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}
natural logarithm
e
x
{\displaystyle e^x}
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}
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 :
e
=
∑
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}}
y
=
e
x
{\displaystyle y=e^x}
y
′
=
y
{\displaystyle y'=y }
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}
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}
ln
x
{\displaystyle \ln x}
e times the initial principal after one year.
