Exploring the Intricacies of Euler's Number, e

Introduction to Euler's Number

Euler's number, commonly denoted as e, serves as a pivotal constant in mathematics, particularly in understanding continuous growth rates and decay processes that relate closely to the sizes of the entities under consideration. It appears in various calculations, such as:

Compound interest
Population dynamics
Radioactive decay
Growth of bacteria
Carbon dioxide levels in the atmosphere

A striking example of its occurrence can be observed in the logarithmic spiral formation of the nautilus shell, where each chamber's size is proportional to its predecessor:

Nautilus shell illustrating logarithmic spiral growth

Historical Context and Development

During the late 17th century, European bankers recognized that the interest on loans increased more rapidly with more frequent compounding. The formula for simple interest is given as:

A = P (1 + r)

For instance, if a $1000 loan is taken out with an annual interest rate of 20%, the total repayment after one year would amount to:

$1000 (1 + 0.20) = $1200

However, with semi-annual compounding, the calculations adjust as follows:

After six months:

$1000 (1 + 0.10) = $1100
After another six months:

$1100 (1 + 0.10) = $1210

As such, the total at the end of the year with compound interest exceeds that of simple interest by $10.

To illustrate further, if compounded quarterly, the total becomes:

A = P(1 + r/t)^t

Applying this to our previous example results in:

$1000 (1 + 0.05)^4 = $1215.51.

A compelling inquiry arises: How much can be accrued at 100% interest? Setting r = 1 (i.e., 100% interest) and examining the output as t increases leads to the following calculation:

Limit calculations for interest compounding

This results in a consistent amount owed at the end of one year, regardless of how frequently compounding occurs. Ultimately, we arrive at the expression:

$1000 * e ≈ $2718.28,

which is exactly what Jacob Bernoulli uncovered in 1683—the limit definition of e:

Unique Characteristics of e

The number e possesses several remarkable traits:

The exponential function e^x is the only function that equals its own derivative, indicating that the slope of the tangent line at any point matches the function's value there:

Tangent line to the exponential function e^x

The area beneath the curve of e^x from x = -∞ to x = n equals e^n:

Additionally, e is the sole number n for which the area under the hyperbola y = 1/x from x = 1 to x = n is exactly 1:

Furthermore, similar to π, e is irrational and cannot be expressed as a fraction of two integers, with its decimal representation extending infinitely without repetition. It is also transcendental, meaning it is not a solution to any polynomial equation with integer coefficients: