Module:Math

This module provides a number of mathematical functions. These functions can be used from #invoke or from other Lua modules.

Use from other Lua modules

To use the module from normal wiki pages, no special preparation is needed. If you are using the module from another Lua module, first you need to load it, like this:

local mm = require('Module:Math')


(The mm variable stands for Module Math; you can choose something more descriptive if you prefer.)

Most functions in the module have a version for Lua and a version for #invoke. It is possible to use the #invoke functions from other Lua modules, but using the Lua functions has the advantage that you do not need to access a Lua frame object. Lua functions are preceded by _, whereas #invoke functions are not.

random

{{#invoke:math|random}}
{{#invoke:math|random|max_value}}
{{#invoke:math|random|min_value|max_value}}

mm._random()
mm._random(max_value)
mm._random(min_value, max_value)


Generates a random number.

• If no arguments are specified, the number produced is greater than or equal to 0 and less than 1.
• If one argument is provided, the number produced is an integer between 1 and that argument. The argument must be a positive integer.
• If two arguments are provided, the number produced is an integer between the first and second arguments. Both arguments must be integers, but can be negative.

This function will not work properly for numbers less than −232 and greater than 232 − 1. If you need to use numbers outside of this range, it is recommended that you use Module:Random.

order

{{#invoke:math|order|n}}

mm._order(n)


Determines the order of magnitude of a number.

precision

{{#invoke:math|precision|n}}
{{#invoke:math|precision|x=n}}

mm._precision(number_string)


Detemines the precision of a number. For example, for "4" it will return "0", for "4.567" it will return "3", and for "100" it will return "-2".

The function attempts to parse the string representation of the number, and detects whether the number uses E notation. For this reason, when called from Lua, very large numbers or very precise numbers should be directly input as strings to get accurate results. If they are input as numbers, the Lua interpreter will change them to E notation and this function will return the precision of the E notation rather than that of the original number. This is not a problem when the number is called from #invoke, as all input from #invoke is in string format.

max

{{#invoke:math|max|v1|v2|v3|...}}

mm._max(v1, v2, v3, ...)


Returns the maximum value from the values specified. Values that cannot be converted to numbers are ignored.

median

{{#invoke:math|median|v1|v2|v3|...}}

mm._median(v1, v2, v3, ...)


Returns the median value from the values specified. Values that cannot be converted to numbers are ignored.

min

{{#invoke:math|min|v1|v2|v3|...}}

mm._min(v1, v2, v3, ...)


Returns the minimum value from the values specified. Values that cannot be converted to numbers are ignored.

sum

{{#invoke:math|sum|v1|v2|v3|...}}

mm._sum(v1, v2, v3, ...)


Returns the sum of the values specified. Values that cannot be converted to numbers are ignored.

average

{{#invoke:math|average|v1|v2|v3|...}}

mm._average(v1, v2, v3, ...)


Returns the average of the values specified. (More precisely, the value returned is the arithmetic mean.) Values that cannot be converted to numbers are ignored.

round

{{#invoke:math|round|value|precision}}
{{#invoke:math|round|value=value|precision=precision}}

mm._round(value, precision)


Rounds a number to the specified precision.

Note: As of October 2019, there is a bug in the display of some rounded numbers. When trying to round a number that rounds to "n.0", like "1.02", to the nearest tenth of a digit (i.e. |r=1), this function should display "1.0", but it unexpectedly displays "1". Use the |precision_format= parameter instead.

log10

{{#invoke:math | log10 | x}}

mm._log10(x)


Returns log10(x), the logarithm of x using base 10.

mod

{{#invoke:math|mod|x|y}}

mm._mod(x, y)


Gets x moduled by y, or the remainder after x has been divided by y. This is accurate for integers up to 253; for larger integers Lua's modulation operator may return an erroneous value. This function deals with this problem by returning 0 if the modulation given by Lua's modulation operator is less than 0 or greater than y.

gcd

{{#invoke:math|gcd|v1|v2|...}}

mm._gcd(v1, v2, ...)


Finds the greatest common divisor of the values specified. Values that cannot be converted to numbers are ignored.

precision_format

{{#invoke:math|precision_format|value_string|precision}}

mm._precision_format(value_string, precision)


Rounds a number to the specified precision and formats according to rules originally used for {{Rnd}}. Output is a string.

Parameter precision should be an integer number of digits after the decimal point. Negative values are permitted. Non-integers give unexpected results. Positive values greater than the input precision add zero-padding, negative values greater than the input order can consume all digits.

Formatting 8,765.567 with {{#invoke:Math|precision_format|8765.567|precision}} gives:

precision Result
2 8,765.57
-2 8,800
6 8,765.567000
-6 0
2.5 8,765.5680426633
-2.5 8,854.3774484715

divide

{{#invoke:Math|divide|x|y|round=|precision=}}

mm._divide(x, y, round, precision)


Divide x by y.

• If y if not a number, it is returned.
• Otherwise, if x is not a number, it is returned.
• If round is true ("yes" for #invoke), the result has no decimals
• Precision indicates how many digits of precision the result should have

If any of the arguments contain HTML tags, they are returned unchanged, allowing any errors in calculating the arguments to the division function to be propagated to the calling template.

cleanNumber

local number, number_string = mm._cleanNumber(number_string)


A helper function that can be called from other Lua modules, but not from #invoke. This takes a string or a number value as input, and if the value can be converted to a number, cleanNumber returns the number and the number string. If the value cannot be converted to a number, cleanNumber returns nil, nil.

--[[

This module provides a number of basic mathematical operations.

]]

local yesno, getArgs -- lazily initialized

local p = {} -- Holds functions to be returned from #invoke, and functions to make available to other Lua modules.
local wrap = {} -- Holds wrapper functions that process arguments from #invoke. These act as intemediary between functions meant for #invoke and functions meant for Lua.

--[[
Helper functions used to avoid redundant code.
]]

local function err(msg)
-- Generates wikitext error messages.
return mw.ustring.format('<strong class="error">Formatting error: %s</strong>', msg)
end

local function unpackNumberArgs(args)
-- Returns an unpacked list of arguments specified with numerical keys.
local ret = {}
for k, v in pairs(args) do
if type(k) == 'number' then
table.insert(ret, v)
end
end
return unpack(ret)
end

local function makeArgArray(...)
-- Makes an array of arguments from a list of arguments that might include nils.
local args = {...} -- Table of arguments. It might contain nils or non-number values, so we can't use ipairs.
local nums = {} -- Stores the numbers of valid numerical arguments.
local ret = {}
for k, v in pairs(args) do
v = p._cleanNumber(v)
if v then
nums[#nums + 1] = k
args[k] = v
end
end
table.sort(nums)
for i, num in ipairs(nums) do
ret[#ret + 1] = args[num]
end
return ret
end

local function fold(func, ...)
-- Use a function on all supplied arguments, and return the result. The function must accept two numbers as parameters,
-- and must return a number as an output. This number is then supplied as input to the next function call.
local vals = makeArgArray(...)
local count = #vals -- The number of valid arguments
if count == 0 then return
-- Exit if we have no valid args, otherwise removing the first arg would cause an error.
nil, 0
end
local ret = table.remove(vals, 1)
for _, val in ipairs(vals) do
ret = func(ret, val)
end
return ret, count
end

--[[
Fold arguments by selectively choosing values (func should return when to choose the current "dominant" value).
]]
local function binary_fold(func, ...)
local value = fold((function(a, b) if func(a, b) then return a else return b end end), ...)
return value
end

--[[
random

Generate a random number

Usage:
{{#invoke: Math | random }}
{{#invoke: Math | random | maximum value }}
{{#invoke: Math | random | minimum value | maximum value }}
]]

function wrap.random(args)
local first = p._cleanNumber(args[1])
local second = p._cleanNumber(args[2])
return p._random(first, second)
end

function p._random(first, second)
math.randomseed(mw.site.stats.edits + mw.site.stats.pages + os.time() + math.floor(os.clock() * 1000000000))
-- math.random will throw an error if given an explicit nil parameter, so we need to use if statements to check the params.
if first and second then
if first <= second then -- math.random doesn't allow the first number to be greater than the second.
return math.random(first, second)
end
elseif first then
return math.random(first)
else
return math.random()
end
end

--[[
order

Determine order of magnitude of a number

Usage:
{{#invoke: Math | order | value }}
]]

function wrap.order(args)
local input_string = (args[1] or args.x or '0');
local input_number = p._cleanNumber(input_string);
if input_number == nil then
return err('order of magnitude input appears non-numeric')
else
return p._order(input_number)
end
end

function p._order(x)
if x == 0 then return 0 end
return math.floor(math.log10(math.abs(x)))
end

--[[
precision

Detemines the precision of a number using the string representation

Usage:
{{ #invoke: Math | precision | value }}
]]

function wrap.precision(args)
local input_string = (args[1] or args.x or '0');
local trap_fraction = args.check_fraction;
local input_number;

if not yesno then
yesno = require('Module:Yesno')
end
if yesno(trap_fraction, true) then -- Returns true for all input except nil, false, "no", "n", "0" and a few others. See [[Module:Yesno]].
local pos = string.find(input_string, '/', 1, true);
if pos ~= nil then
if string.find(input_string, '/', pos + 1, true) == nil then
local denominator = string.sub(input_string, pos+1, -1);
local denom_value = tonumber(denominator);
if denom_value ~= nil then
return math.log10(denom_value);
end
end
end
end

input_number, input_string = p._cleanNumber(input_string);
if input_string == nil then
return err('precision input appears non-numeric')
else
return p._precision(input_string)
end
end

function p._precision(x)
if type(x) == 'number' then
x = tostring(x)
end
x = string.upper(x)

local decimal = x:find('%.')
local exponent_pos = x:find('E')
local result = 0;

if exponent_pos ~= nil then
local exponent = string.sub(x, exponent_pos + 1)
x = string.sub(x, 1, exponent_pos - 1)
result = result - tonumber(exponent)
end

if decimal ~= nil then
result = result + string.len(x) - decimal
return result
end

local pos = string.len(x);
while x:byte(pos) == string.byte('0') do
pos = pos - 1
result = result - 1
if pos <= 0 then
return 0
end
end

return result
end

--[[
max

Finds the maximum argument

Usage:
{{#invoke:Math| max | value1 | value2 | ... }}

Note, any values that do not evaluate to numbers are ignored.
]]

function wrap.max(args)
return p._max(unpackNumberArgs(args))
end

function p._max(...)
local max_value = binary_fold((function(a, b) return a > b end), ...)
if max_value then
return max_value
end
end

--[[
median

Find the median of set of numbers

Usage:
{{#invoke:Math | median | number1 | number2 | ...}}
OR
{{#invoke:Math | median }}
]]

function wrap.median(args)
return p._median(unpackNumberArgs(args))
end

function p._median(...)
local vals = makeArgArray(...)
local count = #vals
table.sort(vals)

if count == 0 then
return 0
end

if p._mod(count, 2) == 0 then
return (vals[count/2] + vals[count/2+1])/2
else
return vals[math.ceil(count/2)]
end
end

--[[
min

Finds the minimum argument

Usage:
{{#invoke:Math| min | value1 | value2 | ... }}
OR
{{#invoke:Math| min }}

When used with no arguments, it takes its input from the parent
frame.  Note, any values that do not evaluate to numbers are ignored.
]]

function wrap.min(args)
return p._min(unpackNumberArgs(args))
end

function p._min(...)
local min_value = binary_fold((function(a, b) return a < b end), ...)
if min_value then
return min_value
end
end

--[[
sum

Finds the sum

Usage:
{{#invoke:Math| sum | value1 | value2 | ... }}
OR
{{#invoke:Math| sum }}

Note, any values that do not evaluate to numbers are ignored.
]]

function wrap.sum(args)
return p._sum(unpackNumberArgs(args))
end

function p._sum(...)
local sums, count = fold((function(a, b) return a + b end), ...)
if not sums then
return 0
else
return sums
end
end

--[[
average

Finds the average

Usage:
{{#invoke:Math| average | value1 | value2 | ... }}
OR
{{#invoke:Math| average }}

Note, any values that do not evaluate to numbers are ignored.
]]

function wrap.average(args)
return p._average(unpackNumberArgs(args))
end

function p._average(...)
local sum, count = fold((function(a, b) return a + b end), ...)
if not sum then
return 0
else
return sum / count
end
end

--[[
round

Rounds a number to specified precision

Usage:
{{#invoke:Math | round | value | precision }}

--]]

function wrap.round(args)
local value = p._cleanNumber(args[1] or args.value or 0)
local precision = p._cleanNumber(args[2] or args.precision or 0)
if value == nil or precision == nil then
return err('round input appears non-numeric')
else
return p._round(value, precision)
end
end

function p._round(value, precision)
local rescale = math.pow(10, precision or 0);
return math.floor(value * rescale + 0.5) / rescale;
end

--[[
log10

returns the log (base 10) of a number

Usage:
{{#invoke:Math | log10 | x }}
]]

function wrap.log10(args)
return math.log10(args[1])
end

--[[
mod

Implements the modulo operator

Usage:
{{#invoke:Math | mod | x | y }}

--]]

function wrap.mod(args)
local x = p._cleanNumber(args[1])
local y = p._cleanNumber(args[2])
if not x then
return err('first argument to mod appears non-numeric')
elseif not y then
return err('second argument to mod appears non-numeric')
else
return p._mod(x, y)
end
end

function p._mod(x, y)
local ret = x % y
if not (0 <= ret and ret < y) then
ret = 0
end
return ret
end

--[[
gcd

Calculates the greatest common divisor of multiple numbers

Usage:
{{#invoke:Math | gcd | value 1 | value 2 | value 3 | ... }}
--]]

function wrap.gcd(args)
return p._gcd(unpackNumberArgs(args))
end

function p._gcd(...)
local function findGcd(a, b)
local r = b
local oldr = a
while r ~= 0 do
local quotient = math.floor(oldr / r)
oldr, r = r, oldr - quotient * r
end
if oldr < 0 then
oldr = oldr * -1
end
return oldr
end
local result, count = fold(findGcd, ...)
return result
end

--[[
precision_format

Rounds a number to the specified precision and formats according to rules
originally used for {{template:Rnd}}.  Output is a string.

Usage:
{{#invoke: Math | precision_format | number | precision }}
]]

function wrap.precision_format(args)
local value_string = args[1] or 0
local precision = args[2] or 0
return p._precision_format(value_string, precision)
end

function p._precision_format(value_string, precision)
local lang = mw.getContentLanguage();

local value
value, value_string = p._cleanNumber(value_string)
precision = p._cleanNumber(precision)

-- Check for non-numeric input
if value == nil or precision == nil then
return err('invalid input when rounding')
end

local current_precision = p._precision(value)
local order = p._order(value)

-- Due to round-off effects it is neccesary to limit the returned precision under
-- some circumstances because the terminal digits will be inaccurately reported.
if order + precision >= 14 then
if order + p._precision(value_string) >= 14 then
precision = 13 - order;
end
end

-- If rounding off, truncate extra digits
if precision < current_precision then
value = p._round(value, precision)
current_precision = p._precision(value)
end

local formatted_num = lang:formatNum(math.abs(value))
local sign

-- Use proper unary minus sign rather than ASCII default
if value < 0 then
sign = '−'
else
sign = ''
end

-- Handle cases requiring scientific notation
if string.find(formatted_num, 'E', 1, true) ~= nil or math.abs(order) >= 9 then
value = value * math.pow(10, -order)
current_precision = current_precision + order
precision = precision + order
formatted_num = lang:formatNum(math.abs(value))
else
order = 0;
end
formatted_num = sign .. formatted_num

-- Pad with zeros, if needed
if current_precision < precision then
if current_precision <= 0 then
if precision > 0 then
local zero_sep = lang:formatNum(1.1)
formatted_num = formatted_num .. zero_sep:sub(2,2)

end

formatted_num = formatted_num .. string.rep('0', padding)
end
else
end
formatted_num = formatted_num .. string.rep('0', padding)
end
end

-- Add exponential notation, if necessary.
if order ~= 0 then
-- Use proper unary minus sign rather than ASCII default
if order < 0 then
order = '−' .. lang:formatNum(math.abs(order))
else
order = lang:formatNum(order)
end

formatted_num = formatted_num .. '<span style="margin:0 .15em 0 .25em">×</span>10<sup>' .. order .. '</sup>'
end

return formatted_num
end

--[[
divide

Implements the division operator

Usage:
{{#invoke:Math | divide | x | y | round= | precision= }}

--]]
function wrap.divide(args)
local x = args[1]
local y = args[2]
local round = args.round
local precision =  args.precision
if not yesno then
yesno = require('Module:Yesno')
end
return p._divide(x, y, yesno(round), precision)
end

function p._divide(x, y, round, precision)
if y == nil or y == "" then
return err("Empty divisor")
elseif not tonumber(y) then
if type(y) == 'string' and string.sub(y, 1, 1) == '<' then
return y
else
return err("Not a number: " .. y)
end
elseif x == nil or x == "" then
return err("Empty dividend")
elseif not tonumber(x) then
if type(x) == 'string' and string.sub(x, 1, 1) == '<' then
return x
else
return err("Not a number: " .. x)
end
else
local z = x / y
if round then
return p._round(z, 0)
elseif precision then
return p._round(z, precision)
else
return z
end
end
end

--[[
Helper function that interprets the input numerically.  If the
input does not appear to be a number, attempts evaluating it as
a parser functions expression.
]]

function p._cleanNumber(number_string)
if type(number_string) == 'number' then
-- We were passed a number, so we don't need to do any processing.
return number_string, tostring(number_string)
elseif type(number_string) ~= 'string' or not number_string:find('%S') then
-- We were passed a non-string or a blank string, so exit.
return nil, nil;
end

-- Attempt basic conversion
local number = tonumber(number_string)

-- If failed, attempt to evaluate input as an expression
if number == nil then
local success, result = pcall(mw.ext.ParserFunctions.expr, number_string)
if success then
number = tonumber(result)
number_string = tostring(number)
else
number = nil
number_string = nil
end
else
number_string = number_string:match("^%s*(.-)%s*$") -- String is valid but may contain padding, clean it. number_string = number_string:match("^%+(.*)$") or number_string -- Trim any leading + signs.
if number_string:find('^%-?0[xX]') then
-- Number is using 0xnnn notation to indicate base 16; use the number that Lua detected instead.
number_string = tostring(number)
end
end

return number, number_string
end

--[[
Wrapper function that does basic argument processing. This ensures that all functions from #invoke can use either the current
frame or the parent frame, and it also trims whitespace for all arguments and removes blank arguments.
]]

local mt = { __index = function(t, k)
return function(frame)
if not getArgs then
getArgs = require('Module:Arguments').getArgs
end
return wrap[k](getArgs(frame))  -- Argument processing is left to Module:Arguments. Whitespace is trimmed and blank arguments are removed.
end
end }

return setmetatable(p, mt)