## 814 Discrete and Continuous Cash Flows

The above formulas suppose that all costs and revenues occur in discrete intervals. That is common engineering practice, in accord with the fact that bills are paid in discrete installments. Thus, growth rates are quoted as annual changes even if growth is continuous. It is instructive to consider the continuous case.

Let us establish the connection between continuous and discrete growth by way of an apocryphal story about the discovery of e, the basis of natural logarithms. Before the days of compound interest, a mathematician who was an inveterate penny pincher thought about the possibilities of increasing the interest he earned on his money. He realized that if the bank gives interest at a rate of r per year, he could get even more by taking the money out after half a year and reinvesting it to earn interest on the interest as well. With m such compounding intervals per year, the money would grow by a factor

(1 + r/m)m and the larger the m, the larger this factor. Of course, he looked at the limit m ^ ^ and found the result lim V1m ^ + -)m = e (8.23)

At the end of one year, the growth factor is (1 + rann) with annual compounding at a rate rann, while with continuous compounding at a rate rcont, the growth factor is exp(rcont). If the two growth factors are to be the same, the growth rates must be related by

Compounding with m compounding intervals at rate rm is equivalent to annual compounding if one takes

## Solar Stirling Engine Basics Explained

The solar Stirling engine is progressively becoming a viable alternative to solar panels for its higher efficiency. Stirling engines might be the best way to harvest the power provided by the sun. This is an easy-to-understand explanation of how Stirling engines work, the different types, and why they are more efficient than steam engines.

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