Module:Math

For formatting mathematical expressions, LATEX-style, see Template: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.

Related pages and templates

---

local z = {}
require( "mw.language" );

-- Clean numeric value
function z._cleanNumber( frame, number_string )
    -- Attempt basic conversion
    local number = tonumber( number_string )
    
    -- If failed, attempt to evaluate input as an expression
    if number == nil then
        local attempt = frame:preprocess( '{{#expr: ' .. number_string .. '}}' );
        attempt = tonumber( attempt );
        if attempt ~= nil then
            number = attempt;
            number_string = tostring( number );
        else
            number = nil;
            number_string = nil;
        end
    else
    -- String is valid but may contain padding, clean it.
        number_string = number_string:match( "^%s*(.-)%s*$" );
    end
    
    return number, number_string;
end

-- Generate random number
function z.random( frame )
    first = tonumber(frame.args[1]) -- if it doesn't exist it's NaN, if not a number it's nil
    second = tonumber(frame.args[2])

    if first then -- if NaN or nil, will skip down to final return
        if first <= second then -- could match if both nil, but already checked that first is a number in last line
            return math.random(first, second)
        end
        return math.random(first)
    end   
    return math.random()
end

-- Determine order of magnitude
function z.order(frame)
    local input_string = (frame.args[1] or frame.args.x or '0');
    local input_number;
    
    input_number = z._cleanNumber( frame, input_string );
    if input_number == nil then
        return '<strong class="error">Formatting error: Order of magnitude input appears non-numeric</strong>'
    else
        return z._order( input_number )
    end    
end
function z._order(x)
    if x == 0 then return 0 end
    return math.floor(math.log10(math.abs(x)))
end

-- Determines precision of a number using the string representation
function z.precision( frame )
    local input_string = (frame.args[1] or frame.args.x or '0');
    local input_number;
    
    input_number, input_string = z._cleanNumber( frame, input_string );
    if input_string == nil then
        return '<strong class="error">Formatting error: Precision input appears non-numeric</strong>'
    else
        return z._precision( input_string )
    end    
end
function z._precision( x )    
    x = string.upper( x )

    local decimal = string.find( x, '.', 1, true )
    local exponent_pos = string.find( x, 'E', 1, true )
    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

-- Finds maximum argument
function z.max( frame )
    if frame.args[1] == nil then
        return ''
    end
    local max_value = nil;
    
    local i = 1;
    while frame.args[i] ~= nil do
        local val = z._cleanNumber( frame, frame.args[i] );
        if val ~= nil then
            if max_value == nil or val > max_value then
                max_value = val;
            end
        end        
        i = i + 1;
    end
  
    return max_value
end

-- Finds minimum argument
function z.min( frame )
    if frame.args[1] == nil then
        return ''
    end
    local min_value = nil;
    
    local i = 1;
    while frame.args[i] ~= nil do
        local val = z._cleanNumber( frame, frame.args[i] );
        if val ~= nil then
            if min_value == nil or val < min_value then
                min_value = val;
            end
        end        
        i = i + 1;
    end
  
    return min_value
end

-- Rounds a number to specified precision
function z.round(frame)
    local value, precision;
    
    value = z._cleanNumber( frame, frame.args[1] or frame.args.value or 0 );
    precision = z._cleanNumber( frame, frame.args[2] or frame.args.precision or 0 );
    
    if value == nil or precision == nil then
        return '<strong class="error">Formatting error: Round input appears non-numeric</strong>'
    else
        return z._round( value, precision );
    end    
end
function z._round( value, precision )
    local rescale = math.pow( 10, precision );
    return math.floor( value * rescale + 0.5 ) / rescale;
end

-- Rounds a number to the specified precision and formats according to rules 
-- originally used for {{template:Rnd}}.  Output is a string.
function z.precision_format( frame )
    -- For access to Mediawiki built-in formatter.
    local lang = mw.getContentLanguage();
    
    local value_string, value, precision;
    value, value_string = z._cleanNumber( frame, frame.args[1] or 0 );
    precision = z._cleanNumber( frame, frame.args[2] or 0 );
    
    -- Check for non-numeric input
    if value == nil or precision == nil then
        return '<strong class="error">Formatting error: invalid input when rounding</strong>'
    end
    
    local current_precision = z._precision( value );

    local order = z._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
        orig_precision = z._precision( value_string );
        if order + orig_precision >= 14 then
            precision = 13 - order;        
        end        
    end

    -- If rounding off, truncate extra digits
    if precision < current_precision then
        value = z._round( value, precision );
        current_precision = z._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

return z