If you start with 1 dollar and compound it with 100% continuous annual growth for 3 years, how much would you have?
Complex Space and Euler’s Identity
We ended up in a pretty strange place last time. It’s a bit like going down the rabbit hole of mathematics. I wrote about a domain called the complex plane where the up and down axis (y axis) is imaginary (ie based on the imaginary number i which is the square root of -1) and the x axis is real. This gives us Complex numbers of the form a+ib with a real part and an imaginary part. It is a plane that is all about defining rotations and phase. Multiplying by complex numbers in complex space gives us a rotation.
If you look at the diagram you can see that a number can be defined in 2 ways in a plane. The first is normal (Cartesian ) coordinates. In complex space the number might be 4+3i for example. This can also be expressed in Polar Coordinates as (R=5, ϑ = 36.9 deg). Here we look at the radius which in this case is 5 (as it is the hypotenuse of the right triangle with the other sides 3 and 4 units) and the angle it forms.
Polar coordinates are good at looking at rotations and are just as useful in Real 2D space as well. Here we have the basic equations from Pythagoras and trigonometry
R² = a² + b², cos ϑ = a/R, sin ϑ = b/R
And therefore a = Rcosϑ and b = Rsinϑ
And a complex number is written as
z = a+bi =Rcosϑ +((Rsinϑ) * I)
The right hand side of the equation is how we define numbers in the complex plane in Polar coordinates.
And we know from our series expansion that
cosϑ+isinϑ = (Eulers formula)
We now have z = a+bi =R and if ϑ = π (half a turn) we have
Log 100 = 2 and log 1000 = 3 by definition. Log in base 10 (our normal counting system) is the power that you need to raise 10 to get the desired number (10² =100, 10³ =1000). Not much to it but they were awful things back when I went to school. This was before computers and calculators.
We were confronted by a book of tables that looked like this
Adding logarithms equates to multiplying the original numbers. So we would look up the 2 logarithms of numbers we wished to multiply (or divide) add the two together (or subtract) and then find the anti logarithm to find the answer. Calculators and spreadsheets are a hell of a lot easier!
As an equation we have
(xy) = (x) + (y)
where the b designates the base counting system. Normal logs are base ten. These give us the logarithmic scales that we are used to like decibels for measuring amplitude (dB) and Richter scale for measuring earthquake size and of course pH measurements (another log base 10 scale). It also explains why we can add the pH and pOH scales as logs to get the log of the disassociation constant of water Kw.
Logarithmic scales are very useful as they show a large range with small numbers.
Computer geeks use logs with base 2 called . Our old mate Leonhard Euler first used binary logs in music analysis. The binary logarithm of a frequency ratio of two musical tones gives the number of octaves by which the tones differ. These days it’s more likely they will be used to determine the number of bits needed to encode a message.
Power curves (red curve) show exponential growth while log curves (blue) are the inverse and show slow growth. Exponential growth is common in biology, eg population growth. In reality however there is always a cap to the exponential growth (eg food supply) and where this is applied the curve is called a logistic curve.
It so happens that one of the most useful logs is the one with a base e, called the natural log or ln. Again it was Euler who showed us how to use this one. Ln (x) is the inverse of and the derivative of ln(x)(ie it’s rate of change) is 1/x. This gives an easy way of defining the natural log (x) as the area under the curve of the graph 1/x (called a hyperbola).
We originally found e by looking at compound interest and can be defined as a measure of growth at 100% compounding interest. Ln(x) is then the time taken to achieve that growth. For example is 20.08. After 3 units of time, we end up with 20.08 times what we started with. On the other hand ln(20.08) is about 3. If we want growth of 20.08, we need to wait 3 units of time (again, assuming a 100% continuous growth rate).
Radioactive decay, for example follows a natural log curve such that
ln(N) = -kt + ln ().
If we plot this on a natural log scale on the y axis and a regular time scale on the x axis we get a straight line with the rate of decay as the slope (-k). The ln () just defines the starting point (original number of atoms).
We can rearrange the terms to get ln (N) – ln () = -kt or
which is called exponential decay. It gives you the number of atoms left after time t if decay is at a rate of k and you start with atoms.
That may be good if you are a nuclear physicist, but it also pops up all over the place. Most importantly it has been shown to be true for the process of beer froth decay. First order chemical reactions follow it, as do capacitors in electric circuits, atmospheric pressure as a function of height, eliminating toxins from the body and decay in vibrations (or noise).
It is also the rate of decay on my retirement fund as well as in many other financial and economic considerations.
So, while we might consider 10 or 2 as a more natural base to count in, physics seems to like e instead.
Logs are used in a lot of statistical analysis. Here is a plot someone put together to analyse Olympic medals with regard to a country’s population and GDP. Australia is a standout from the regression line (a line of best fit) as are Cuba and Russia and USA with a lot more medals than expected. We can probably guess why Russia also does!
These are called outliers (without them the regression line would fit better). R squared is a measure of how well the line fits (should be closer to 1 for a good correlation).
Coming up we will have a look at statistics and how they work. One area that gets a lot of attention is the use of mathematical models and their reliance on data and statistics. These include climate models and their predictions, stock market investment schemes, housing market forecasts etc.
You just need to remember Nils Bohr’s (a famous physicist) quote
“Prediction is very difficult, especially if it’s about the future.”