Module:Math - Wikipedia
Module:Math
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For formatting mathematical expressions, LATEX-style, see Template:Math.
This Lua module is used on 1,760,000+ pages, or roughly 3% of all pages.
To avoid major disruption and server load, any changes should be tested in the module's /sandbox or /testcases subpages, or in your own module sandbox. The tested changes can be added to this page in a single edit. Consider discussing changes on the talk page before implementing them.
This module is subject to page protection. It is a highly visible module in use by a very large number of pages, or is substituted very frequently. Because vandalism or mistakes would affect many pages, and even trivial editing might cause substantial load on the servers, it is protected from editing.
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
See also: Module: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 modulo y, or the remainder after x has been divided by y. This is accurate for integers up to 253; for larger integers Lua's modulo operator may return an erroneous value. This function deals with this problem by returning 0 if the modulo given by Lua's modulo 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:
precisionResult
28,765.57
-28,800
68,765.567000
-60
2.58,765.5680426633
-2.58,854.3774484715
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.
See also
The above documentation is transcluded from Module:Math/doc. (edit | history)
Editors can experiment in this module's sandbox (edit | diff) and testcases (edit | run) pages.
Subpages of this module.
--[[
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 retend
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, countend
--[[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​) endend
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​) endend
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 resultend
--[[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​)] endend
--[[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 parentframe. 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 endend
--[[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) endend
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) endend
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 rulesoriginally 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) -- For access to Mediawiki built-in formatter. 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 local padding 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)
padding = precision if padding > 20 then padding = 20 end
formatted_num = formatted_num .. string.rep('0', padding) end else padding = precision - current_precision if padding > 20 then padding = 20 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 endend
--[[Helper function that interprets the input numerically. If theinput does not appear to be a number, attempts evaluating it asa 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 currentframe 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. endend }
return setmetatable(p, mt)
Last edited on 11 March 2021, at 22:23
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