Download Basic symbols and algebra notations

Basic symbols and algebra notations
Background mathematics review
David Miller
Basic symbols and algebra notations
Basic mathematical symbols
Background mathematics review
David Miller
Elementary arithmetic symbols

Equals

Addition or ”plus”
Subtraction, “minus” or “less”

 or  Multiplication
23  6
 or / Division
63  2
23 5
3 2 1
23  6
6/3 2
 dividend 
6
or (numerator ) 
  quotient   2 6 3  2
(demonimator )  divisor 
3
6
3
2
Relational symbols

“Is equivalent to”
 or  “Is approximately equal to”







“Is proportional to”
“Is greater than”
“Is greater than or equal to”
“Is less than”
“Is less than or equal to”
“Is much greater than”
“Is much less than”
x
x/ y 
y
1
 0.33
3
a x  x
32
1  x2  1
23
1  1  x2
100  1
1  100
Greek characters used as symbols
Upper
case
Lower
case
Name
Roman
equiv.
Keyboard
Upper
case
Lower
case
Name
Roman
equiv.
Keyboard


alpha
a
a


nu
n
n


beta
b
b


xi
x
x


gamma
g
g


omicron
(o)
o


delta
d
d


pi
p
p


epsilon
e
e


rho
r
r


zeta
z
z


sigma
s
s


eta
(e)
h


tau
t
t


theta
th
q


upsilon
u
u


iota
i
i


phi
ph
f


kappa
k
k


chi
ch
c


lambda
l
l


psi
psy
y


mu
m
m


omega
o
w
Basic symbols and algebra notations
Basic mathematical operations
Background mathematics review
David Miller
Conventions for multiplication
For multiplying numbers
We explicitly use the multiplication sign “”
23  6
For multiplying variables
We can use the multiplication sign
But where there is no confusion
We drop it
ab  c
might be simply replaced by
ab  c
Use of parentheses and brackets
When we want to group numbers or variables
We can use parentheses (or brackets)
2  (3  4)  2  7  14
For such grouping, we can alternatively use
square brackets
2  3  4  2  7  14
or curly brackets
2  3  4  2  7  14
When used this way, there is no difference in
the mathematical meaning of these brackets
Associative property
Operations are associative if it does not matter
how we group them
e.g., addition of numbers is associative
 a  b   c  a  b  c 
e.g., multiplication of numbers is associative
 a  b   c  a  b  c 
But
division of numbers is not associative
8 / 4  / 2  2 / 2  1 but 8 /  4 / 2   8 / 2  4
Distributive property
Property where terms within parentheses can
be “distributed” to remove the parentheses
a  b  c   a  b  a  c
Here, multiplication is said to be
distributive over addition
Many other conceivable operations are
not distributive, however
E.g., addition is not distributive over
multiplication
3   2  5   13   3  2    3  5   40
Commutative property
Property where the order can be switched round
e.g., addition of numbers is commutative
ab  ba
e.g., multiplication of numbers is commutative
ab  ba
But
e.g., subtraction is not commutative
5  3  2  3  5  2
e.g., division is not commutative
6 / 3  2  3 / 6  12
Basic symbols and algebra notations
Algebra notation and functions
Background mathematics review
David Miller
Parentheses and functions
A function is something that
relates or “maps”
One set of values
Such as an “input”
variable or “argument” x
To another set of values
which we could think of
as an “output”
For example, the function
1
f  x  x 
4
3
f  x
2
1
2
x
1
0
1
2
1
2
Parentheses and functions
3
Conventionally, we say
“f of x” when we read f  x 
Here obviously
f  x  is not “f times x”
Most commonly
Only parentheses are used
around the argument x
not square [ ] or curly { }
brackets
f  x
2
1
2
x
1
0
1
2
1
2
Parentheses and functions
For a few very commonly used
functions
Such as the trigonometric
functions
The parentheses are
optionally omitted when
the argument is simple
sin  instead of sin  
Note, incidentally,
sin      sin  
1
sin 


1

Parentheses and functions
For a few very commonly used
functions
Such as the trigonometric
functions
The parentheses are
optionally omitted when
the argument is simple
cos  instead of cos  
Note, incidentally
cos     cos  
cos 
1


1

Sine, cosine, and tangent
r or
hypotenuse
angle 

x, base or
“adjacent” side
y, height or
“opposite”
side
Defined using a right-angled
triangle
x
y
y
sin  
cos  
tan  
r
x
r
sin 
tan  
cos 
Natural units for angles in
mathematics are radians
2 radians in a circle
1 radian ~ 57.3 degrees
Cosecant, secant, and cotangent
r or
hypotenuse
Cosecant
r
1
cosec  csc  
y sin 
y, height or
“opposite” Secant
angle 

side
r
1
sec  
x cos
x, base or
“adjacent” side
Cotangent
x
1
cos 
cotan   cot   

y tan  sin 
Inverse sine function
The inverse sine function sin 1  a 
or arcsine function
Pronounced “arc-sine”
works backwards to give the
angle from the sine value
If a  sin  then

sin 1  a 
2
a
1
arcsin  a   asin  a   sin 1  a   
Note sin 1  a  does not mean 1/ sin  a 
0


2
The “-1” here means “inverse function” not “reciprocal”
1
sin2 and cos2 functions
1
sin 
2
However
2
2
sin   sin   sin    sin  
Not
sin  sin  

2
2
cos


cos

Similarly


0
1

0

cos 2 
Only trigonometric functions
and their close relatives
commonly use this notation
