# 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 Operations on Arithmatic numbers

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 ${\displaystyle A^{n}=C}$ ${\displaystyle 2^{3}=2\times 2\times 2=8}$ Root ${\displaystyle {\sqrt {A}}=C}$ ${\displaystyle {\sqrt {9}}=3}$ Logarithm ${\displaystyle LogA=C}$ ${\displaystyle Log100=2}$ Ln ${\displaystyle LnA=C}$ ${\displaystyle Ln9\approx 2.2}$

## Arithmetic Expression

Example of arithmetic expression

${\displaystyle ax,y^{2}}$

### Operation Arithmetic Expression

Example

${\displaystyle (x-y)^{2}+y=z}$
${\displaystyle x+y^{2}=6}$

Order of performing mathematical operation on expression follows

1. Parenthesis . {}, [] , ()
2. Power .
3. Plus, Minus,Multiply,Divide . +, -, x , /

## Arithmetic Function

### Definitiion

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

### Example

• ${\displaystyle f(x)=x}$
 x -2 -1 0 1 2 f(x) -2 -1 0 1 2

Plot those points above in rectangular coordinate we obtain a graph of a straight line passing through origin point (0,0)

• ${\displaystyle f(x)=2x}$
 x -2 -1 0 1 2 f(x) -4 -2 0 2 4

Plot those points above in rectangular coordinate we obtain a graph of a straight line passing through origin point (0,0) with a steeper slope

• ${\displaystyle f(x)=2x+3}$
 x -2 -1 0 1 2 f(x) -7 -5 0 5 7

Plot those points above in rectangular coordinate we obtain a graph of a straight line

Cuts y axis at (0,3) . This point is called x intercept
Cuts x axis at (-3/2,0) . This point is called y intercept

### Calculus , mathematical operations 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

Arithmetic Equation is a expression of a function of variable that has value equal to zero

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

Equation can be solve to find value of variable that satisfy equation . The process finding value of variable is called Root finding . All value of variable that would make its function equal to zero is called Root of the equation

### Example

Equation . ${\displaystyle 2x+5=9}$
Root . ${\displaystyle x={\frac {9-5}{2}}={\frac {4}{2}}=2}$
${\displaystyle x=2}$ is the root of the equation ${\displaystyle 2x+5=9}$ since substitution the value of x in the equation we have
${\displaystyle 2(2)+5=9}$