background mathematics review david miller mathematics review david miller parentheses and functions...
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Basic symbols and algebra notations
Basic mathematical symbols
Background mathematics review David Miller
Elementary arithmetic symbols
EqualsAddition or ”plus”Subtraction, “minus” or “less”MultiplicationDivision
2 3 5
3 2 1
/or 6 3 2 6 / 3 2 2 3 6 or 2 3 6
or ( )
( )dividendnumerator quotient
demonimator divisor 6 23 6 2
3 6
3 2
Relational symbols
“Is equivalent to”
“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”
/ xx yy
a x x
or 1 0.333
3 221 1x
2 3
21 1 x 100 1 1 100
Greek characters used as symbols
Upper case
Lower case
Name Roman equiv.
Key-board
alpha a a
beta b b
gamma g g
delta d d
epsilon e e
zeta z z
eta (e) h
theta th q
iota i i
kappa k k
lambda l l
mu m m
Upper case
Lower case
Name Roman equiv.
Key-board
nu n n
xi x x
omicron (o) o
pi p p
rho r r
sigma s s
tau t t
upsilon u u
phi ph f
chi ch c
psi psy y
omega o w
Basic symbols and algebra notations
Basic mathematical operations
Background mathematics review David Miller
Conventions for multiplication
For multiplying numbersWe explicitly use the multiplication sign “”
For multiplying variablesWe can use the multiplication sign
But where there is no confusionWe drop it
might be simply replaced by
2 3 6
a b c
ab c
Use of parentheses and brackets
When we want to group numbers or variablesWe can use parentheses (or brackets)
For such grouping, we can alternatively usesquare brackets
or curly brackets
When used this way, there is no difference in the mathematical meaning of these brackets
2 (3 4) 2 7 14
2 3 4 2 7 14
2 3 4 2 7 14
Associative property
Operations are associative if it does not matter how we group them
e.g., addition of numbers is associative
e.g., multiplication of numbers is associative
But division of numbers is not associative
but
a b c a b c
a b c a b c
8 / 4 / 2 2 / 2 1 8 / 4 / 2 8 / 2 4
Distributive property
Property where terms within parentheses can be “distributed” to remove the parentheses
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
a b c a b a c
3 2 5 13 3 2 3 5 40
Commutative property
Property where the order can be switched rounde.g., addition of numbers is commutative
e.g., multiplication of numbers is commutative
Bute.g., subtraction is not commutative
e.g., division is not commutative
a b b a
a b b a
5 3 2 3 5 2
126 / 3 2 3 / 6
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 valuesSuch as an “input”
variable or “argument” xTo another set of values
which we could think of as an “output”
For example, the function
14
f x x
2 1 0 1 2
2
1
1
2
3
x
f x
Parentheses and functions
Conventionally, we say“f of x” when we read
Here obviously is not “f times x”
Most commonlyOnly parentheses are used around the argument x
not square [ ] or curly { } brackets
2 1 0 1 2
2
1
1
2
3
x
f x f x
f x
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
instead ofNote, incidentally,
sin
sin sin 1
1
sin sin
1
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
instead ofNote, incidentally
cos
cos cos
cos cos
Sine, cosine, and tangent
Defined using a right-angled triangle
Natural units for angles in mathematics are radians
2 radians in a circle1 radian ~ 57.3 degrees
x, base or “adjacent” side
y, height or “opposite”
side
r or hypotenuse
sin yr
cos xr
tan yx
angle sintancos
Cosecant, secant, and cotangent
Cosecant
Secant
Cotangent
x, base or “adjacent” side
y, height or “opposite”
side
r or hypotenuse
1cosec cscsin
ry
1seccosx
r
1 coscotan cottan siny
x
angle
Inverse sine function
The inverse sine functionor arcsine function
Pronounced “arc-sine”works backwards to give the angle from the sine value
If then
Note does not mean
1 0 1
2
2
1arcsin asin sina a a
sina
1sin a 1sin a
a
1sin a 1/ sin aThe “-1” here means “inverse function” not “reciprocal”
sin2 and cos2 functions
However
Not
Similarly
Only trigonometric functions and their close relatives commonly use this notation
22sin sin sin sin
sin sin 0
12sin
22cos cos
0
12cos