# Fundamental Mathematics/Arithmetic

## Numbers

 Natural number A non-negative integer. Also referred to as counting numbers. $\mathbb {N} =0,1,2,3,4,5,6,7,8,9$ , Even number A number which can be divided by 2 without remainder. $\mathbb {2N} =0,2,4,6,8,...$ Odd number A number which cannot be divided by 2 without remainder. $\mathbb {2N+1} =1,3,5,7,9,...$ Prime number A number above 1 which can be divided by 1 and itself without remainder. $\mathbb {P} =2,3,5,7...$ Integer A signed whole number. $\mathbb {I} =(-I,0,+I)=(I<0,I=0,I>0)$ Fraction ${\frac {a}{b}}$ Complex Number A number made up of real and imaginary numbers. $\mathbb {Z} =a+ib={\sqrt {a^{2}+b^{2}}}\angle {\frac {b}{a}}$ Imaginary Number $\mathbb {i} ={\sqrt {-1}}$ $i9$ ## 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 $\log {A}=C$ $\log {100}=2$ Natural Logarithm $\ln {A}=C$ $\ln {9}\approx 2.2$ ## Expressions

An expression is a mathematical construct which evaluates to something. $2\times 10$ , $2^{8}$ , and ${\sqrt {64}}$ are all expressions.

• Expression

## Order of operations

Order of performing mathematical operation on expressions are as follows

1. Evaluate expressions within parenthesis {}, [] , ()
2. Exponentiation.
3. Multiply and divide, from left to right.
4. Add and subtract, from left to right.

Example

$(4-2)^{2}+2=6$ 1. $2^{2}+2=6$ (parenthesis)
2. $4+2=6$ (exponentiation)
3. $6=6$ (addition)
$20+6^{2}=56$ 1. $20+36=56$ (exponentiation)
2. $56=56$ (addition)

• Order of operations

## Coordinate system

### Cartesian coordinates ### Polar coordinates ### 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

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

### Definition

Functions are an arithmetical expression which relates 2 variables. Functions are usually 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 $x$ .

### Graphs of functions

$f(x)=x$ 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)

### Operations on function

• Function

## Equations

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

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

### Examples

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

• Equation
• Arithmetic