Euler's number

*Not to be confused with Euler's constant. See E for other **e**'s.*
**e** is a number commonly used as base in logarithmic and exponential functions.

The letter **e** in mathematics usually stands for the so-called "natural base" for logarithms and exponential functions.

log

e

⁡
x
≡
ln
⁡
x

{\displaystyle \log\_{e}x \equiv \ln x}

, the natural logarithm

e
x

{\displaystyle e^x}

, the exponential function with base *e*
## Value

Euler's number is an irrational number (and a transcendental number), but it can be approximated as 2.71828 18284 59045 23536...

* Decimal: 2.71828182845904523536... (non-repeating, non-terminating)
* Limits:
	+ lim

	n
	→
	∞

	(
	1

	1
	n

	)
	n

	=
	e

	{\displaystyle \lim\_{n \to \infty} (1 + {1\over n})^n = e}

	 (this is the formal definition)
	+ lim

	n
	→
	−
	∞

	(
	1

	1
	n

	)
	n

	=
	e

	{\displaystyle \lim\_{n \to -\infty} (1 + {1\over n})^n = e}
	+ lim

	n
	→
	±
	∞

	(
	1
	−

	1
	n

	)

	−
	n

	=
	e

	{\displaystyle \lim\_{n \to \pm\infty} (1 - {1\over n})^{-n} = e}
	+ lim

	n
	→
	±
	∞

	(
	1
	−

	1
	n

	)
	n

	=

	1
	e

	{\displaystyle \lim\_{n \to \pm\infty} (1 - {1\over n})^n = \frac{1}{e}}
	+ lim

	n
	→
	±
	∞

	(
	1

	1
	n

	)

	−
	n

	=

	1
	e

	{\displaystyle \lim\_{n \to \pm\infty} (1 + {1\over n})^{-n} = \frac{1}{e}}
	+ lim

	n
	→
	0

	(
	1
	n

	)

	1

	/

	n

	=
	e

	{\displaystyle \lim\_{n \to 0} (1 + n)^{1/n} = e}
	+ lim

	n
	→
	0

	(
	1
	−
	n

	)

	−
	1

	/

	n

	=
	e

	{\displaystyle \lim\_{n \to 0} (1 - n)^{-1/n} = e}
	+ lim

	n
	→
	0

	(
	1
	−
	n

	)

	1

	/

	n

	=

	1
	e

	{\displaystyle \lim\_{n \to 0} (1 - n)^{1/n} = \frac{1}{e}}
	+ lim

	n
	→
	0

	(
	1
	n

	)

	−
	1

	/

	n

	=

	1
	e

	{\displaystyle \lim\_{n \to 0} (1 + n)^{-1/n} = \frac{1}{e}}
* Continued fraction: e = [2;1,2,1,1,4,1,1,6,1,1,8,...,1,1,2k,...]
* Infinite series:

# n
0

∞

1

n
!

{\displaystyle e = \sum\_{n = 0}^{\infty} \frac{1}{n!}}

	+ e
	x

	=

	∑

	n
	=
	0

	∞

	x
	n

	n
	!

	{\displaystyle e^x = \sum\_{n=0}^{\infty} \frac{x^n}{n!} }

## Applications

Euler's number has many practical uses, particularly in higher level mathematics such as calculus, differential equations, discrete mathematics, trigonometry, complex analysis, statistics, among others.

## Properties

The reason Euler's number is such an important constant is that is has unique properties that simplify many equations and patterns.

Some of the defining relationships include:

* d

d
x

e

x

=

e

x

{\displaystyle \frac{d}{dx} e^{x} = e^{x}}

 (most useful in calculus)
# * y

e
x

{\displaystyle y=e^x}

 is that function such that

y
′

=
y

{\displaystyle y'=y }

 (useful in differential equations)
* d

d
x

e
x

# x
0

=
1

{\displaystyle \frac{d}{dx} e^x |\_{x=0} = 1}
* ∫

e
x

d
# x

e
x

C

{\displaystyle \int e^x\,dx = e^x + C}
* ∫

−
∞

0

e
x

d
# x
1

{\displaystyle \int\_{-\infty}^0 e^x\,dx = 1}
* d

d
x

ln
⁡
(
x
# )

1
x

{\displaystyle \frac{d}{dx} \ln(x) = \frac{1}{x}}
* d

d
x

ln
⁡
(
x
)

# x
1

=
1

{\displaystyle \frac{d}{dx} \ln(x) |\_{x=1} = 1}
* ∫

1
x

d
# x
ln
⁡
(
x
)
C

{\displaystyle \int \frac{1}{x} \,dx = \ln(x) + C}
* ∫

1

e

1
x

d
# x
1

{\displaystyle \int\_{1}^{e} \frac{1}{x} \,dx = 1}

* cos
⁡
θ
i
sin
⁡
# θ

e

i
θ

{\displaystyle \cos \theta + i \sin \theta = e^{i \theta}}

 (Euler's formula, angle

θ

{\displaystyle \theta}

 is to be measured in radians)
* ln
⁡
(
−
1
# )
i
π

{\displaystyle \ln(-1) = i\pi}

One of the original defining attributes of **e** is the fact any bank account having a 100% APR interest rate which is compounded continuously, will grow at the exponential rate et, where t is time in years, discovered by Jacob Bernoulli. To get

x

{\displaystyle x}

 times the initial principal, leave it in there for

ln
⁡
x

{\displaystyle \ln x}

 years. Intuitively, compounding an initial account will yield **e** times the initial principal after one year.

ja:ネイピア数