# big.js

A small, fast, easy-to-use library for arbitrary-precision decimal arithmetic.

Hosted on GitHub.

## API

In all examples below, `var` and semicolons are not shown, and if a commented-out value is in quotes it means `toString` has been called on the preceding expression.

### CONSTRUCTOR

##### Big`Big( value ) ⇒ Big`

`value`
  number|string|Big
  A decimal value.
  String values may be in exponential, as well as normal (non-exponential) notation.
  There is no limit to the number of digits of a string value (other than that of Javascript's maximum array size), but the largest recommended exponent magnitude is 1e+6.
  `Infinity`, `NaN` and hexadecimal literal strings, e.g. '0xff', are not valid.
  String values in octal literal form will be interpreted as decimals, e.g. '011' is 11, not 9.

Returns a new instance of a Big number object.

Throws `NaN` on an invalid `value`.

    x = new Big(9)                       // '9'
    y = new Big(x)                       // '9'
    Big(435.345)                         // 'new' is optional
    new Big('5032485723458348569331745.33434346346912144534543')
    new Big('4.321e+4')                  // '43210'
    new Big('-735.0918e-430')            // '-7.350918e-428'

#### Properties

##### DP

number : integer, 0 to 1e+6 inclusive
Default value: `20`

The maximum number of decimal places of the results of operations involving division.
It is relevant only to the `div` and `sqrt` methods, and the `pow` method when the exponent is negative.

The value will be checked for validity when one of the above methods is called.
!Big.DP! will be thrown if the value is found to be invalid.

    Big.DP = 40

##### RM

number : 0, 1 or 2
Default value: `1`

The rounding mode used in the above operations and by `round`, `toExponential`, `toFixed` and `toPrecision`.

Value | Description | BigDecimal equivalent
--- | --- | ---
0 | Rounds towards zero. I.e. truncate, no rounding. | ROUND_DOWN
1 | Rounds towards nearest neighbour. If equidistant, rounds away from zero. | ROUND_HALF_UP
2 | Rounds towards nearest neighbour. If equidistant, rounds towards even neighbour. | ROUND_HALF_EVEN

The value will be checked for validity when one of the above methods is called.
!Big.RM! will be thrown if the value is found to be invalid.

    Big.RM = 0

### INSTANCE

#### Methods

The methods inherited by a Big number instance from its constructor's prototype object.

A Big number is immutable in the sense that it is not changed by its methods.

##### abs`.abs() ⇒ Big`

Returns a Big number whose value is the absolute value, i.e. the magnitude, of this Big number.

    x = new Big(-0.8)
    x.abs()                     // '0.8'

##### div`.div(n) ⇒ Big`

`n` : number|string|Big

Returns a Big number whose value is the value of this Big number divided by `n`.

If the result has more fraction digits than is specified by `Big.DP`, it will be rounded to `Big.DP` decimal places using rounding mode `Big.RM`.

Throws NaN if `n` is invalid.
Throws `±Infinity` on division by zero.
Throws `NaN` on division of zero by zero.

    x = new Big(355)
    y = new Big(113)
    x.div(y)                   // '3.14159292035398230088'
    Big.DP = 2
    x.div(y)                   // '3.14'
    x.div(5)                   // '71'

##### eq`.eq(n) ⇒ boolean`

`n` : number|string|Big

Returns `true` if the value of this Big equals the value of `n`, otherwise returns `false`.

Throws NaN if `n` is invalid.

    0 === 1e-324               // true
    x = new Big(0)
    x.eq('1e-324')             // false
    Big(-0).eq(x)              // true  ( -0 === 0 )

##### gt`.gt(n) ⇒ boolean`

`n` : number|string|Big

Returns `true` if the value of this Big is greater than the value of `n`, otherwise returns `false`.

Throws NaN if `n` is invalid.

    0.1 > 0.3 - 0.2              // true
    x = new Big(0.1)
    x.gt(Big(0.3).minus(0.2))    // false
    Big(0).gt(x)                 // false

##### gte`.gte(n) ⇒ boolean`

`n` : number|string|Big

Returns `true` if the value of this Big is greater than or equal to the value of `n`, otherwise returns `false`.

Throws NaN if `n` is invalid.

    0.3 - 0.2 >= 0.1               // false
    x = new Big(0.3).minus(0.2)
    x.gte(0.1)                     // true
    Big(1).gte(x)                  // true

##### lt`.lt(n) ⇒ boolean`

`n` : number|string|Big

Returns `true` if the value of this Big is less than the value of `n`, otherwise returns `false`.

Throws NaN if `n` is invalid.

    0.3 - 0.2 < 0.1                // true
    x = new Big(0.3).minus(0.2)
    x.lt(0.1)                      // false
    Big(0).lt(x)                   // true

##### lte`.lte(n) ⇒ boolean`

`n` : number|string|Big

Returns `true` if the value of this Big is less than or equal to the value of `n`, otherwise returns `false`.

Throws NaN if `n` is invalid.

    0.1 <= 0.3 - 0.2               // false
    x = new Big(0.1)
    x.lte(Big(0.3).minus(0.2))     // true
    Big(-1).lte(x)                 // true

##### minus`.minus(n) ⇒ Big`

`n` : number|string|Big

Returns a Big number whose value is the value of this Big number minus `n`.

Throws NaN if `n` is invalid.

    0.3 - 0.1                  // 0.19999999999999998
    x = new Big(0.3)
    x.minus(0.1)               // '0.2'

##### mod`.mod(n) ⇒ Big`

`n` : number|string|Big

Returns a Big number whose value is the value of this Big number modulo `n`, i.e. the integer remainder of dividing this Big number by `n`.

The result will have the same sign as this Big number, and it will match that of Javascript's % operator (within the limits of its precision) and BigDecimal's remainder method.

Throws NaN if `n` is negative or otherwise invalid.

    1 % 0.9                    // 0.09999999999999998
    x = new Big(1)
    x.mod(0.9)                 // '0.1'

##### plus`.plus(n) ⇒ Big`

`n` : number|string|Big

Returns a Big number whose value is the value of this Big number plus `n`.

Throws NaN if `n` is invalid.

    0.1 + 0.2                  // 0.30000000000000004
    x = new Big(0.1)
    y = x.plus(0.2)            // '0.3'
    Big(0.7).plus(x).plus(y)   // '1'

##### pow`.pow( exp ) ⇒ Big`

`exp` : number : integer, -1e+6 to 1e+6 inclusive

Returns a Big number whose value is the value of this Big number raised to the power `exp`.

If `exp` is negative and the result has more fraction digits than is specified by `Big.DP`, it will be rounded to `Big.DP` decimal places using rounding mode `Big.RM`.

Throws !pow! if `exp` is invalid.

Note: High value exponents may cause this method to be slow to return.

    Math.pow(0.7, 2)           // 0.48999999999999994
    x = new Big(0.7)
    x.pow(2)                   // '0.49'
    Big.DP = 20
    Big(3).pow(-2)             // '0.11111111111111111111'

    new Big(123.456).pow(1000).toString().length     // 5099
    new Big(2).pow(1e+6)       // Time taken (Node.js): 9 minutes 34 secs.

##### round`.round( [dp [, rm]] ) ⇒ Big`

`dp` : number : integer, 0 to 1e+6 inclusive
`rm` : number : 0, 1 or 2

Returns a Big number whose value is the value of this Big number rounded using rounding mode `rm` to a maximum of `dp` decimal places.

if `dp` is omitted or is `null` or undefined, the return value is `n` rounded to a whole number.
if `rm` is omitted or is `null` or undefined, the current `Big.RM` setting is used.

Throws !round! if `dp` is invalid.
Throws !Big.RM! if `rm` is invalid.

    x = 123.45
    Math.round(x)              // 123
    y = new Big(x)
    y.round()                  // '123'
    y.round(2)                 // '123.45'
    y.round(10)                // '123.45'
    y.round(1, 0)              // '123.4'
    y.round(1, 1)              // '123.5'
    y.round(1, 2)              // '123.4'
    y                          // '123.45'

##### sqrt`.sqrt() ⇒ Big`

Returns a Big number whose value is the square root of this Big number.

If the result has more fraction digits than is specified by `Big.DP`, it will be rounded to `Big.DP` decimal places using rounding mode `Big.RM`.

Throws `NaN` if this Big number is negative.

    x = new Big(16)
    x.sqrt()                   // '4'
    y = new Big(3)
    y.sqrt()                   // '1.73205080756887729353'

##### times`.times(n) ⇒ Big`

`n` : number|string|Big

Returns a Big number whose value is the value of this Big number times `n`.

Throws NaN if `n` is invalid.

    0.6 * 3                    // 1.7999999999999998
    x = new Big(0.6)
    y = x.times(3)             // '1.8'
    Big('7e+500').times(y)     // '1.26e+501'

##### toExponential`.toExponential( [dp] ) ⇒ string`

`dp` : number : integer, 0 to 1e+6 inclusive

Returns a string representing the value of this Big number in exponential notation to a fixed number of decimal places `dp`.

If the value of this Big number in exponential notation has more digits to the right of the decimal point than is specified by `dp`, the return value will be rounded to `dp` decimal places using rounding mode `Big.RM`.

If the value of this Big number in exponential notation has fewer digits to the right of the decimal point than is specified by `dp`, the return value will be appended with zeros accordingly.

If `dp` is omitted, or is `null` or undefined, the number of digits after the decimal point defaults to the minimum number of digits necessary to represent the value exactly.

Throws !toExp! if `dp` is invalid.

    x = 45.6
    y = new Big(x)
    x.toExponential()          // '4.56e+1'
    y.toExponential()          // '4.56e+1'
    x.toExponential(0)         // '5e+1'
    y.toExponential(0)         // '5e+1'
    x.toExponential(1)         // '4.6e+1'
    y.toExponential(1)         // '4.6e+1'
    x.toExponential(3)         // '4.560e+1'
    y.toExponential(3)         // '4.560e+1'

##### toFixed`.toFixed( [dp] ) ⇒ string`

`dp` : number : integer, 0 to 1e+6 inclusive

Returns a string representing the value of this Big number in normal notation to a fixed number of decimal places `dp`.

If the value of this Big number in normal notation has more digits to the right of the decimal point than is specified by `dp`, the return value will be rounded to `dp` decimal places using rounding mode `Big.RM`.

If the value of this Big number in normal notation has fewer fraction digits then is specified by `dp`, the return value will be appended with zeros accordingly.

Unlike `Number.prototype.toFixed`, which returns exponential notation if a number is greater or equal to 1021, this method will always return normal notation.

If `dp` is omitted, or is `null` or undefined, then the return value is simply the value in normal notation. This is also unlike `Number.prototype.toFixed`, which returns the value to zero decimal places.

Throws !toFix! if `dp` is invalid.

    x = 45.6
    y = new Big(x)
    x.toFixed()                // '46'
    y.toFixed()                // '45.6'
    y.toFixed(0)               // '46'
    x.toFixed(3)               // '45.600'
    y.toFixed(3)               // '45.600'

##### toPrecision`.toPrecision( [sd] ) ⇒ string`

`sd` : number : integer, 1 to 1e+6 inclusive

Returns a string representing the value of this Big number to the specified number of significant digits `sd`.

If the value of this Big number has more digits than is specified by `sd`, the return value will be rounded to `sd` significant digits using rounding mode `Big.RM`.

If the value of this Big number has fewer digits than is specified by `sd`, the return value will be appended with zeros accordingly.

If `sd` is less than the number of digits necessary to represent the integer part of the value in normal notation, then exponential notation is used.

If `sd` is omitted, or is `null` or undefined, then the return value is the same as `.toString()`.

Throws !toPre! if `sd` is invalid.

    x = 45.6
    y = new Big(x)
    x.toPrecision()            // '45.6'
    y.toPrecision()            // '45.6'
    x.toPrecision(1)           // '5e+1'
    y.toPrecision(1)           // '5e+1'
    x.toPrecision(5)           // '45.600'
    y.toPrecision(5)           // '45.600'

##### toString`.toString() ⇒ string`

Returns a string representing the value of this Big number.

If this Big number has a positive exponent that is equal to or greater than 21, or a negative exponent equal to or less than -7, then exponential notation is returned.

The point at which `toString` returns exponential rather than normal notation can be adjusted by changing the value of `TO_EXP_POS` and `TO_EXP_NEG` in the EDITABLE DEFAULTS section at the top of the source code file. By default, Big numbers correspond to Javascript's number type in this regard.

    x = new Big('9.99e+20')
    x.toString()               // '999000000000000000000'
    y = new Big('1E21')
    x.toString()               // '1e+21'

##### valueOf`.valueOf() ⇒ string`

As `toString`.

    x = new Big('177.7e+457')
    x.valueOf()                // '1.777e+459'

#### Properties

A Big number is an object with three properties:

Property | Description | Type | Value
--- | --- | --- | ---
c | coefficient* | number`[]` | Array of single digits
e | exponent | number | Integer, -1e+6 to 1e+6 inclusive
s | sign | number | -1 or 1

*significand

The value of a Big number is stored in a normalised decimal floating point format which corresponds to the value's `toExponential` form, with the decimal point to be positioned after the most significant (left-most) digit of the coefficient.

Note that, as with Javascript numbers, the original exponent and fractional trailing zeros are not preserved.

    x = new Big(0.123)                 // '0.123'
    x.toExponential()                  // '1.23e-1'
    x.c                                // '1,2,3'
    x.e                                // -1
    x.s                                // 1

    y = new Number(-123.4567000e+2)    // '-12345.67'
    y.toExponential()                  // '-1.234567e+4'
    z = new Big('-123.4567000e+2')     // '-12345.67'
    z.toExponential()                  // '-1.234567e+4'
    z.c                                // '1,2,3,4,5,6,7'
    z.e                                // 4
    z.s                                // -1

A Big number is mutable in the sense that the value of its properties can be changed.
For example, to rapidly shift a value by a power of 10:

    x = new Big('1234.000')    // '1234'
    x.toExponential()          // '1.234e+3'
    x.c                        // '1,2,3,4'
    x.e                        // 3

    x.e = -5
    x                          // '0.00001234'

If changing the coefficient array directly, which is not recommended, be careful to avoid leading or trailing zeros (unless zero itself is being represented).

Minus zero is a valid Big number value, but like Javascript numbers the minus sign is not shown.

    x = new Number(-0)         // 0
    1 / x == -Infinity         // true

    y = new Big(-0)            // '0'
    y.c                        // '0'    [0].toString()
    y.e                        // 0
    y.s                        // -1

#### Errors

The table below summarises the errors that may be be thrown.

Method(s) | Throws | On
--- | --- | ---
`Big div eq gt gte lt lte minus mod plus times` | `NaN` | Invalid number
`div` | `±Infinity` | Division by zero
`NaN` | Division of zero by zero
!Big.DP! | Invalid Big.DP
!Big.RM! | Invalid Big.RM
`mod` | `NaN` | Modulo zero
`pow` | !pow! | Invalid exponent
!Big.DP! | Invalid Big.DP
!Big.RM! | Invalid Big.RM
`round` | !round! | Invalid dp
!Big.RM! | Invalid rm/Big.RM
`sqrt` | `NaN` | Negative number
!Big.DP! | Invalid Big.DP
!Big.RM! | Invalid Big.RM
`toExponential` | !toExp! | Invalid dp
!Big.RM! | Invalid Big.RM
`toFixed` | !toFix! | Invalid dp
!Big.RM! | Invalid Big.RM
`toPrecision` | !toPre! | Invalid sd
!Big.RM! | Invalid Big.RM

#### FAQ

###### Why are trailing fractional zeros removed from Big numbers?

Many arbitrary-precision libraries retain trailing fractional zeros as they can indicate the precision of a value. This can be useful but the results of arithmetic operations can be misleading.

    x = new BigDecimal("1.0")
    y = new BigDecimal("1.1000")
    z = x.add(y)                      // 2.1000

    x = new BigDecimal("1.20")
    y = new BigDecimal("3.45000")
    z = x.multiply(y)                 // 4.1400000

To specify the precision of a value is to specify that the value lies within a certain range.

In the first example, `x` has a value of 1.0. The trailing zero shows the precision of the value, implying that the value is in the range 0.95 to 1.05. Similarly, the precision indicated by the trailing zeros of `y` indicates that the value is in the range 1.09995 to 1.10005. If we add the two lowest values in the ranges we get 0.95 + 1.09995 = 2.04995 and if we add the two highest values we get 1.05 + 1.10005 = 2.15005, so the range of the result of the addition implied by the precision of its operands is 2.04995 to 2.15005. The result given by BigDecimal of 2.1000 however, indicates that the value is in the range 2.09995 to 2.10005 and therefore the precision implied by its trailing zeros is misleading.

In the second example, the true range is 4.122744 to 4.157256 yet the BigDecimal answer of 4.1400000 indicates a range of 4.13999995 to 4.14000005. Again, the precision implied by the trailing zeros is misleading.

This library, like binary floating-point and most calculators, does not retain trailing fractional zeros.
Instead, the `toExponential`, `toFixed` and `toPrecision` methods enable trailing zeros to be added if and when required.