Problem 41 - Calculus

Question

Euler charges \(c=100 \%\) interest on his $$\$ 1$$ fee for discovering \(e .\) What do you owe (including the $$\$ 1$$ ) after a year with (a) no compounding; (b) compounding every week; (c) continuous compounding?

Step-by-Step Solution

Answer

(a) $2; (b) $2.69; (c) $e \(\approx 2.72\).

Step 1: Calculate with No Compounding

When interest is charged with no compounding, the formula is simply the principal amount plus the interest rate multiplied by the principal. Here, the principal \( P \) is \( 1 \), and the interest rate \( r \) is \( 100\% = 1 \). The formula for the total amount \( A \) is:\[ A = P + r \times P \]Substituting the given values:\[ A = 1 + 1 \times 1 = 2 \]So, you would owe \( \$2 \) after one year with no compounding.

Step 2: Calculate with Weekly Compounding

With weekly compounding, the interest formula is: \[ A = P \times \left(1 + \frac{r}{n}\right)^{n \times t} \] where \( n \) is the number of times compounding occurs in a year, and \( t \) is time in years. For weekly compounding, \( n = 52 \), \( r = 1 \), and \( t = 1 \). Thus,\[ A = 1 \times \left(1 + \frac{1}{52}\right)^{52 \times 1} \]Calculating this gives:\[ A = 1 \times \left(1.019230769\right)^{52} \approx 2.6922 \]Thus, you would owe approximately \( \$2.69 \) after one year with weekly compounding.

Step 3: Calculate with Continuous Compounding

For continuous compounding, the formula is \[ A = P \times e^{r \times t} \] where \( e \) is the base of natural logarithms (approximately 2.71828). Here, \( P = 1 \), \( r = 1 \), and \( t = 1 \). Substituting these values gives us:\[ A = 1 \times e^{1} = e \]Hence, you would owe \( e \approx 2.71828 \) dollars after one year with continuous compounding.

Key Concepts

Euler's number (e)

Euler's number, commonly denoted as \( e \), is a mathematical constant approximately equal to 2.71828. It plays a pivotal role in various areas of mathematics, especially in calculus and complex analysis. This number arises naturally in the process of calculating compound interest over time.
This constant is the base of natural logarithms and it allows us to express exponential growth and decay processes, such as bacteria growth or radioactive decay, in a continuous manner.
Unlike whole numbers, \( e \) is an irrational number, meaning that it cannot be expressed precisely as a simple fraction. It has an infinite number of decimals and does not repeat. Understanding \( e \) is fundamental for those studying finance, as it helps in calculating continuous compound interest, which models how investments grow when they are continually compounded. Thus, \( e \) is essential for representing real-world growth scenarios.

Continuous Compounding

Continuous compounding is a method of calculating interest where it is added to the principal balance an infinite number of times within a time period. This contrasts with traditional compounding methods where interest is added on a fixed schedule: daily, monthly, or annually.
The formula for continuous compounding is \( A = P \times e^{r \times t} \), where:

• \( A \) is the total amount after interest.
• \( P \) is the initial principal balance.
• \( r \) is the annual nominal interest rate (expressed as a decimal).
• \( t \) is the time in years.

Continuous compounding effectively allows for exponential growth. The more frequently interest is compounded, the greater the amount of interest accrued, making continuous compounding result in the highest possible return on investment over time. Understanding how \( e \) integrates into this formula is key to mastering financial growth calculations, ensuring that one can predict future investment values accurately.

Interest Calculation Methods

Interest calculation methods vary, with the main options being simple interest and compound interest. Simple interest is straightforward: it adds interest to the principal based on an explicit rate and time period, without considering previously accrued interest.
Compound interest, on the other hand, calculates interest on both the initial principal and the accumulated interest from previous periods. This can be done on various schedules - annually, quarterly, monthly, weekly, or even continuously.
The formula for compound interest when interest is not continuous but applies to a specified number of compounding intervals is:

• The total amount \( A \) is given by \( A = P \times \left(1 + \frac{r}{n}\right)^{n \times t} \).
• \( n \) represents the number of compounding periods per year.

Continuous compounding takes this to the theoretical extreme, using Euler's number to calculate growth as an ongoing, seamless process. Each method has its applications, with compound and continuous compounding providing valuable insights, especially to those in finance and investment planning.