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(Top)
1
Distributive property
2
Sophie Germain's identity
3
Quadratic equation
4
Exponentiation
5
Logarithm
6
Arithmetic progression
7
Geometric progression
8
Combinatorics
9
Trigonometry
10
See also
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Basic math formulas
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From Wikipedia, the free encyclopedia
The list of the simplest
elementary mathematics
formulas presented in a concise form.
Distributive property
[
edit
]
a
⋅
(
b
+
c
)
=
(
a
⋅
b
)
+
(
a
⋅
c
)
{\displaystyle \ a\cdot (b+c)=(a\cdot b)+(a\cdot c)}
Sophie Germain's identity
[
edit
]
(
a
±
b
)
2
=
a
2
±
2
a
b
+
b
2
{\displaystyle (a\pm b)^{2}=a^{2}\pm 2ab+b^{2}}
a
2
−
b
2
=
(
a
+
b
)
(
a
−
b
)
{\displaystyle a^{2}-b^{2}=(a+b)(a-b)}
Quadratic equation
[
edit
]
a
x
2
+
b
x
+
c
=
0
,
a
≠
0
{\displaystyle ax^{2}+bx+c=0,\quad a\neq 0}
D
=
b
2
−
4
a
c
,
x
1
,
2
=
−
b
±
D
2
a
{\displaystyle D=b^{2}-4ac,\qquad x_{1,2}={\frac {-b\pm {\sqrt {D}}}{2a}}}
Vieta's formulas
a
(
x
−
x
1
)
(
x
−
x
2
)
=
0
,
x
1
+
x
2
=
−
b
a
,
x
1
⋅
x
2
=
c
a
{\displaystyle a(x-x_{1})(x-x_{2})=0,\qquad x_{1}+x_{2}={\frac {-b}{a}},\qquad x_{1}\cdot x_{2}={\frac {c}{a}}}
.
Exponentiation
[
edit
]
a
−
n
=
1
a
n
a
m
n
=
a
m
n
{\displaystyle a^{-n}={\frac {1}{a^{n}}}\qquad \qquad \quad {\sqrt[{n}]{a^{m}}}=a^{\frac {m}{n}}}
a
m
⋅
a
n
=
a
m
+
n
,
(
a
m
)
n
=
a
m
⋅
n
{\displaystyle a^{m}\cdot a^{n}=a^{m+n},\qquad (a^{m})^{n}=a^{m\cdot n}}
a
m
:
a
n
=
a
m
−
n
,
(
a
⋅
b
)
n
=
a
n
⋅
b
n
{\displaystyle a^{m}:a^{n}=a^{m-n},\qquad (a\cdot b)^{n}=a^{n}\cdot b^{n}}
Logarithm
[
edit
]
a
log
a
b
=
b
log
a
a
=
1
log
a
1
=
0
{\displaystyle a^{\log _{a}b}=b\qquad \log _{a}a=1\qquad \log _{a}1=0}
log
a
(
m
⋅
n
)
=
log
a
m
+
log
a
n
,
log
a
(
n
k
)
=
k
⋅
log
a
n
{\displaystyle \log _{a}(m\cdot n)=\log _{a}m+\log _{a}n,\qquad \log _{a}(n^{k})=k\cdot \log _{a}n}
log
a
(
m
:
n
)
=
log
a
m
−
log
a
n
log
a
k
(
n
)
=
1
k
⋅
log
a
n
{\displaystyle \log _{a}(m:n)=\log _{a}m-\log _{a}n\qquad \log _{a^{k}}(n)={\frac {1}{k}}\cdot \log _{a}n}
Arithmetic progression
[
edit
]
a
n
=
a
1
+
d
(
n
−
1
)
{\displaystyle a_{n}=a_{1}+d(n-1)}
S
n
=
a
1
+
…
+
a
n
=
a
1
+
a
n
2
n
=
2
a
1
+
d
(
n
−
1
)
2
n
{\displaystyle S_{n}=a_{1}+\ldots +a_{n}={a_{1}+a_{n} \over 2}n={2a_{1}+d(n-1) \over 2}n}
Geometric progression
[
edit
]
b
n
=
b
1
⋅
q
n
−
1
{\displaystyle b_{n}=b_{1}\cdot q^{n-1}}
S
n
=
b
1
q
n
−
1
q
−
1
,
(
q
≠
1
)
{\displaystyle S_{n}=b_{1}{\frac {q^{n}-1}{q-1}},\quad (q\neq 1)}
Combinatorics
[
edit
]
P
n
=
1
⋅
2
⋅
3
⋅
…
⋅
n
=
n
!
C
n
k
=
n
!
k
!
(
n
−
k
)
!
A
n
k
=
n
!
(
n
−
k
)
!
{\displaystyle P_{n}=1\cdot 2\cdot 3\cdot \ldots \cdot n=n!\qquad C_{n}^{k}={\frac {n!}{k!(n-k)!}}\qquad A_{n}^{k}={\frac {n!}{(n-k)!}}}
Trigonometry
[
edit
]
Main article:
List of trigonometric identities
sin
α
=
cos
(
90
∘
−
α
)
{\displaystyle \sin \alpha =\cos(90^{\circ }-\alpha )}
tg
α
=
ctg
(
90
∘
−
α
)
{\displaystyle \operatorname {tg} \alpha =\operatorname {ctg} (90^{\circ }-\alpha )}
sin
2
α
+
cos
2
α
=
1
{\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha =1}
tg
α
=
sin
α
cos
α
{\displaystyle \operatorname {tg} \alpha ={\frac {\sin \alpha }{\cos \alpha }}}
ctg
α
=
cos
α
sin
α
{\displaystyle \operatorname {ctg} \alpha ={\frac {\cos \alpha }{\sin \alpha }}}
tg
α
⋅
ctg
α
=
1
{\displaystyle \operatorname {tg} \alpha \cdot \operatorname {ctg} \alpha =1}
1
+
tg
2
α
=
1
cos
2
α
{\displaystyle 1+\operatorname {tg} ^{2}\alpha ={\frac {1}{\cos ^{2}\alpha }}}
1
+
ctg
2
α
=
1
sin
2
α
{\displaystyle 1+\operatorname {ctg} ^{2}\alpha ={\frac {1}{\sin ^{2}\alpha }}}
sin
2
α
=
2
sin
α
cos
α
{\displaystyle \sin 2\alpha =2\sin \alpha \cos \alpha }
cos
2
α
=
cos
2
α
−
sin
2
α
{\displaystyle \cos 2\alpha =\cos ^{2}\alpha -\sin ^{2}\alpha }
sin
3
α
=
3
sin
α
−
4
sin
3
α
{\displaystyle \sin 3\alpha =3\sin \alpha -4\sin ^{3}\alpha }
cos
3
α
=
4
cos
3
α
−
3
cos
α
{\displaystyle \cos 3\alpha =4\cos ^{3}\alpha -3\cos \alpha }
See also
[
edit
]
Binomial theorem
Elementary arithmetic
Long division
Differentiation rules
Lists of integrals
Hidden categories:
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