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

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