Tag: exponential

Posted in Science & Nature

Compound Interest

Posted on by Jinavie

When is the best time to invest? Is it when you have sufficient income and savings that you feel that you have a surplus to invest with?

The correct answer is much simpler: yesterday, with the second best time being today. Because of the magic of compound interest, investing early is the best strategy possible.

Thanks to a simple mathematic rule, compound interest rewards early, small investments more than late, large investments.

The way compound interest works is that after a given time interval (e.g. year), interest (as a percentage of the original investment) is paid out. The next year, interest is paid out again but as a percentage of the new amount. As an example:
1000 x 1.08 = 1080 (end of year 1)
1080 x 1.08 = 1166.40 (end of year 2)
1166.40 x 1.08 = 1259.71 (end of year 3)
…until end of year 10

If we use mathematical shortcuts and convert this into a formula, we can express it as:

(A = future value, P = present value, r = interest rate as decimal, n = number of periods/years)

For example, if we invest $1000 (PV) at an interest rate of 5% (r=0.05) for 10 years, then:

$1000 x 1.08^10 = $2158.92,

meaning we have earned $1158.92 over 10 years. Taking it further, in 30 years our investment would have grown to $10062.66 – ten times our original investment.

Because the formula uses exponents (or powers) for the time, your investments grow exponentially with time. This means that the earlier you invest, the greater your returns become disproportionately. This is why within 10 years, we have more than doubled our initial investment despite a reasonable interest rate and not doing anything else.

A rule of thumb for calculating how long it will take your investment to double is to divide 72 by the interest rate in % (e.g. 7). This is the number of years it will take for your investment to double (e.g. 72/7 = 10.3 years).

On top of this, if we invest small amounts every year, then we can benefit even more from the exponential growth of our investment. For example, just by adding in $100 every year, we end up with an additional $564.55 of investment earnings at the end of 10 years – a 50% increase in returns.

Unfortunately, mathematics works both ways and compound interest also applies to certain loans, such as credit cards. This means that your debt will grow exponentially unless you aggressively pay it back, making it seem impossible to pay off your credit card debt sometimes.

(This graph shows that investing early and consistently is the best strategy to maximise your eventual earnings. Compare the grey and purple line and you will see that despite investing a third of what Lyla invests total, Quincy ends up with a higher portfolio by retirement.)
Posted in Science & Nature

Folding Paper

Posted on by Jinavie

Take any piece of paper and fold it in half. Then fold it in half again. Chances are, you will not be able to fold the paper more than seven times. Try it. No matter how thin the piece of paper is, it is extremely difficult to fold a piece of paper in half more than seven times. The reason? Mathematics.

A standard sheet of office paper is less than 0.1mm thick. By folding it in half, the thickness doubles and becomes 0.2mm. Another fold increases it to 0.4mm. Already, the problem can be seen. Folding a paper in half doubles the thickness, meaning every fold increases the thickness exponentially (2ⁿ). By seven folds, the thickness is 2 x 2 x 2 x 2 x 2 x 2 x 2 = 128 times the original thickness. This makes the piece of paper so thick that it is “unfoldable”.

Another limitation is that folding the paper using the traditional method means the area also halves, decreasing exponentially. With a standard piece of paper, the area of the paper is so small after seven folds that it is mechanically impossible to fold it. Furthermore, the distortion caused by the folds is too great for you to apply enough leverage for folding the paper.

Could these limitations be overcome by using a larger piece of paper? Sadly, no matter how large the piece of paper, it is impossible (or at least extremely difficult) to fold a piece of paper over seven times. This has been a mathematical conundrum for ages, until it was solved in 2002 by a high school student named Britney Gallivan. Gallivan demonstrated that using maths, she could fold a piece of paper 12 times. The solution was not simple though. To fold the paper 12 times, she had to use a special, single piece of toilet paper 1200m in length. She calculated that instead of folding in half every other direction (the traditional way), the least volume of paper to get 12 folds would be to fold in the same direction using a very long sheet of paper.

Mathematics, along with science, is what makes something that seems so simple, impossible.