# overview of mathematical structures
## expressions
expressions represent values and consist of variables, constants, and operators. they do not include an equals sign or assert equality.

* algebraic expressions:
  * monomial: a single term (eg '5 x')
  * binomial: two terms (eg 'x ** 2 + 2 x')
  * polynomial: multiple terms (eg, '4x ** 3 - 3x ** 2 + 2 x - 1')
* rational expressions: ratios of polynomials (eg '(x ** 2 - 1) / (x + 1)')
* exponential expressions: expressions involving exponents (eg '2 ** x')
* logarithmic expressions: expressions involving logarithms (eg 'log(x)')
* trigonometric expressions: expressions involving trigonometric functions (eg 'sin(x) + cos(x)')

## equations
equations state that two expressions are equal and include an equals sign. they can be solved to find the values of the variables.

* linear equations: first-degree equations (eg '2 x + 3 = 7')
* quadratic equations: second-degree equations (eg 'ax ** 2 + bx + c = 0')
* polynomial equations: higher-degree equations (eg 'x ** 3 - 4x ** 2 + x + 6 = 0')
* rational equations: equations involving rational expressions (eg '1 / (x + 1) = 2')
* exponential equations: equations involving exponential expressions (eg '2 ** x = 8')
* logarithmic equations: equations involving logarithms (eg 'lox(x) = 3')
* trigonometric equations: equations involving trigonometric functions (eg 'sin(x) = 0.5')
* differential equations: equations involving derivatives (eg '(d ** 2 * y) / (d x ** 2) + 3 * ((d y) / (d x)) + 2 y = 0')

## functions
functions describe a relationship between inputs and outputs. each input corresponds to exactly one output.

* linear functions: 'f(x) = m x + b'
* quadratic functions: 'f(x) = a x ** 2 + b x + c'
* polynomial functions: 'f(x) = x ** 3 - 4 x ** 2 + x + 6'
* rational functions: ''
* exponential functions: ''
* logarithmic functions: ''
* trigonometric functions: ''
* piecewise functions
  * defined by different expressions over different intervals
  * ''
* implicit functions: ''

## inequalities
inequalities express a relationship where two expressions are not necessarily equal, but one is greater or less than the other.

* linear inequalities: '2 x + 3 > 7'
* polynomial inequalities: ''
* rational inequalities: ''
* exponential inequalities: ''
* logarithmic inequalities: ''
* trigonometric inequalities: ''

# characteristics of functions
## arity
the number of arguments or inputs a function takes.

* types:
  * unary: a function with one input (eg '')
  * binary: a function with two inputs (eg '')
  * n-ary: a function with nn inputs (eg '')

## domain
the set of all possible inputs for the function.

example: for 'f(x) = sqrt(x)' the domain is 'x > 0'.

## range
the set of all possible outputs of the function.

example: for 'f(x) = x ** 2', the range is 'y > 0'.

## codomain
the set of values that could potentially be outputs, not necessarily all of which are achieved.

example: for 'f(real_numbers) -> real_numbers' the codomain is 'real_numbers' even if the range is only non-negative reals.

## injectivity (one-to-one)
a function ff is injective if different inputs map to different outputs.

example: ''

## surjectivity (onto)
a function ff is surjective if every element in the codomain is mapped by some element in the domain.

example: ''

## bijectivity
a function is bijective if it is both injective and surjective.

example: ''

## continuity
a function ff is continuous if small changes in the input result in small changes in the output.

example: 'f(x) = x ** 2' is continuous everywhere.

## differentiability
a function ff is differentiable if it has a derivative at each point in its domain.

example: 'f(x) = x ** 3' is differentiable everywhere.

## periodicity
a function ff is periodic if there exists a positive number tt such that f(x+t)=f(x)f(x+t)=f(x) for all xx.

example: 'f(x) = sin(x)' has a period of '2 pi'.

## symmetry
* even function: 'f(-x) = f(x)' for all 'x' in the domain
  * example: 'f(x) = x ** 2'
* odd function: 'f(-x) = -f(x)' for all 'x' in the domain
  * example: 'f(x) = x ** 3'

## monotonicity
a function is monotonic if it is either entirely non-increasing or non-decreasing.

types:
* monotonically increasing: ''
* monotonically decreasing: ''

## boundedness
a function ff is bounded if there exists a number mm such that ∣f(x)∣≤m∣f(x)∣≤m for all xx in the domain.

types:
* bounded above: ''
* bounded below: ''

## inverse function
a function 'f' has an inverse 'f ** -1' if 'f((f ** -1)(y)) = y' and '(f ** -1)(f(x)) = x'

example: the inverse of 'f(x) = 2 x' is '(f ** -1)(x) = x / 2'

# plaintext ascii math notation
* readable plaintext mathematics notation similar to what is used in programming languages
* avoids symbol proliferation
* operators, values, and other expressions must be separated by spaces

* multiplication: *
* exponentiation: **
* division: /
* subtraction: -
* addition: +
* application: function_name(argument, ...)
* definition: function_name(parameter, ...) = ...
* standard library
  * srqt()
  * expt()

## see also
* [asciimath](http://asciimath.org/)
  * rendered
  * notation is harder to read