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 $\mathbb {N} =0,1,2,3,4,5,6,7,8,9$ , Even number Number divides by 2 without remainder . Even number is denotes as 2N $\mathbb {2N} =0,2,4,6,8,...$ Odd number Number divides by 2 with remainder Odd number is denotes as 2N+1 $\mathbb {2N+1} =1,3,5,7,9,...$ Prime number Number divides by 1 and itself without remainder . Prime number is denoted as P $\mathbb {P} =1,3,5,7...$ Integer Signed numbers $\mathbb {I} =(-I,0,+I)=(I<0,I=0,I>0)$ Fraction ${\frac {a}{b}}$ Complex Number number made up of real and imaginary number $\mathbb {Z} =a+ib={\sqrt {a^{2}+b^{2}}}\angle {\frac {b}{a}}$ Imaginary Number $\mathbb {i} ={\sqrt {-1}}$ $i9$ Arithmetic Operations

Mathematical Operations on arithmetic numbers

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

Example

$ax,y^{2}$ Operation Arithmetic Expression

Order of performing mathematical operation on expression follows

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

Example

$(x-y)^{2}+y=z$ $x+y^{2}=6$ Coordinate system

Cartesian Coordinate Polar Coordinate Real Number Coordination

A point in XY co ordinate can be presented as ($X,Y$ ) and ($R,\theta$ ) in R θ co ordinate

A , ($X,Y$ ) , ($R,\theta$ )
 Scalar maths Vector Maths $R\angle \theta ={\sqrt {X^{2}+Y^{2}}}\angle Tan^{-1}{\frac {Y}{X}}$ $R={\sqrt {X^{2}+Y^{2}}}$ $\theta =\angle Tan^{-1}{\frac {Y}{X}}$ $X(\theta )=RCos\theta$ $Y(\theta )=RSin\theta$ $R(\theta )=X(\theta )+Y(\theta )=R(Cos\theta +Sin\theta )$ $\nabla \cdot R(\theta )=X(\theta )=RCos\theta$ $\nabla \times R(\theta )=Y(\theta )=RSin\theta$ Complex Number Coordination

 $Z.(X,jY),(Z,\theta )$ $Z^{*}.(X,-jY),(R,-\theta )$ $X(\theta )=ZCos\theta$ $jY(\theta )=jZSin\theta$ $-jY(\theta )=-jZSin\theta$ $Z\angle \theta ={\sqrt {X^{2}+Y^{2}}}\angle Tan^{-1}{\frac {Y}{X}}$ $Z\angle -\theta ={\sqrt {X^{2}+Y^{2}}}\angle -Tan^{-1}{\frac {Y}{X}}$ $Z={\sqrt {X^{2}+Y^{2}}}$ $\theta =\angle Tan^{-1}{\frac {Y}{X}}$ $Z(\theta )=X(\theta )+jY(\theta )=Z(Cos\theta +jSin\theta )$ $\nabla \cdot Z(\theta )=X(\theta )=ZCos\theta$ $\nabla \times Z(\theta )=jY(\theta )=jZSin\theta$ $Z^{*}(\theta )=X(\theta )-jY(\theta )=Z(Cos\theta -jSin\theta )$ $\nabla \cdot Z(\theta )=X(\theta )=Z^{*}Cos\theta$ $\nabla \times Z(\theta )=-jY(\theta )=jZ^{*}Sin\theta$ $Cos\theta ={\frac {Z(\theta )+Z^{*}(\theta )}{2}}$ $Sin\theta ={\frac {Z(\theta )-Z^{*}(\theta )}{2j}}$ $-Sin\theta ={\frac {Z^{*}(\theta )-Z^{(}\theta )}{2j}}$ Arithematic Function

Definition

Function is an arithmetical expression which relates 2 variables . Function is denoted as

$f(x)=y$ meaning for any value of x there is a corresponding value y=f(x)

Where

x - independent variable
y - dependent variable
f(x) - function of x

Graph of function

$f(x)=x$ x -2 -1 0 1 2 f(x) -2 -1 0 1 2
Straight line passing through origin point (0,0) with slope equals 1

$f(x)=2x$ x -2 -1 0 1 2 f(x) -4 -2 0 2 4
Straight line passing through origin point (0,0) with slope equals 2

$f(x)=2x+3$ x -2 -1 0 1 2 f(x) -1 1 3 5 7

Straight line with slope equals 2 has x intercept (-3/2,0) and y intercept (0,3)

Arithmetic Equation

An arithmetic equation is an expression of a function of variable that has a value equal to zero

$f(x)=0$ Arithmetic equations can be solved to find the value of variable that satisfies the equation. The process of finding this value is called root finding. All values of variable that make its function equal to zero are called roots of the equation.

Example

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