Natural number A non-negative integer. Also referred to as counting numbers.
N
=
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
{\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.
2
N
=
0
,
2
,
4
,
6
,
8
,
.
.
.
{\displaystyle \mathbb {2N} =0,2,4,6,8,...}
Odd number A number which cannot be divided by 2 without remainder.
2
N
+
1
=
1
,
3
,
5
,
7
,
9
,
.
.
.
{\displaystyle \mathbb {2N+1} =1,3,5,7,9,...}
Prime number A number above 1 which can be divided by 1 and itself without remainder.
P
=
2
,
3
,
5
,
7...
{\displaystyle \mathbb {P} =2,3,5,7...}
Integer A signed whole number.
I
=
(
−
I
,
0
,
+
I
)
=
(
I
<
0
,
I
=
0
,
I
>
0
)
{\displaystyle \mathbb {I} =(-I,0,+I)=(I<0,I=0,I>0)}
Fraction
a
b
{\displaystyle {\frac {a}{b}}}
Complex Number A number made up of real and imaginary numbers.
Z
=
a
+
i
b
=
a
2
+
b
2
∠
b
a
{\displaystyle \mathbb {Z} =a+ib={\sqrt {a^{2}+b^{2}}}\angle {\frac {b}{a}}}
Imaginary Number
i
=
−
1
{\displaystyle \mathbb {i} ={\sqrt {-1}}}
i
9
{\displaystyle i9}
Mathematical Operations on arithmetic numbers
Mathematical Operation Symbol Example
Addition
A
+
B
=
C
{\displaystyle A+B=C}
2
+
3
=
5
{\displaystyle 2+3=5}
Subtraction
A
−
B
=
C
{\displaystyle A-B=C}
2
−
3
=
−
1
{\displaystyle 2-3=-1}
Multiplication
A
×
B
=
C
{\displaystyle A\times B=C}
2
×
3
=
6
{\displaystyle 2\times 3=6}
Division
A
B
=
C
{\displaystyle {\frac {A}{B}}=C}
2
3
≈
0.667
{\displaystyle {\frac {2}{3}}\approx 0.667}
Exponentiation
A
n
=
C
{\displaystyle A^{n}=C}
2
3
=
2
×
2
×
2
=
8
{\displaystyle 2^{3}=2\times 2\times 2=8}
Root
A
=
C
{\displaystyle {\sqrt {A}}=C}
9
=
3
{\displaystyle {\sqrt {9}}=3}
Logarithm
log
A
=
C
{\displaystyle \log {A}=C}
log
100
=
2
{\displaystyle \log {100}=2}
Natural Logarithm
ln
A
=
C
{\displaystyle \ln {A}=C}
ln
9
≈
2.2
{\displaystyle \ln {9}\approx 2.2}
An expression is a mathematical construct which evaluates to something.
2
×
10
{\displaystyle 2\times 10}
2
8
{\displaystyle 2^{8}}
64
{\displaystyle {\sqrt {64}}}
Order of performing mathematical operation on expressions are as follows
Evaluate expressions within parenthesis {}, [] , ()
Exponentiation.
Multiply and divide, from left to right.
Add and subtract, from left to right. Example
(
4
−
2
)
2
+
2
=
6
{\displaystyle (4-2)^{2}+2=6}
2
2
+
2
=
6
{\displaystyle 2^{2}+2=6}
4
+
2
=
6
{\displaystyle 4+2=6}
6
=
6
{\displaystyle 6=6}
20
+
6
2
=
56
{\displaystyle 20+6^{2}=56}
20
+
36
=
56
{\displaystyle 20+36=56}
56
=
56
{\displaystyle 56=56}
A point in XY co ordinate can be presented as (
X
,
Y
{\displaystyle X,Y}
R
,
θ
{\displaystyle R,\theta }
A , (
X
,
Y
{\displaystyle X,Y}
R
,
θ
{\displaystyle R,\theta }
Scalar maths Vector Maths
R
∠
θ
=
X
2
+
Y
2
∠
T
a
n
−
1
Y
X
{\displaystyle R\angle \theta ={\sqrt {X^{2}+Y^{2}}}\angle Tan^{-1}{\frac {Y}{X}}}
R
=
X
2
+
Y
2
{\displaystyle R={\sqrt {X^{2}+Y^{2}}}}
θ
=
∠
T
a
n
−
1
Y
X
{\displaystyle \theta =\angle Tan^{-1}{\frac {Y}{X}}}
X
(
θ
)
=
R
C
o
s
θ
{\displaystyle X(\theta )=RCos\theta }
Y
(
θ
)
=
R
S
i
n
θ
{\displaystyle Y(\theta )=RSin\theta }
R
(
θ
)
=
X
(
θ
)
+
Y
(
θ
)
=
R
(
C
o
s
θ
+
S
i
n
θ
)
{\displaystyle R(\theta )=X(\theta )+Y(\theta )=R(Cos\theta +Sin\theta )}
∇
⋅
R
(
θ
)
=
X
(
θ
)
=
R
C
o
s
θ
{\displaystyle \nabla \cdot R(\theta )=X(\theta )=RCos\theta }
∇
×
R
(
θ
)
=
Y
(
θ
)
=
R
S
i
n
θ
{\displaystyle \nabla \times R(\theta )=Y(\theta )=RSin\theta }
Complex number coordination [ edit | edit source ]
Z
.
(
X
,
j
Y
)
,
(
Z
,
θ
)
{\displaystyle Z.(X,jY),(Z,\theta )}
Z
∗
.
(
X
,
−
j
Y
)
,
(
R
,
−
θ
)
{\displaystyle Z^{*}.(X,-jY),(R,-\theta )}
X
(
θ
)
=
Z
C
o
s
θ
{\displaystyle X(\theta )=ZCos\theta }
j
Y
(
θ
)
=
j
Z
S
i
n
θ
{\displaystyle jY(\theta )=jZSin\theta }
−
j
Y
(
θ
)
=
−
j
Z
S
i
n
θ
{\displaystyle -jY(\theta )=-jZSin\theta }
Z
∠
θ
=
X
2
+
Y
2
∠
T
a
n
−
1
Y
X
{\displaystyle Z\angle \theta ={\sqrt {X^{2}+Y^{2}}}\angle Tan^{-1}{\frac {Y}{X}}}
Z
∠
−
θ
=
X
2
+
Y
2
∠
−
T
a
n
−
1
Y
X
{\displaystyle Z\angle -\theta ={\sqrt {X^{2}+Y^{2}}}\angle -Tan^{-1}{\frac {Y}{X}}}
Z
=
X
2
+
Y
2
{\displaystyle Z={\sqrt {X^{2}+Y^{2}}}}
θ
=
∠
T
a
n
−
1
Y
X
{\displaystyle \theta =\angle Tan^{-1}{\frac {Y}{X}}}
Z
(
θ
)
=
X
(
θ
)
+
j
Y
(
θ
)
=
Z
(
C
o
s
θ
+
j
S
i
n
θ
)
{\displaystyle Z(\theta )=X(\theta )+jY(\theta )=Z(Cos\theta +jSin\theta )}
∇
⋅
Z
(
θ
)
=
X
(
θ
)
=
Z
C
o
s
θ
{\displaystyle \nabla \cdot Z(\theta )=X(\theta )=ZCos\theta }
∇
×
Z
(
θ
)
=
j
Y
(
θ
)
=
j
Z
S
i
n
θ
{\displaystyle \nabla \times Z(\theta )=jY(\theta )=jZSin\theta }
Z
∗
(
θ
)
=
X
(
θ
)
−
j
Y
(
θ
)
=
Z
(
C
o
s
θ
−
j
S
i
n
θ
)
{\displaystyle Z^{*}(\theta )=X(\theta )-jY(\theta )=Z(Cos\theta -jSin\theta )}
∇
⋅
Z
(
θ
)
=
X
(
θ
)
=
Z
∗
C
o
s
θ
{\displaystyle \nabla \cdot Z(\theta )=X(\theta )=Z^{*}Cos\theta }
∇
×
Z
(
θ
)
=
−
j
Y
(
θ
)
=
j
Z
∗
S
i
n
θ
{\displaystyle \nabla \times Z(\theta )=-jY(\theta )=jZ^{*}Sin\theta }
C
o
s
θ
=
Z
(
θ
)
+
Z
∗
(
θ
)
2
{\displaystyle Cos\theta ={\frac {Z(\theta )+Z^{*}(\theta )}{2}}}
S
i
n
θ
=
Z
(
θ
)
−
Z
∗
(
θ
)
2
j
{\displaystyle Sin\theta ={\frac {Z(\theta )-Z^{*}(\theta )}{2j}}}
−
S
i
n
θ
=
Z
∗
(
θ
)
−
Z
(
θ
)
2
j
{\displaystyle -Sin\theta ={\frac {Z^{*}(\theta )-Z^{(}\theta )}{2j}}}
Functions are an arithmetical expression which relates 2 variables. Functions are usually denoted as
f
(
x
)
=
y
{\displaystyle f(x)=y}
meaning for any value of
x
{\displaystyle x}
y
=
f
(
x
)
{\displaystyle y=f(x)}
x
{\displaystyle x}
y
{\displaystyle y}
f
(
x
)
{\displaystyle f(x)}
x
{\displaystyle x}
f
(
x
)
=
x
{\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
f
(
x
)
=
2
x
{\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
f
(
x
)
=
2
x
+
3
{\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) An equation is an expression of a function of a variable that has a value equal to zero
f
(
x
)
=
0
{\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.
Equation .
2
x
+
5
=
9
{\displaystyle 2x+5=9}
Root .
x
=
9
−
5
2
=
4
2
=
2
{\displaystyle x={\frac {9-5}{2}}={\frac {4}{2}}=2}
x
=
2
{\displaystyle x=2}
2
x
+
5
=
9
{\displaystyle 2x+5=9}
2
(
2
)
+
5
=
9
{\displaystyle 2(2)+5=9}
