Question

John Napier's Answer John Napier \((1550-1617),\) the Scottish laird who invented logarithms, was the first person to answer the question, "What happens if you invest an amount of money at 100\(\%\) yearly interest, compounded continuously?" (a) Writing to Learn What does happen? Explain. (b) How long does it take to triple your money? (c) Writing to Learn How much can you earn in a year?

Short answer

The money after a year would be approximately \(e\) times the initial investment. It would take about \(1.0986\) years to triple the money. The amount earned in one year would be roughly \(1.71828\) times the principal amount.

Step-by-step solution

Apply the Formula for Continuous Compounding

Napier essentially asked about the calculation of continuous compound interest. This uses the formula \(A = Pe^{rt}\), where \(A\) is the amount of money accumulated after n years, including interest, \(P\) is the principal amount (the initial amount of money), \(r\) is the annual interest rate (in decimal), and \(t\) is the time in years.

Calculate for (a)

To know what happens if you invest an amount of money at 100% yearly interest, compounded continuously, we plug in \(r=1\), as 100% interest means the rate is 1 when expressed in decimal form. Also assume \(P=1\) and \(t=1\) year. So \(A = 1e^{1*1} = e\). Thus, the amount of money after a year would be equal to the value of \(e\) (approximately \(2.71828\)) times the principal.

Calculate for (b)

To find when your money triples, we set \(A=3\) times the principal amount. We assume \(P=1\) for simpler calculations. So, we solve for \(t\) in \(3 = e^t\). Taking natural logarithms on both sides, \(ln(3) = t\). Thus, it will take approximately \(1.0986\) years to triple your investment.

Calculate for (c)

The amount earned in a year would be the final amount minus the principal. This will be equal to \(A - 1 = e - 1\). So, if you invest \(1\) unit of your currency, you'll earn roughly \(e - 1\) or \(1.71828\) units in a year.