%path = "maths/functions/exponential" %kind = kinda["texts"] %level = 11
In the exponential function
\(x\) is the exponent
\(a\) is the basis
\(y\) is the exponential function of \(x\) on the basis \(a\)
The exponent tells how often multiplication with \(a\) is repeated. \(a\) must be a positive real number: \(a\in\mathbb{R}\).
Multiplication
Multiplication is an operation happening in the real world and we encode it with a number. In the number set \(\mathbb{Q}\) the operation is part of the number: \(2\) means \(\cdot 2\), and \(1/2\) mean \(/2\). \(\cdot\) stands for the multiplication operation and \(/\) stands for the inverse operation, the division. But the inverse operation is made part of the number by the inclusion of the fractions in \(\mathbb{Q}\). So we only speak of multiplication and mean the application of the operation of \(\mathbb{Q}\subset\mathbb{R}\).
If \(a\) is bigger than \(1\), then \(y\) will increase (grow) with \(x\) strictly monotonically: \(x_1<x_2 \Rightarrow y_1<y_2\).
If \(a\) is smaller than \(1\), then \(y\) will decrease (diminish) with \(x\) strictly monotonically: \(x_1<x_2 \Rightarrow y_1>y_2\).
Let’s compare the number of combinations of n bits:
with the growing processes, like with accruing of capital with annual compounding
or the especially interesting natural growing
\(e\) is Euler’s Number whose importance is founded on the given relation.
The key to compare them is to understand information in the shape of bits as a growing process. Every bit increases the size by \(1\) times what is there already. Let’s denote this aspect of the bit by \((1+1)\) to emphasize that an additional \(1\) is added to the one there already. The parentheses make this an operator, an element of the number set \(\mathbb Q\). \(n\) repeated applications of \((1+1)\) produces a multitude of size
Every new bit is compounded to the existing combinations.
The information measure for a real variable of size \(C\) is the number of bits \(n=\log_2 C\) needed to grow \(C\) combinations.
Which other variable to compare to?
Instead of bits we could as well use the considered variable itself because that is there physically. But combinations are also physically there and the selection of values, which ultimately gives birth to variables, is physical, and the number of involved variables plays a role. First this means that information is physical and secondly, considering quantum mechanics, the physical number of involved variables is huge and their individual contributions are tiny.
If we start from a number of variables, the exponential function gives the number of value combinations. If we start from a number of values, the logarithm gives the number of variables needed to represent it.
For interest calculation we look at an amount of money (the \(1\)), which is deposited in the bank with interest \(i\). After \(n\) years the \(1\) has grown to
The growth factor \(q\) is not \(2\), normally just a little above \(1\). The corresponding “information” measure in this financial context would be the number of years.
The essential difference with respect to bit information is that, what is added, is a fraction of what is there. But then, fraction is actually just a matter of units.
The units of living organisms are cells and the ultimate units in the real world are the quantum particles. Both of them are small compared to the things around us. And with such small units one can also compound arbitrarily (infinitely) often:
In the first equality we see that, given a certain growth, varying the compounding steps amounts to varying the growth factor. Due to the importance of \(e^x\) one often moves the growth factor \(q\) in \(y=q^x\) to the exponent of \(e\) (\(y=e^{kx}\)). \(k=\ln q\) is called the gowth constant.
Natural compounding in the finantial world
Actually in the financial world the real compounding takes place in very small steps, just that the bank forwards them to the customer in larger units of time.
\(x\) is the information in the natural information unit nat. Basically we split up the size of the variable to infinitely many infinitely small variables, such that the growth factor per step is almost \(1\).