# Fundamental Mathematics/Arithmetic

## Numbers

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

## Operations

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 ${\displaystyle \log {A}=C}$ ${\displaystyle \log {100}=2}$ Natural Logarithm ${\displaystyle \ln {A}=C}$ ${\displaystyle \ln {9}\approx 2.2}$

## Expressions

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

## 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

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

## Coordinate system

### Real number coordination

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

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

### Complex number coordination

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

## Functions

### Definition

Functions are an arithmetical expression which relates 2 variables. Functions are usually denoted as

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

meaning for any value of ${\displaystyle x}$  there is a corresponding value ${\displaystyle y=f(x)}$  where

${\displaystyle x}$  - independent variable.
${\displaystyle y}$  - dependent variable.
${\displaystyle f(x)}$  - function of ${\displaystyle x}$ .

### Graphs of functions

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

 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

${\displaystyle f(x)=2x}$

 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

${\displaystyle f(x)=2x+3}$

 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)

## Equations

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

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

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 . ${\displaystyle 2x+5=9}$
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 ${\displaystyle 2(2)+5=9}$

## Further reading

 Go to the School of Mathematics