# Fundamental Mathematics/Arithmetic

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## Arithmetic Number

 Natural number The numbers which are generally used in our day to day life for counting are termed as natural numbers. They are also referred to as "counting" numbers ${\displaystyle \mathbb {N} =0,1,2,3,4,5,6,7,8,9}$, Even number Number divides by 2 without remainder . Even number is denotes as 2N ${\displaystyle \mathbb {2N} =0,2,4,6,8,...}$ Odd number Number divides by 2 with remainder Odd number is denotes as 2N+1 ${\displaystyle \mathbb {2N+1} =1,3,5,7,9,...}$ Prime number Number divides by 1 and itself without remainder . Prime number is denoted as P ${\displaystyle \mathbb {P} =1,3,5,7...}$ Integer Signed numbers ${\displaystyle \mathbb {I} =(-I,0,+I)=(I<0,I=0,I>0)}$ Fraction ${\displaystyle {\frac {a}{b}}}$ Complex Number number made up of real and imaginary number ${\displaystyle \mathbb {Z} =a+ib={\sqrt {a^{2}+b^{2}}}\angle {\frac {b}{a}}}$ Imaginary Number ${\displaystyle \mathbb {i} ={\sqrt {-1}}}$ ${\displaystyle i9}$

## Arithmetic Operation

Mathematical Operations on arithmetic numbers

 Mathematical Operation Symbol Example Addition ${\displaystyle A+B=C}$ ${\displaystyle 2+3=5}$ Subtraction ${\displaystyle A-B=C}$ ${\displaystyle 2-3=-1}$ Multiplication ${\displaystyle A\times B=C}$ ${\displaystyle 2\times 3=6}$ Division ${\displaystyle {\frac {A}{B}}=C}$ ${\displaystyle {\frac {2}{3}}\approx 0.667}$ Exponentiation ${\displaystyle A^{n}=C}$ ${\displaystyle 2^{3}=2\times 2\times 2=8}$ Radical ${\displaystyle {\sqrt {A}}=C}$ ${\displaystyle {\sqrt {9}}=3}$ Log ${\displaystyle LogA=C}$ ${\displaystyle Log100=2}$ Ln ${\displaystyle LnA=C}$ ${\displaystyle Ln9\approx 2.2}$

## Arithmetic Function

Function is an arithmetical expression relate 2 variables . Function is denoted as

${\displaystyle f(x)=y}$

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

#### Polynomial Functions

 Straight Line function ${\displaystyle f(x)=x}$ Parabola function ${\displaystyle f(x)=x^{2}}$ Hyperbolic function ${\displaystyle f(x)=x^{3}}$

#### Trigonometry Functions

 Sine Function ${\displaystyle f(x)=Sin\theta }$ Cosine Function ${\displaystyle f(x)=Cos\theta }$ Secant Function ${\displaystyle f(x)=Sec\theta }$ CoSecant Function ${\displaystyle f(x)=Cosec\theta }$ Tan Function ${\displaystyle f(x)=Tan\theta }$ Cotan Function ${\displaystyle f(x)=Cotan\theta }$

### Mathematical operation on arithmetic function

 Mathematical Operations Symbol Example Change in variables ${\displaystyle \Delta x}$ , ${\displaystyle \Delta f(x)=\Delta y}$ ${\displaystyle x-x_{o}}$ , ${\displaystyle y-y_{o}}$ Rate of change ${\displaystyle {\frac {\Delta f(x)}{\Delta x}}}$ ${\displaystyle {\frac {y-y_{o}}{x-x_{o}}}}$ Limit ${\displaystyle Limf(t)}$ Differentiation ${\displaystyle {\frac {d}{dt}}f(t)}$ ${\displaystyle {\frac {d}{dt}}x^{n}=nx^{n-1}}$ Integration ${\displaystyle \int f(t)dt}$

## Arithmetic Equation

${\displaystyle f(x)=0}$

### Polynomial Equation

1st ordered polynomial equation

1st ordered polynomial equation has general form ${\displaystyle Ax+B=0}$ . The solution to the 1st ordered polynomial equation
${\displaystyle x=-{\frac {B}{A}}}$


Divide the quadratic equation by a, which is allowed because a is non-zero:

${\displaystyle x+{\frac {B}{A}}=0}$
${\displaystyle x=-{\frac {B}{A}}}$

2nd ordered polynomial equation

1st ordered polynomial equation has general form ${\displaystyle Ax^{2}+Bx+C=0}$ . The solution of the 2nd ordered polynomial equation ${\displaystyle x={\frac {-B\pm {\sqrt {B^{2}-4AC}}}{2A}}\ \dots \ (2)}$


Divide the quadratic equation by a, which is allowed because a is non-zero:

${\displaystyle x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}=0.}$

Subtract c/a from both sides of the equation, yielding:

${\displaystyle x^{2}+{\frac {b}{a}}x=-{\frac {c}{a}}.}$

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:

${\displaystyle x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}=-{\frac {c}{a}}+\left({\frac {b}{2a}}\right)^{2},}$

which produces:

${\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}=-{\frac {c}{a}}+{\frac {b^{2}}{4a^{2}}}.}$

Accordingly, after rearranging the terms on the right hand side to have a common denominator, we obtain:

${\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}.}$

The square has thus been completed. Taking the square root of both sides yields the following equation:

${\displaystyle x+{\frac {b}{2a}}=\pm {\frac {\sqrt {b^{2}-4ac\ }}{2a}}.}$

Isolating x gives the quadratic formula:

${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.}$

The plus-minus symbol "±" indicates that both

${\displaystyle x={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\quad {\text{and}}\quad x={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}}$

are solutions of the quadratic equation.[1] There are many alternatives of this derivation with minor differences, mostly concerning the manipulation of a.

Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as ax2 − 2bx + c = 0[2] or ax2 + 2bx + c = 0,[3] where b has a magnitude one half of the more common one. These result in slightly different forms for the solution, but are otherwise equivalent.

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:

${\displaystyle x={\frac {2c}{-b\mp {\sqrt {b^{2}-4ac}}}}.}$

### Differential Equation

1st ordered differential equation

${\displaystyle A{\frac {d}{dx}}f(x)+Bf(x)=0}$ . Solution of the 1st ordered differential equation above is ${\displaystyle f(x)=Ae^{st}=Ae^{-{\frac {B}{A}}t}}$


${\displaystyle A{\frac {d}{dx}}f(x)+Bf(x)=0}$
${\displaystyle {\frac {d}{dx}}f(x)=-{\frac {B}{A}}f(x)}$
${\displaystyle sf(x)=-{\frac {B}{A}}f(x)}$
${\displaystyle s=-{\frac {B}{A}}}$
${\displaystyle f(x)=Ae^{st}=Ae^{-{\frac {B}{A}}t}}$

2nd ordered differential equation

${\displaystyle A{\frac {d^{2}}{dx^{2}}}f(x)+B{\frac {d}{dx}}f(x)+C=0}$
${\displaystyle s^{2}f(x)+2\alpha sf(x)+\beta f(x)=0}$

The solution of the 2nd ordered polynomial equation above

 One real root ${\displaystyle s=-\alpha }$ ${\displaystyle i(t)=Ae^{-\alpha t}}$ Two real roots ${\displaystyle s=-\alpha \pm {\sqrt {\beta -\alpha }}}$ ${\displaystyle i(t)=Ae^{(\alpha \pm {\sqrt {\alpha -\beta }})t}}$ One complex roots ${\displaystyle s=-\alpha \pm j{\sqrt {\beta -\alpha }}}$ ${\displaystyle i(t)=Ae^{(-\alpha \pm j{\sqrt {\beta -\alpha )}}t}}$

Special nth derivative differential equation

${\displaystyle {\frac {d^{n}}{dt^{n}}}f(t)=-{\frac {1}{T}}f(t)}$
${\displaystyle s^{n}f(t)=-{\frac {1}{T}}f(t)}$
${\displaystyle s=\pm jn{\sqrt {\frac {1}{T}}}f(x)}$
${\displaystyle f(t)=Ae^{st}=Ae^{\pm jn{\sqrt {\frac {1}{T}}}t}=ASin\omega t}$
${\displaystyle \omega =n{\sqrt {\frac {1}{T}}}}$

1. Sterling, Mary Jane (2010), Algebra I For Dummies, Wiley Publishing, ISBN 978-0-470-55964-2
2. Kahan, Willian (November 20, 2004), On the Cost of Floating-Point Computation Without Extra-Precise Arithmetic