Which irrational number will 1 dollar which has continuously compounding annual interest of 100% (r=1) approach after a year? Answer alternatives: (a) Pi, (b) e (Euler's number), (c) squareroot of 2, (d) Theta (The golden ratio)

To find the value that $1 will approach with continuously compounding annual interest of 100% after one year, we use the formula for continuous compounding:

\[
A = P \times e^{rt}
\]

where:
- $ A $ is the amount after time $ t $,
- $ P $ is the principal (initial amount),
- $ r $ is the annual interest rate,
- $ t $ is the time in years,
- $ e $ is Euler's number, an irrational number approximately equal to 2.71828.

Given the problem:
- $ P = 1 $ (starting with $1),
- $ r = 1 $ (100% annual interest rate),
- $ t = 1 $ year.

Plugging in the values:

\[
A = 1 \times e^{1 \times 1} = e
\]

After one year, with continuous compounding at a 100% interest rate, the amount approaches **$ e $ (Euler's number)**.

So, the correct answer is:

**(b) $ e $**

Would you like further details or have any questions?

### Follow-Up Questions:
1. What would be the amount if the interest rate was 50% instead of 100%?
2. How does continuous compounding differ from annual compounding?
3. What is the significance of the number $ e $ in mathematics and finance?
4. How would the amount change if the time period was 2 years instead of 1 year?
5. What if the interest rate was compounded quarterly instead of continuously?

**Tip:** Continuous compounding is an idealized way of calculating interest that assumes interest is being added constantly, leading to the mathematical constant $ e $, which is central to many areas of mathematics and natural growth processes.

Discover how to calculate the future value of $1 with continuous compounding at 100% annual interest rate using the formula A = P * e^(rt). Understand the significance of Euler's number (e) in finance and mathematics. Learn the correct answer to the problem and related concepts.

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