Fundamental Mathematics/Arithmetic

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 on Arithmatic numbers

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	$LogA=C$	$Log100=2$
Ln	$LnA=C$	$Ln9\approx 2.2$

Arithmetic Expression

Example of arithmetic expression

$ax,y^{2}$

Operation Arithmetic Expression

Example

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

$x+y^{2}=6$

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

$f(x)=y$

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

• $f(x)=x$

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)

• $f(x)=2x$

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

• $f(x)=2x+3$

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

Types and Graphs of function

• Polynomial Function
• Trigonometry Function

Calculus , mathematical operations on arithmetic function

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

Arithmetic Equation

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

$f(x)=0$

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 . $2x+5=9$

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

$2(2)+5=9$

Types of equations

• Polynomial Equation
• Differential Equation
• Partial Differential Equation
• Inequality Equation

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