Arithmetic Number [ edit ]
Natural number The numbers which are generally used in our day to day life for counting are
N
=
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
{\displaystyle \mathbb {N} =0,1,2,3,4,5,6,7,8,9}
Even number Number divides by 2 without remainder . Even number is denotes as 2N
2
N
=
0
,
2
,
4
,
6
,
8
,
.
.
.
{\displaystyle \mathbb {2N} =0,2,4,6,8,...}
Odd number Number divides by 2 with remainder Odd number is denotes as 2N+1
2
N
+
1
=
1
,
3
,
5
,
7
,
9
,
.
.
.
{\displaystyle \mathbb {2N+1} =1,3,5,7,9,...}
Prime number Number divides by 1 and itself without remainder . Prime number is denoted as P
P
=
1
,
3
,
5
,
7...
{\displaystyle \mathbb {P} =1,3,5,7...}
Integer Signed numbers
I
=
(
−
I
,
0
,
+
I
)
=
(
I
<
0
,
I
=
0
,
I
>
0
)
{\displaystyle \mathbb {I} =(-I,0,+I)=(I<0,I=0,I>0)}
Fraction
a
b
{\displaystyle {\frac {a}{b}}}
Complex Number number made up of real and imaginary number
Z
=
a
+
i
b
=
a
2
+
b
2
∠
b
a
{\displaystyle \mathbb {Z} =a+ib={\sqrt {a^{2}+b^{2}}}\angle {\frac {b}{a}}}
Imaginary Number
i
=
−
1
{\displaystyle \mathbb {i} ={\sqrt {-1}}}
i
9
{\displaystyle i9}
Arithmetic Operation [ edit ] Mathematical Operations on arithmetic numbers
Mathematical Operation Symbol Example
Addition
A
+
B
=
C
{\displaystyle A+B=C}
2
+
3
=
5
{\displaystyle 2+3=5}
Subtraction
A
−
B
=
C
{\displaystyle A-B=C}
2
−
3
=
−
1
{\displaystyle 2-3=-1}
Multiplication
A
×
B
=
C
{\displaystyle A\times B=C}
2
×
3
=
6
{\displaystyle 2\times 3=6}
Division
A
B
=
C
{\displaystyle {\frac {A}{B}}=C}
2
3
≈
0.667
{\displaystyle {\frac {2}{3}}\approx 0.667}
Exponentiation
A
n
=
C
{\displaystyle A^{n}=C}
2
3
=
2
×
2
×
2
=
8
{\displaystyle 2^{3}=2\times 2\times 2=8}
Radical
A
=
C
{\displaystyle {\sqrt {A}}=C}
9
=
3
{\displaystyle {\sqrt {9}}=3}
Log
L
o
g
A
=
C
{\displaystyle LogA=C}
L
o
g
100
=
2
{\displaystyle Log100=2}
Ln
L
n
A
=
C
{\displaystyle LnA=C}
L
n
9
≈
2.2
{\displaystyle Ln9\approx 2.2}
Function is an arithmetical expression relate 2 variables . Function is denoted as
f
(
x
)
=
y
{\displaystyle f(x)=y}
meaning for any value of x there is a corresponding value y=f(x)
Where
x - indepent variable
x - depent variable
f(x) - function of x
Types and Graphs of function [ edit ] Polynomial Functions [ edit ]
Straight Line function
f
(
x
)
=
x
{\displaystyle f(x)=x}
Parabola function
f
(
x
)
=
x
2
{\displaystyle f(x)=x^{2}}
Hyperbolic function
f
(
x
)
=
x
3
{\displaystyle f(x)=x^{3}}
Trigonometry Functions [ edit ]
Mathematical operation on arithmetic function [ edit ]
Mathematical Operations Symbol Example
Change in variables
Δ
x
{\displaystyle \Delta x}
Δ
f
(
x
)
=
Δ
y
{\displaystyle \Delta f(x)=\Delta y}
x
−
x
o
{\displaystyle x-x_{o}}
y
−
y
o
{\displaystyle y-y_{o}}
Rate of change
Δ
f
(
x
)
Δ
x
{\displaystyle {\frac {\Delta f(x)}{\Delta x}}}
y
−
y
o
x
−
x
o
{\displaystyle {\frac {y-y_{o}}{x-x_{o}}}}
Limit
L
i
m
f
(
t
)
{\displaystyle Limf(t)}
Differentiation
d
d
t
f
(
t
)
{\displaystyle {\frac {d}{dt}}f(t)}
d
d
t
x
n
=
n
x
n
−
1
{\displaystyle {\frac {d}{dt}}x^{n}=nx^{n-1}}
Integration
∫
f
(
t
)
d
t
{\displaystyle \int f(t)dt}
f
(
x
)
=
0
{\displaystyle f(x)=0}
Polynomial Equation [ edit ] 1st ordered polynomial equation
1st ordered polynomial equation has general form
A
x
+
B
=
0
{\displaystyle Ax+B=0}
x
=
−
B
A
{\displaystyle x=-{\frac {B}{A}}}
Divide the quadratic equation by a a
x
+
B
A
=
0
{\displaystyle x+{\frac {B}{A}}=0}
x
=
−
B
A
{\displaystyle x=-{\frac {B}{A}}}
2nd ordered polynomial equation
1st ordered polynomial equation has general form
A
x
2
+
B
x
+
C
=
0
{\displaystyle Ax^{2}+Bx+C=0}
x
=
−
B
±
B
2
−
4
A
C
2
A
…
(
2
)
{\displaystyle x={\frac {-B\pm {\sqrt {B^{2}-4AC}}}{2A}}\ \dots \ (2)}
Divide the quadratic equation by a a
x
2
+
b
a
x
+
c
a
=
0.
{\displaystyle x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}=0.}
Subtract c / a
x
2
+
b
a
x
=
−
c
a
.
{\displaystyle x^{2}+{\frac {b}{a}}x=-{\frac {c}{a}}.}
The quadratic equation is now in a form to which the method of completing the square can be applied. Thus, add a constant to both sides of the equation such that the left hand side becomes a complete square:
x
2
+
b
a
x
+
(
b
2
a
)
2
=
−
c
a
+
(
b
2
a
)
2
,
{\displaystyle x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}=-{\frac {c}{a}}+\left({\frac {b}{2a}}\right)^{2},}
which produces:
(
x
+
b
2
a
)
2
=
−
c
a
+
b
2
4
a
2
.
{\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}=-{\frac {c}{a}}+{\frac {b^{2}}{4a^{2}}}.}
Accordingly, after rearranging the terms on the right hand side to have a common denominator, we obtain:
(
x
+
b
2
a
)
2
=
b
2
−
4
a
c
4
a
2
.
{\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}.}
The square has thus been completed. Taking the square root of both sides yields the following equation:
x
+
b
2
a
=
±
b
2
−
4
a
c
2
a
.
{\displaystyle x+{\frac {b}{2a}}=\pm {\frac {\sqrt {b^{2}-4ac\ }}{2a}}.}
Isolating x
x
=
−
b
±
b
2
−
4
a
c
2
a
.
{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.}
The plus-minus symbol "±" indicates that both
x
=
−
b
+
b
2
−
4
a
c
2
a
and
x
=
−
b
−
b
2
−
4
a
c
2
a
{\displaystyle x={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\quad {\text{and}}\quad x={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}}
are solutions of the quadratic equation.[1] a
Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as ax 2 − 2bx + c = 0[2] ax 2 + 2bx + c = 0[3] b
A lesser known quadratic formula, as used in Muller's method , and which can be found from Vieta's formulas , provides the same roots via the equation:
x
=
2
c
−
b
∓
b
2
−
4
a
c
.
{\displaystyle x={\frac {2c}{-b\mp {\sqrt {b^{2}-4ac}}}}.}
Differential Equation [ edit ] 1st ordered differential equation
A
d
d
x
f
(
x
)
+
B
f
(
x
)
=
0
{\displaystyle A{\frac {d}{dx}}f(x)+Bf(x)=0}
f
(
x
)
=
A
e
s
t
=
A
e
−
B
A
t
{\displaystyle f(x)=Ae^{st}=Ae^{-{\frac {B}{A}}t}}
A
d
d
x
f
(
x
)
+
B
f
(
x
)
=
0
{\displaystyle A{\frac {d}{dx}}f(x)+Bf(x)=0}
d
d
x
f
(
x
)
=
−
B
A
f
(
x
)
{\displaystyle {\frac {d}{dx}}f(x)=-{\frac {B}{A}}f(x)}
s
f
(
x
)
=
−
B
A
f
(
x
)
{\displaystyle sf(x)=-{\frac {B}{A}}f(x)}
s
=
−
B
A
{\displaystyle s=-{\frac {B}{A}}}
f
(
x
)
=
A
e
s
t
=
A
e
−
B
A
t
{\displaystyle f(x)=Ae^{st}=Ae^{-{\frac {B}{A}}t}}
2nd ordered differential equation
A
d
2
d
x
2
f
(
x
)
+
B
d
d
x
f
(
x
)
+
C
=
0
{\displaystyle A{\frac {d^{2}}{dx^{2}}}f(x)+B{\frac {d}{dx}}f(x)+C=0}
s
2
f
(
x
)
+
2
α
s
f
(
x
)
+
β
f
(
x
)
=
0
{\displaystyle s^{2}f(x)+2\alpha sf(x)+\beta f(x)=0}
The solution of the 2nd ordered polynomial equation above
One real root
s
=
−
α
{\displaystyle s=-\alpha }
i
(
t
)
=
A
e
−
α
t
{\displaystyle i(t)=Ae^{-\alpha t}}
Two real roots
s
=
−
α
±
β
−
α
{\displaystyle s=-\alpha \pm {\sqrt {\beta -\alpha }}}
i
(
t
)
=
A
e
(
α
±
α
−
β
)
t
{\displaystyle i(t)=Ae^{(\alpha \pm {\sqrt {\alpha -\beta }})t}}
One complex roots
s
=
−
α
±
j
β
−
α
{\displaystyle s=-\alpha \pm j{\sqrt {\beta -\alpha }}}
i
(
t
)
=
A
e
(
−
α
±
j
β
−
α
)
t
{\displaystyle i(t)=Ae^{(-\alpha \pm j{\sqrt {\beta -\alpha )}}t}}
Special nth derivative differential equation
d
n
d
t
n
f
(
t
)
=
−
1
T
f
(
t
)
{\displaystyle {\frac {d^{n}}{dt^{n}}}f(t)=-{\frac {1}{T}}f(t)}
s
n
f
(
t
)
=
−
1
T
f
(
t
)
{\displaystyle s^{n}f(t)=-{\frac {1}{T}}f(t)}
s
=
±
j
n
1
T
f
(
x
)
{\displaystyle s=\pm jn{\sqrt {\frac {1}{T}}}f(x)}
f
(
t
)
=
A
e
s
t
=
A
e
±
j
n
1
T
t
=
A
S
i
n
ω
t
{\displaystyle f(t)=Ae^{st}=Ae^{\pm jn{\sqrt {\frac {1}{T}}}t}=ASin\omega t}
ω
=
n
1
T
{\displaystyle \omega =n{\sqrt {\frac {1}{T}}}}
↑ Sterling, Mary Jane (2010), Algebra I For Dummies 978-0-470-55964-2 , https://books.google.com/books?id=2toggaqJMzEC&pg=PA219&dq=quadratic+formula#v=onepage&q=quadratic%20formula&f=false ↑ Kahan, Willian (November 20, 2004), On the Cost of Floating-Point Computation Without Extra-Precise Arithmetic , http://www.cs.berkeley.edu/~wkahan/Qdrtcs.pdf ↑ "Quadratic Formula" , Proof Wiki , http://www.proofwiki.org/wiki/Quadratic_Formula
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