1. Limit Definition:
e is defined as the limit:
e = limn → ∞ (1 + 1/n)n
2. Infinite Series Representation:
e can also be represented as the sum of an infinite series:
e = Σn=0∞ 1/n! = 1 + 1/1! + 1/2! + 1/3! + …
1. Irrational Number:
e is an irrational number, meaning it cannot be expressed as a simple fraction. Its decimal expansion is non-terminating and non-repeating.
2. Transcendental Number:
e is also transcendental, meaning it is not a root of any non-zero polynomial equation with rational coefficients.
3. Natural Base:
e serves as the base of natural logarithms. For any positive number x,
ln(x) = loge(x)
1. Compound Interest:
In finance, e appears in continuous compounding of interest. If a principal amount P is invested at an annual interest rate r, the value after t years is:
A = P ert
2. Differential Equations:
The function ex is its own derivative and integral:
d/dx ex = ex, ∫ ex dx = ex + C
3. Probability and Statistics:
In probability theory, e is used in modeling events and exponential distributions.
4. Physics and Engineering:
e arises in calculations involving exponential growth or decay, such as population models, radioactive decay, and capacitor discharge.
5. Complex Numbers (Euler's Formula):
e links trigonometric functions and exponential functions via Euler's formula:
eix = cos(x) + i sin(x)
1. Discovery:
The constant e was discovered in the early 17th century by Jacob Bernoulli while studying compound interest.
2. Name Origin:
It is named after the Swiss mathematician Leonhard Euler, who popularized its use and made significant contributions to its understanding.
3. Notation:
Euler first used the symbol e for this constant in 1731.
e≈2.718281828459045235360287471352
1. The expression e-x describes the natural decay curve.
2. ex can be expanded using its Taylor series:
ex = Σn=0∞ xn/n!
