Wim Ruitenburg's Fall 2010 MATH 1300-101

Last updated: September 2010
Comments and suggestions: Email wimr@mscs.mu.edu

Despite all the money talk, chapter 10 is really about exponential growth, with examples from money management.
From the chapter we can learn:

Increasing an amount like 200 by 7 percent is essentially the same as multiplying 200 by (1 + 7/100) or by 107/100 or by 1.07, and the result equals 214.
Decreasing an amount like 200 by 7 percent is essentially the same as multiplying 200 by (1 - 7/100) or by 93/100 or by 0.93, and the result equals 186.
Increasing an amount like 300 by 5 percent and then reducing the new amount by 5 percent is NOT neutral. For first we multiply 300 by 1.05 to get 315, and then we multiply 315 by 0.95 to get as final result 299.25.
Compound interest is very powerful. When we start with just 400 (dollars, say), and increase the amount by 3 percent each year for 5 years, the new amount equals 400 * 1.03 * 1.03 * 1.03 * 1.03 * 1.03, also written 400 * (1.03 ^ 5), which equals approximately 463.71.
However, if we have 400 at 3 percent each year for 200 years, the new amount equals 400 * (1.03 ^ 200), which equals approximately 147742.33.
Earning interest of 24 percent for one year is not the same as earning interest of 2 percent a month for one year. Interest of 24 percent for one year means multiplying your original amount by only 1.24. However, earning compound interest at 2 percent a month for one year means multiplying you original amount by 1.02 ^ 12, which equals approximately 1.268.

We spent some time on continuously compounded interest. We approximate this through the following example.

If we start with 1 dollar and get 100% interest for the year, we end the year with 2 dollars.
If we start with 1 dollar and get 50% interest for the first 6 months and 50% for the second 6 months, we end the year with 1 * (1 + 1/2) * (1 + 1/2) = (1 + 1/2)^2 dollars, which equals 2.25.
If we start with 1 dollar and get 25% interest for the first 3 months and 25% for the second 3 months and 25% for the third 3 months and 25% for the fourth 3 months, we end the year with 1 * (1 + 1/4) * (1 + 1/4) * (1 + 1/4) * (1 + 1/4) = (1 + 1/4)^4 dollars, which is approximately 2.4414.
And so on. There are 31556952 seconds in the average Gregorian calendar year. If we start with 1 dollar and get 1/31556952 interest each second, then we end the year with 1 * (1 + 1/31556952)^31556952 dollars, which is approximately 2.71828.
And so on. The farther we go, the closer we get to Euler's number 2.71828182845999988...