Mathematics and Motivation with a Compound Case

Have a glimpse:

Many of you might have seen this picture. Whether through internet or magazines, books, etc., it has gained fair amount of popularity. It is really a fascinating picture, an erotic picture for the mathematical guys. James Altucher once termed this picture as weird mathematics in his article. This fascinating picture has some significance too. This picture is often seen in the inspirational articles. While giving motivation people like to use it very much.

Significance:

More often than not it is interpreted as follows:

If you improve yourself everyday by 1% of what you are today you will end up improving by 37.78 times by the end of a year.

Likewise, if you degrade everyday by 1% of what you are today you will end up degrading by 39.21(1/0.0255) times by the end of a year.

[Note: To be precise it is 0.0255, 0.03 is an only approximation. As I am analyzing it mathematically I am taking more precise value than the market value. Taking 0.03 will make error in my calculation and shows result contrary to elementary mathematical laws.]

It is interesting to note here that although your increase and degrade rates are same, final degrading ratio is more than increasing ratio at the end of a year. I will give you a hint on why this happens and you could work out its reason.

Hint: If you increase 4 by 1, you end up increasing by 20% and if you decrease 4 by 1 you will end up decreasing by 25%. Now it is on your hand to find out the reason for the above interesting finding.

Is it true (mathematically)?

In our first case, rate of improvement is 1% and n is 365 days i.e. R=1 and n=365.

Is it practically possible?

What do you think about its practical feasibility? Mathematically we proved it. Increasing by 1% is not a big deal, isn’t it? It is just a percent. We feel we can do it. Then can we increase ourselves by 37.78 times in a year? To be honest we cannot. Really we cannot. Let’s see how the condition can be visualized.

At first let’s see it in a graph.

Though we may feel the rise is like linear but actually the rise is exponential.  The graph shows that. Even we can reduce our equation to exponential equation as shown in equation (1). And it is known fact that improving exponentially is impossible. Therefore the first condition is not practically feasible. Those inspirational speakers were only being philosophical while mentioning first expression ;). They used it to inspire you, to make you optimistic. But what they said is not practically possible. We even can’t go little near to this equation.

Nevertheless the second expression is true. Absolutely true!

Its equation is:

Let’s see it graphically.

You can degrade exponentially. Cost can depreciate exponentially. Radioactive decomposition occurs exponentially. Those speakers and writers have used second equation to suggest you be alert. They are absolutely right in second equation. You never know how you are ripping into. 

You might have remembered this famous quote by Warren Buffet, “It takes 20 years to build a reputation and five minutes to ruin it.”

Other similar cases in Mathematics:

I would like to call it as a compound case. In many places they have their use like compound population, compound interest, compound depreciation, etc. They are all exponential equations.

For an example:

In compound depreciation everything is same except the constant c being <1.

The function becomes increasing or decreasing function depending upon the value of c (whether it is greater than 1 or smaller than 1 respectively). If c = o the function is a constant function as c = 0 comes from rate of increase ‘R’ being zero.

Thank you very much. I will bring more and more mathematics stuffs in coming articles.