-10 logarithm.
The base of log x is 10. If ln x is a logarithmic function, it must have a base as well. Before calculus, the base of ln was already suspected to be around 2.71828.
Once proven that
we can define the exponential function as the inverse of the logarithmic function:
y = ln x
x = ey
The function y = ln x has no holes for x>0. By choosing x carefully, we can make y to be any real number. Therefore the inverse function x=ey is also free of holes, and the argument y can be any real number, including zero, negative, non-integer or irrational.
Well, we have found one case where an exponent can be irrational: when the base is the e number.
Now we use this finding to redefine the power function in a way that it can accept any real number as exponent. The properties of logarithms can help us in this. For example:
10n = en × ln 10
Since ex and ln x have no holes, we know the intermediate value ln 10 exists, as well as the final result.
We could say that 10π "didn't exist" by the original definition of power, but comes into existence as we adopt the new definition based on Euler's number. The definition of power we learn at primary school is a corner case, valid only for rational exponents.
Ok, maybe this is a bit harsh. The "primary school" definition is still the ultimate when we deal with integer numbers exclusively. Modern cryptography and many other tricks in discrete math are based on powers of integers. But, when dealing with real numbers, the "new" definition is the valid one.
Putting the thing in a general form:
an = en × Ln a (a ≠ 0)
The definition above is valid even for negative bases, even though this forces us to deal with complex numbers in intermediate calculations. Note the capital-letter L in "Ln". A negative number has infinitively many logarithms, all of them complex. One of them, exactly one, is considered the "principal". The capital-letter Ln means that we get this one.
One issue is the case a=0. We can appeal to the concept of limit: it is easy to prove that the whole expression tends to zero when a=0.
The new definition works equally well for complex bases and exponents, which gives birth to the intriguing Euler's identity whose exponent is imaginary and irrational:
ejπ = -1
Even if we don't have use for higher mathematics, or even for complex numbers, in our daily lives, the functions ln x and ex are still useful. Since they are based on the simple reciprocal, they can be calculated by infinite series of arithmetic operations. For example:
ln (1+x) = x - x2÷2 + x3÷3 - ...
In spite of the name "infinite series", in practice the calculation goes on until the desired precision is achieved (e.g. 10 digits in pocket calculators), then it stops.
All transcendental functions (logarithm, sine, cosine, power, etc.) can be computed by using infinite series. Or did you think that your pocket calculator used a table?
Or did you think that, when you ask 250, it actually multiplied 2 fifty times? No way, the calculator spends the same effort for any power, regardless of base or exponent.
In particular, to calculate 250, the operation is reinterpreted as
e50 × ln 2
Likewise, when you type 10π the device applies the formula
eπ × ln 10
because the result "exists" in this form, and the intermediate results are rather small numbers:
eπ × ln 10 = eπ × 2.3025 = e7.2337 = 1385.45
Calculating 10π using a rational approximation like 1031415926÷10000000 would be utterly stupid, because the intermediate results would be huge and they would take a long time to calculate, yielding no better precision than using exponentials.