When you trade financial markets you quickly learn that 10% up and 10% down are not the same thing!…

If we were to grow $100 in 1000 incremental steps of 0.0095315% we will arrive at $110. And if we subsequently unwind/decline from this $110 in 1000 incremental steps of 0.0095315% we will arrive back at $100. If however we were to grow the same $100 to $110 in one incremental step of 10%, and then decline from $110 in one incremental step of 10% we will not arrive back at $100 but at $99 instead.

So why is this? The reason is simple. Because we are rounding out very tiny errors on each step, smooth incremental change will synchronize up and down movements; but because large errors are too big to be rounded out, coarse incremental change will not.

__Mathematical Linearization__

In the simplest of mathematical terms an increase of x% followed by a decrease of x% is the same as multiplying $100 by the product of (1 + x) times (1 – x) which can be written mathematically as

$100(1 + x)(1 – x) = $100(1 – x^{2}) = $100 – $100(x^{2})

Now if x is 10% then

$100 – $100(0.10^{2}) = $100 – $1.00 = $99.00

However if x is 0.0095315% then

$100 – $100(0.000095315^{2}) = $100 – $0.00000009 = $100.00

So when x is small we can ignore the (x^{2}) term, because this tiny error is negligible.

The process of linearization basically says that when x is small we can ignore the (x^{2}) term, because this tiny error is negligible. So when the error is small we are effectively resetting to *“no error”* on each step.

__Cornerstone of Mathematical Science__

Now believe it or not, “*ignoring the (x ^{2}) term”* is the cornerstone of all mathematical science. To ignore the (x

^{2}) term is simply to ignore

*“insignificant deviations”*and this idea is referred to as

*“linear approximation”*or

*“linearization”*.

The two primary tools of linearization are

Breaking all incremental change down into an**Pure Continuity:***“infinite number”*of tiny infinitesimal steps (a simple example of which we have seen above).Dealing with an**The Law of Large Numbers:***“infinitely large number”*of things, so that we can confidently compress all information into an*“average”*(a modest example of which is the line-fitting graph shown below).

Linear Approximation is used in virtually every area of science; from physics to engineering, from economics to ecology. By eliminating insignificant deviations we can approximate the behaviour of a *“nonlinear”* system as being *“linear”* – which allows us to *“compress”* this behaviour into neat *“solvable”* mathematical equations.

Without this process of linearization the mathematical description of systems would be unsolvable, which effectively means that *“without linearization nothing would be predictable”. *Thus we can say that

**Linearization and Linear Dynamics have, to-date, been the cornerstone of all of mathematical science.**