# bignumber.js

A JavaScript library for arbitrary-precision arithmetic.

Hosted on GitHub.

## API

See the README on GitHub for a quick-start introduction.

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

##### BigNumber`BigNumber(value [, base]) ⇒ BigNumber`

`value`
  number|string|BigNumber: see RANGE for range.
  A numeric value.
  Legitimate values include ±`0`, ±`Infinity` and `NaN`.
  Values of type number with more than `15` significant digits are considered invalid (if `ERRORS` is true) as calling `toString` or `valueOf` on such numbers may not result in the intended value.

      console.log( 823456789123456.3 );    // 823456789123456.2

  There is no limit to the number of digits of a value of type string (other than that of JavaScript's maximum array size).
  Decimal string values may be in exponential, as well as normal (fixed-point) notation. Non-decimal values must be in normal notation.
  String values in hexadecimal literal form, e.g. `'0xff'`, are valid, as are string values with the octal and binary prefixs `'0o'` and `'0b'`. String values in octal literal form without the prefix will be interpreted as decimals, e.g. `'011'` is interpreted as 11, not 9.
  Values in any base may have fraction digits.
  For bases from `10` to `36`, lower and/or upper case letters can be used to represent values from `10` to `35`.
  For bases above 36, `a-z` represents values from `10` to `35`, `A-Z` from `36` to `61`, and `$` and `_` represent `62` and `63` respectively (this can be changed by editing the `ALPHABET` variable near the top of the source file).

`base`
  number: integer, `2` to `64` inclusive
  The base of `value`.
  If `base` is omitted, or is `null` or `undefined`, base `10` is assumed.

Returns a new instance of a BigNumber object.

If a base is specified, the value is rounded according to the current `DECIMAL_PLACES` and `ROUNDING_MODE` configuration.

See Errors for the treatment of an invalid `value` or `base`.

    x = new BigNumber(9)                       // '9'
    y = new BigNumber(x)                       // '9'

    // 'new' is optional if ERRORS is false
    BigNumber(435.345)                         // '435.345'

    new BigNumber('5032485723458348569331745.33434346346912144534543')
    new BigNumber('4.321e+4')                  // '43210'
    new BigNumber('-735.0918e-430')            // '-7.350918e-428'
    new BigNumber(Infinity)                    // 'Infinity'
    new BigNumber(NaN)                         // 'NaN'
    new BigNumber('.5')                        // '0.5'
    new BigNumber('+2')                        // '2'
    new BigNumber(-10110100.1, 2)              // '-180.5'
    new BigNumber(-0b10110100.1)               // '-180.5'
    new BigNumber('123412421.234324', 5)       // '607236.557696'
    new BigNumber('ff.8', 16)                  // '255.5'
    new BigNumber('0xff.8')                    // '255.5'

The following throws `'not a base 2 number'` if `ERRORS` is true, otherwise it returns a BigNumber with value `NaN`.

    new BigNumber(9, 2)

The following throws `'number type has more than 15 significant digits'` if `errors` is true, otherwise it returns a BigNumber with value `96517860459076820`.

    new BigNumber(96517860459076817.4395)

The following throws `'not a number'` if `ERRORS` is true, otherwise it returns a BigNumber with value `NaN`.

    new BigNumber('blurgh')

A value is only rounded by the constructor if a base is specified.

    BigNumber.config({ DECIMAL_PLACES: 5 })
    new BigNumber(1.23456789)                  // '1.23456789'
    new BigNumber(1.23456789, 10)              // '1.23457'

#### Methods

The static methods of a BigNumber constructor.

##### another`.another([obj]) ⇒ BigNumber constructor`

`obj`: object

Returns a new independent BigNumber constructor with configuration as described by `obj` (see `config`), or with the default configuration if `obj` is `null` or `undefined`.

    BigNumber.config({ DECIMAL_PLACES: 5 })
    BN = BigNumber.another({ DECIMAL_PLACES: 9 })

    x = new BigNumber(1)
    y = new BN(1)

    x.div(3)                        // 0.33333
    y.div(3)                        // 0.333333333

    // BN = BigNumber.another({ DECIMAL_PLACES: 9 }) is equivalent to:
    BN = BigNumber.another()
    BN.config({ DECIMAL_PLACES: 9 })

##### config`config([obj]) ⇒ object`

`obj`: object: an object that contains some or all of the following properties.

Configures the 'global' settings for this particular BigNumber constructor.

Note: the configuration can also be supplied as an argument list, see below.

`DECIMAL_PLACES`
  number: integer, `0` to `1e+9` inclusive
  Default value: `20`
  The maximum number of decimal places of the results of operations involving division, i.e. division, square root and base conversion operations, and power operations with negative exponents.

      BigNumber.config({ DECIMAL_PLACES: 5 })
      BigNumber.config(5)    // equivalent

`ROUNDING_MODE`
  number: integer, `0` to `8` inclusive
  Default value: `4` (`ROUND_HALF_UP`)
  The rounding mode used in the above operations and the default rounding mode of `round`, `toExponential`, `toFixed`, `toFormat` and `toPrecision`.
  The modes are available as enumerated properties of the BigNumber constructor.

      BigNumber.config({ ROUNDING_MODE: 0 })
      BigNumber.config(null, BigNumber.ROUND_UP)    // equivalent

`EXPONENTIAL_AT`
  number: integer, magnitude `0` to `1e+9` inclusive, or
  number[]: [ integer `-1e+9` to `0` inclusive, integer `0` to `1e+9` inclusive ]
  Default value: `[-7, 20]`
  The exponent value(s) at which `toString` returns exponential notation.
  If a single number is assigned, the value is the exponent magnitude.
  If an array of two numbers is assigned then the first number is the negative exponent value at and beneath which exponential notation is used, and the second number is the positive exponent value at and above which the same.
  For example, to emulate JavaScript numbers in terms of the exponent values at which they begin to use exponential notation, use `[-7, 20]`.

      BigNumber.config({ EXPONENTIAL_AT: 2 })
      new BigNumber(12.3)         // '12.3'        e is only 1
      new BigNumber(123)          // '1.23e+2'
      new BigNumber(0.123)        // '0.123'       e is only -1
      new BigNumber(0.0123)       // '1.23e-2'

      BigNumber.config({ EXPONENTIAL_AT: [-7, 20] })
      new BigNumber(123456789)    // '123456789'   e is only 8
      new BigNumber(0.000000123)  // '1.23e-7'

      // Almost never return exponential notation:
      BigNumber.config({ EXPONENTIAL_AT: 1e+9 })

      // Always return exponential notation:
      BigNumber.config({ EXPONENTIAL_AT: 0 })

  Regardless of the value of `EXPONENTIAL_AT`, the `toFixed` method will always return a value in normal notation and the `toExponential` method will always return a value in exponential form.
  Calling `toString` with a base argument, e.g. `toString(10)`, will also always return normal notation.
`RANGE`
  number: integer, magnitude `1` to `1e+9` inclusive, or
  number[]: [ integer `-1e+9` to `-1` inclusive, integer `1` to `1e+9` inclusive ]
  Default value: `[-1e+9, 1e+9]`
  The exponent value(s) beyond which overflow to `Infinity` and underflow to zero occurs.
  If a single number is assigned, it is the maximum exponent magnitude: values wth a positive exponent of greater magnitude become `Infinity` and those with a negative exponent of greater magnitude become zero.
  If an array of two numbers is assigned then the first number is the negative exponent limit and the second number is the positive exponent limit.
  For example, to emulate JavaScript numbers in terms of the exponent values at which they become zero and `Infinity`, use `[-324, 308]`.

      BigNumber.config({ RANGE: 500 })
      BigNumber.config().RANGE     // [ -500, 500 ]
      new BigNumber('9.999e499')   // '9.999e+499'
      new BigNumber('1e500')       // 'Infinity'
      new BigNumber('1e-499')      // '1e-499'
      new BigNumber('1e-500')      // '0'

      BigNumber.config({ RANGE: [-3, 4] })
      new BigNumber(99999)         // '99999'      e is only 4
      new BigNumber(100000)        // 'Infinity'   e is 5
      new BigNumber(0.001)         // '0.01'       e is only -3
      new BigNumber(0.0001)        // '0'          e is -4

  The largest possible magnitude of a finite BigNumber is `9.999...e+1000000000`.
  The smallest possible magnitude of a non-zero BigNumber is `1e-1000000000`.
`ERRORS`
  boolean|number: `true`, `false`, `0` or `1`.
  Default value: `true`
  The value that determines whether BigNumber Errors are thrown.
  If `ERRORS` is false, no errors will be thrown.
  See Errors.

      BigNumber.config({ ERRORS: false })

`CRYPTO`
  boolean|number: `true`, `false`, `0` or `1`.
  Default value: `false`
  The value that determines whether cryptographically-secure pseudo-random number generation is used.
  If `CRYPTO` is set to `true` then the `random` method will generate random digits using `crypto.getRandomValues` in browsers that support it, or `crypto.randomBytes` if using a version of Node.js that supports it.
  If neither function is supported by the host environment then attempting to set `CRYPTO` to `true` will fail, and if `ERRORS` is `true` an exception will be thrown.
  If `CRYPTO` is `false` then the source of randomness used will be `Math.random` (which is assumed to generate at least `30` bits of randomness).
  See `random`.

      BigNumber.config({ CRYPTO: true })
      BigNumber.config().CRYPTO       // true
      BigNumber.random()              // 0.54340758610486147524

`MODULO_MODE`
  number: integer, `0` to `9` inclusive
  Default value: `1` (`ROUND_DOWN`)
  The modulo mode used when calculating the modulus: `a mod n`.
  The quotient, `q = a / n`, is calculated according to the `ROUNDING_MODE` that corresponds to the chosen `MODULO_MODE`.
  The remainder, `r`, is calculated as: `r = a - n * q`.
  The modes that are most commonly used for the modulus/remainder operation are shown in the following table. Although the other rounding modes can be used, they may not give useful results.

  Property | Value | Description
  --- | --- | ---
  ROUND_UP | 0 | The remainder is positive if the dividend is negative, otherwise it is negative.
  ROUND_DOWN | 1 | The remainder has the same sign as the dividend. This uses 'truncating division' and matches the behaviour of JavaScript's remainder operator `%`.
  ROUND_FLOOR | 3 | The remainder has the same sign as the divisor. This matches Python's `%` operator.
  ROUND_HALF_EVEN | 6 | The IEEE 754 remainder function.
  EUCLID | 9 | The remainder is always positive. Euclidian division: `q = sign(n) * floor(a / abs(n))`

  The rounding/modulo modes are available as enumerated properties of the BigNumber constructor.
  See `modulo`.

      BigNumber.config({ MODULO_MODE: BigNumber.EUCLID })
      BigNumber.config({ MODULO_MODE: 9 })          // equivalent

`POW_PRECISION`
  number: integer, `0` to `1e+9` inclusive.
  Default value: `100`
  The maximum number of significant digits of the result of the power operation (unless a modulus is specified).
  If set to `0`, the number of signifcant digits will not be limited.
  See `toPower`.

      BigNumber.config({ POW_PRECISION: 100 })

`FORMAT`
  object
  The `FORMAT` object configures the format of the string returned by the `toFormat` method.
  The example below shows the properties of the `FORMAT` object that are recognised, and their default values.
  Unlike the other configuration properties, the values of the properties of the `FORMAT` object will not be checked for validity. The existing `FORMAT` object will simply be replaced by the object that is passed in. Note that all the properties shown below do not have to be included.
  See `toFormat` for examples of usage.

      BigNumber.config({
          FORMAT: {
              // the decimal separator
              decimalSeparator: '.',
              // the grouping separator of the integer part
              groupSeparator: ',',
              // the primary grouping size of the integer part
              groupSize: 3,
              // the secondary grouping size of the integer part
              secondaryGroupSize: 0,
              // the grouping separator of the fraction part
              fractionGroupSeparator: ' ',
              // the grouping size of the fraction part
              fractionGroupSize: 0
          }
      });

Returns an object with the above properties and their current values.

If the value to be assigned to any of the above properties is `null` or `undefined` it is ignored.

See Errors for the treatment of invalid values.

    BigNumber.config({
        DECIMAL_PLACES: 40,
        ROUNDING_MODE: BigNumber.ROUND_HALF_CEIL,
        EXPONENTIAL_AT: [-10, 20],
        RANGE: [-500, 500],
        ERRORS: true,
        CRYPTO: true,
        MODULO_MODE: BigNumber.ROUND_FLOOR,
        POW_PRECISION: 80,
        FORMAT: {
            groupSize: 3,
            groupSeparator: ' ',
            decimalSeparator: ','
        }
    });

    // Alternatively but equivalently (excluding FORMAT):
    BigNumber.config( 40, 7, [-10, 20], 500, 1, 1, 3, 80 )

    obj = BigNumber.config();
    obj.ERRORS       // true
    obj.RANGE        // [-500, 500]

##### max`.max([arg1 [, arg2, ...]]) ⇒ BigNumber`

`arg1`, `arg2`, ...: number|string|BigNumber
See `BigNumber` for further parameter details.

Returns a BigNumber whose value is the maximum of `arg1`, `arg2`,... .

The argument to this method can also be an array of values.

The return value is always exact and unrounded.

    x = new BigNumber('3257869345.0378653')
    BigNumber.max(4e9, x, '123456789.9')          // '4000000000'

    arr = [12, '13', new BigNumber(14)]
    BigNumber.max(arr)                            // '14'

##### min`.min([arg1 [, arg2, ...]]) ⇒ BigNumber`

`arg1`, `arg2`, ...: number|string|BigNumber
See `BigNumber` for further parameter details.

Returns a BigNumber whose value is the minimum of `arg1`, `arg2`,... .

The argument to this method can also be an array of values.

The return value is always exact and unrounded.

    x = new BigNumber('3257869345.0378653')
    BigNumber.min(4e9, x, '123456789.9')          // '123456789.9'

    arr = [2, new BigNumber(-14), '-15.9999', -12]
    BigNumber.min(arr)                            // '-15.9999'

##### random`.random([dp]) ⇒ BigNumber`

`dp`: number: integer, `0` to `1e+9` inclusive

Returns a new BigNumber with a pseudo-random value equal to or greater than `0` and less than `1`.

The return value will have `dp` decimal places (or less if trailing zeros are produced).
If `dp` is omitted then the number of decimal places will default to the current `DECIMAL_PLACES` setting.

Depending on the value of this BigNumber constructor's `CRYPTO` setting and the support for the `crypto` object in the host environment, the random digits of the return value are generated by either `Math.random` (fastest), `crypto.getRandomValues` (Web Cryptography API in recent browsers) or `crypto.randomBytes` (Node.js).

If `CRYPTO` is `true`, i.e. one of the `crypto` methods is to be used, the value of a returned BigNumber should be cryptographically-secure and statistically indistinguishable from a random value.

    BigNumber.config({ DECIMAL_PLACES: 10 })
    BigNumber.random()              // '0.4117936847'
    BigNumber.random(20)            // '0.78193327636914089009'

#### Properties

The library's enumerated rounding modes are stored as properties of the constructor.
(They are not referenced internally by the library itself.)

Rounding modes `0` to `6` (inclusive) are the same as those of Java's BigDecimal class.

Property | Value | Description
--- | --- | ---
ROUND_UP | 0 | Rounds away from zero
ROUND_DOWN | 1 | Rounds towards zero
ROUND_CEIL | 2 | Rounds towards `Infinity`
ROUND_FLOOR | 3 | Rounds towards `-Infinity`
ROUND_HALF_UP | 4 | Rounds towards nearest neighbour. If equidistant, rounds away from zero
ROUND_HALF_DOWN | 5 | Rounds towards nearest neighbour. If equidistant, rounds towards zero
ROUND_HALF_EVEN | 6 | Rounds towards nearest neighbour. If equidistant, rounds towards even neighbour
ROUND_HALF_CEIL | 7 | Rounds towards nearest neighbour. If equidistant, rounds towards `Infinity`
ROUND_HALF_FLOOR | 8 | Rounds towards nearest neighbour. If equidistant, rounds towards `-Infinity`

    BigNumber.config({ ROUNDING_MODE: BigNumber.ROUND_CEIL })
    BigNumber.config({ ROUNDING_MODE: 2 })     // equivalent

### INSTANCE

#### Methods

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

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

The treatment of ±`0`, ±`Infinity` and `NaN` is consistent with how JavaScript treats these values.

Many method names have a shorter alias.
(Internally, the library always uses the shorter method names.)

##### absoluteValue`.abs() ⇒ BigNumber`

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

The return value is always exact and unrounded.

    x = new BigNumber(-0.8)
    y = x.absoluteValue()           // '0.8'
    z = y.abs()                     // '0.8'

##### ceil`.ceil() ⇒ BigNumber`

Returns a BigNumber whose value is the value of this BigNumber rounded to a whole number in the direction of positive `Infinity`.

    x = new BigNumber(1.3)
    x.ceil()                        // '2'
    y = new BigNumber(-1.8)
    y.ceil()                        // '-1'

##### comparedTo`.cmp(n [, base]) ⇒ number`

`n`: number|string|BigNumber
`base`: number
See BigNumber for further parameter details.

Returns |
--- | ---
`1` | If the value of this BigNumber is greater than the value of `n`
`-1` | If the value of this BigNumber is less than the value of `n`
`0` | If this BigNumber and `n` have the same value
`null` | If the value of either this BigNumber or `n` is `NaN`

    x = new BigNumber(Infinity)
    y = new BigNumber(5)
    x.comparedTo(y)                 // 1
    x.comparedTo(x.minus(1))        // 0
    y.cmp(NaN)                      // null
    y.cmp('110', 2)                 // -1

##### decimalPlaces`.dp() ⇒ number`

Return the number of decimal places of the value of this BigNumber, or `null` if the value of this BigNumber is ±`Infinity` or `NaN`.

    x = new BigNumber(123.45)
    x.decimalPlaces()               // 2
    y = new BigNumber('9.9e-101')
    y.dp()                          // 102

##### dividedBy`.div(n [, base]) ⇒ BigNumber`

`n`: number|string|BigNumber
`base`: number
See BigNumber for further parameter details.

Returns a BigNumber whose value is the value of this BigNumber divided by `n`, rounded according to the current `DECIMAL_PLACES` and `ROUNDING_MODE` configuration.

    x = new BigNumber(355)
    y = new BigNumber(113)
    x.dividedBy(y)                  // '3.14159292035398230088'
    x.div(5)                        // '71'
    x.div(47, 16)                   // '5'

##### dividedToIntegerBy`.divToInt(n [, base]) ⇒ BigNumber`

`n`: number|string|BigNumber
`base`: number
See BigNumber for further parameter details.

Return a BigNumber whose value is the integer part of dividing the value of this BigNumber by `n`.

    x = new BigNumber(5)
    y = new BigNumber(3)
    x.dividedToIntegerBy(y)         // '1'
    x.divToInt(0.7)                 // '7'
    x.divToInt('0.f', 16)           // '5'

##### equals`.eq(n [, base]) ⇒ boolean`

`n`: number|string|BigNumber
`base`: number
See BigNumber for further parameter details.

Returns `true` if the value of this BigNumber equals the value of `n`, otherwise returns `false`.
As with JavaScript, `NaN` does not equal `NaN`.

Note: This method uses the `comparedTo` method internally.

    0 === 1e-324                    // true
    x = new BigNumber(0)
    x.equals('1e-324')              // false
    BigNumber(-0).eq(x)             // true  ( -0 === 0 )
    BigNumber(255).eq('ff', 16)     // true

    y = new BigNumber(NaN)
    y.equals(NaN)                   // false

##### floor`.floor() ⇒ BigNumber`

Returns a BigNumber whose value is the value of this BigNumber rounded to a whole number in the direction of negative `Infinity`.

    x = new BigNumber(1.8)
    x.floor()                       // '1'
    y = new BigNumber(-1.3)
    y.floor()                       // '-2'

##### greaterThan`.gt(n [, base]) ⇒ boolean`

`n`: number|string|BigNumber
`base`: number
See BigNumber for further parameter details.

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

Note: This method uses the `comparedTo` method internally.

    0.1 > (0.3 - 0.2)                           // true
    x = new BigNumber(0.1)
    x.greaterThan(BigNumber(0.3).minus(0.2))    // false
    BigNumber(0).gt(x)                          // false
    BigNumber(11, 3).gt(11.1, 2)                // true

##### greaterThanOrEqualTo`.gte(n [, base]) ⇒ boolean`

`n`: number|string|BigNumber
`base`: number
See BigNumber for further parameter details.

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

Note: This method uses the `comparedTo` method internally.

    (0.3 - 0.2) >= 0.1                   // false
    x = new BigNumber(0.3).minus(0.2)
    x.greaterThanOrEqualTo(0.1)          // true
    BigNumber(1).gte(x)                  // true
    BigNumber(10, 18).gte('i', 36)       // true

##### isFinite`.isFinite() ⇒ boolean`

Returns `true` if the value of this BigNumber is a finite number, otherwise returns `false`.

The only possible non-finite values of a BigNumber are `NaN`, `Infinity` and `-Infinity`.

    x = new BigNumber(1)
    x.isFinite()                    // true
    y = new BigNumber(Infinity)
    y.isFinite()                    // false

Note: The native method `isFinite()` can be used if `n <= Number.MAX_VALUE`.

##### isInteger`.isInt() ⇒ boolean`

Returns `true` if the value of this BigNumber is a whole number, otherwise returns `false`.

    x = new BigNumber(1)
    x.isInteger()                   // true
    y = new BigNumber(123.456)
    y.isInt()                       // false

##### isNaN`.isNaN() ⇒ boolean`

Returns `true` if the value of this BigNumber is `NaN`, otherwise returns `false`.

    x = new BigNumber(NaN)
    x.isNaN()                       // true
    y = new BigNumber('Infinity')
    y.isNaN()                       // false

Note: The native method `isNaN()` can also be used.

##### isNegative`.isNeg() ⇒ boolean`

Returns `true` if the value of this BigNumber is negative, otherwise returns `false`.

    x = new BigNumber(-0)
    x.isNegative()                  // true
    y = new BigNumber(2)
    y.isNeg()                       // false

Note: `n < 0` can be used if `n <= -Number.MIN_VALUE`.

##### isZero`.isZero() ⇒ boolean`

Returns `true` if the value of this BigNumber is zero or minus zero, otherwise returns `false`.

    x = new BigNumber(-0)
    x.isZero() && x.isNeg()         // true
    y = new BigNumber(Infinity)
    y.isZero()                      // false

Note: `n == 0` can be used if `n >= Number.MIN_VALUE`.

##### lessThan`.lt(n [, base]) ⇒ boolean`

`n`: number|string|BigNumber
`base`: number
See BigNumber for further parameter details.

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

Note: This method uses the `comparedTo` method internally.

    (0.3 - 0.2) < 0.1                     // true
    x = new BigNumber(0.3).minus(0.2)
    x.lessThan(0.1)                       // false
    BigNumber(0).lt(x)                    // true
    BigNumber(11.1, 2).lt(11, 3)          // true

##### lessThanOrEqualTo`.lte(n [, base]) ⇒ boolean`

`n`: number|string|BigNumber
`base`: number
See BigNumber for further parameter details.

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

Note: This method uses the `comparedTo` method internally.

    0.1 <= (0.3 - 0.2)                                // false
    x = new BigNumber(0.1)
    x.lessThanOrEqualTo(BigNumber(0.3).minus(0.2))    // true
    BigNumber(-1).lte(x)                              // true
    BigNumber(10, 18).lte('i', 36)                    // true

##### minus`.minus(n [, base]) ⇒ BigNumber`

`n`: number|string|BigNumber
`base`: number
See BigNumber for further parameter details.

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

The return value is always exact and unrounded.

    0.3 - 0.1                       // 0.19999999999999998
    x = new BigNumber(0.3)
    x.minus(0.1)                    // '0.2'
    x.minus(0.6, 20)                // '0'

##### modulo`.mod(n [, base]) ⇒ BigNumber`

`n`: number|string|BigNumber
`base`: number
See BigNumber for further parameter details.

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

The value returned, and in particular its sign, is dependent on the value of the `MODULO_MODE` setting of this BigNumber constructor. If it is `1` (default value), the result will have the same sign as this BigNumber, and it will match that of Javascript's `%` operator (within the limits of double precision) and BigDecimal's `remainder` method.

The return value is always exact and unrounded.

See `MODULO_MODE` for a description of the other modulo modes.

    1 % 0.9                         // 0.09999999999999998
    x = new BigNumber(1)
    x.modulo(0.9)                   // '0.1'
    y = new BigNumber(33)
    y.mod('a', 33)                  // '3'

##### negated`.neg() ⇒ BigNumber`

Returns a BigNumber whose value is the value of this BigNumber negated, i.e. multiplied by `-1`.

    x = new BigNumber(1.8)
    x.negated()                     // '-1.8'
    y = new BigNumber(-1.3)
    y.neg()                         // '1.3'

##### plus`.plus(n [, base]) ⇒ BigNumber`

`n`: number|string|BigNumber
`base`: number
See BigNumber for further parameter details.

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

The return value is always exact and unrounded.

    0.1 + 0.2                       // 0.30000000000000004
    x = new BigNumber(0.1)
    y = x.plus(0.2)                 // '0.3'
    BigNumber(0.7).plus(x).plus(y)  // '1'
    x.plus('0.1', 8)                // '0.225'

##### precision`.sd([z]) ⇒ number`

`z`: boolean|number: `true`, `false`, `0` or `1`

Returns the number of significant digits of the value of this BigNumber.

If `z` is `true` or `1` then any trailing zeros of the integer part of a number are counted as significant digits, otherwise they are not.

    x = new BigNumber(1.234)
    x.precision()                   // 4
    y = new BigNumber(987000)
    y.sd()                          // 3
    y.sd(true)                      // 6

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

`dp`: number: integer, `0` to `1e+9` inclusive
`rm`: number: integer, `0` to `8` inclusive

Returns a BigNumber whose value is the value of this BigNumber rounded by 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`, `ROUNDING_MODE` is used.

See Errors for the treatment of other non-integer or out of range `dp` or `rm` values.

    x = 1234.56
    Math.round(x)                             // 1235

    y = new BigNumber(x)
    y.round()                                 // '1235'
    y.round(1)                                // '1234.6'
    y.round(2)                                // '1234.56'
    y.round(10)                               // '1234.56'
    y.round(0, 1)                             // '1234'
    y.round(0, 6)                             // '1235'
    y.round(1, 1)                             // '1234.5'
    y.round(1, BigNumber.ROUND_HALF_EVEN)     // '1234.6'
    y                                         // '1234.56'

##### shift`.shift(n) ⇒ BigNumber`

`n`: number: integer, `-9007199254740991` to `9007199254740991` inclusive

Returns a BigNumber whose value is the value of this BigNumber shifted `n` places.

The shift is of the decimal point, i.e. of powers of ten, and is to the left if `n` is negative or to the right if `n` is positive.

The return value is always exact and unrounded.

    x = new BigNumber(1.23)
    x.shift(3)                      // '1230'
    x.shift(-3)                     // '0.00123'

##### squareRoot`.sqrt() ⇒ BigNumber`

Returns a BigNumber whose value is the square root of the value of this BigNumber, rounded according to the current `DECIMAL_PLACES` and `ROUNDING_MODE` configuration.

The return value will be correctly rounded, i.e. rounded as if the result was first calculated to an infinite number of correct digits before rounding.

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

##### times`.times(n [, base]) ⇒ BigNumber`

`n`: number|string|BigNumber
`base`: number
See BigNumber for further parameter details.

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

The return value is always exact and unrounded.

    0.6 * 3                         // 1.7999999999999998
    x = new BigNumber(0.6)
    y = x.times(3)                  // '1.8'
    BigNumber('7e+500').times(y)    // '1.26e+501'
    x.times('-a', 16)               // '-6'

##### toDigits`.toDigits([sd [, rm]]) ⇒ BigNumber`

`sd`: number: integer, `1` to `1e+9` inclusive.
`rm`: number: integer, `0` to `8` inclusive.

Returns a BigNumber whose value is the value of this BigNumber rounded to `sd` significant digits using rounding mode `rm`.

If `sd` is omitted or is `null` or `undefined`, the return value will not be rounded.
If `rm` is omitted or is `null` or `undefined`, `ROUNDING_MODE` will be used.

See Errors for the treatment of other non-integer or out of range `sd` or `rm` values.

    BigNumber.config({ precision: 5, rounding: 4 })
    x = new BigNumber(9876.54321)

    x.toDigits()                          // '9876.5'
    x.toDigits(6)                         // '9876.54'
    x.toDigits(6, BigNumber.ROUND_UP)     // '9876.55'
    x.toDigits(2)                         // '9900'
    x.toDigits(2, 1)                      // '9800'
    x                                     // '9876.54321'

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

`dp`: number: integer, `0` to `1e+9` inclusive
`rm`: number: integer, `0` to `8` inclusive

Returns a string representing the value of this BigNumber in exponential notation rounded using rounding mode `rm` to `dp` decimal places, i.e with one digit before the decimal point and `dp` digits after it.

If the value of this BigNumber in exponential notation has fewer than `dp` fraction digits, 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.
If `rm` is omitted or is `null` or `undefined`, `ROUNDING_MODE` is used.

See Errors for the treatment of other non-integer or out of range `dp` or `rm` values.

    x = 45.6
    y = new BigNumber(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'
    y.toExponential(1, 1)           // '4.5e+1'  (ROUND_DOWN)
    x.toExponential(3)              // '4.560e+1'
    y.toExponential(3)              // '4.560e+1'

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

`dp`: number: integer, `0` to `1e+9` inclusive
`rm`: number: integer, `0` to `8` inclusive

Returns a string representing the value of this BigNumber in normal (fixed-point) notation rounded to `dp` decimal places using rounding mode `rm`.

If the value of this BigNumber in normal notation has fewer than `dp` fraction digits, 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`, the return value will be unrounded and in normal notation. This is also unlike `Number.prototype.toFixed`, which returns the value to zero decimal places.
It is useful when fixed-point notation is required and the current `EXPONENTIAL_AT` setting causes `toString` to return exponential notation.
If `rm` is omitted or is `null` or `undefined`, `ROUNDING_MODE` is used.

See Errors for the treatment of other non-integer or out of range `dp` or `rm` values.

    x = 3.456
    y = new BigNumber(x)
    x.toFixed()                     // '3'
    y.toFixed()                     // '3.456'
    y.toFixed(0)                    // '3'
    x.toFixed(2)                    // '3.46'
    y.toFixed(2)                    // '3.46'
    y.toFixed(2, 1)                 // '3.45'  (ROUND_DOWN)
    x.toFixed(5)                    // '3.45600'
    y.toFixed(5)                    // '3.45600'

##### toFormat`.toFormat([dp [, rm]]) ⇒ string`

`dp`: number: integer, `0` to `1e+9` inclusive
`rm`: number: integer, `0` to `8` inclusive

Returns a string representing the value of this BigNumber in normal (fixed-point) notation rounded to `dp` decimal places using rounding mode `rm`, and formatted according to the properties of the `FORMAT` object.

See the examples below for the properties of the `FORMAT` object, their types and their usage.

If `dp` is omitted or is `null` or `undefined`, then the return value is not rounded to a fixed number of decimal places.
If `rm` is omitted or is `null` or `undefined`, `ROUNDING_MODE` is used.

See Errors for the treatment of other non-integer or out of range `dp` or `rm` values.

    format = {
        decimalSeparator: '.',
        groupSeparator: ',',
        groupSize: 3,
        secondaryGroupSize: 0,
        fractionGroupSeparator: ' ',
        fractionGroupSize: 0
    }
    BigNumber.config({ FORMAT: format })

    x = new BigNumber('123456789.123456789')
    x.toFormat()                    // '123,456,789.123456789'
    x.toFormat(1)                   // '123,456,789.1'

    // If a reference to the object assigned to FORMAT has been retained,
    // the format properties can be changed directly
    format.groupSeparator = ' '
    format.fractionGroupSize = 5
    x.toFormat()                    // '123 456 789.12345 6789'

    BigNumber.config({
        FORMAT: {
            decimalSeparator = ',',
            groupSeparator = '.',
            groupSize = 3,
            secondaryGroupSize = 2
        }
    })

    x.toFormat(6)                   // '12.34.56.789,123'

##### toFraction`.toFraction([max]) ⇒ [string, string]`

`max`: number|string|BigNumber: integer >= `1` and < `Infinity`

Returns a string array representing the value of this BigNumber as a simple fraction with an integer numerator and an integer denominator. The denominator will be a positive non-zero value less than or equal to `max`.

If a maximum denominator, `max`, is not specified, or is `null` or `undefined`, the denominator will be the lowest value necessary to represent the number exactly.

See Errors for the treatment of other non-integer or out of range `max` values.

    x = new BigNumber(1.75)
    x.toFraction()                  // '7, 4'

    pi = new BigNumber('3.14159265358')
    pi.toFraction()                 // '157079632679,50000000000'
    pi.toFraction(100000)           // '312689, 99532'
    pi.toFraction(10000)            // '355, 113'
    pi.toFraction(100)              // '311, 99'
    pi.toFraction(10)               // '22, 7'
    pi.toFraction(1)                // '3, 1'

##### toJSON`.toJSON() ⇒ string`

As `valueOf`.

    x = new BigNumber('177.7e+457')
    y = new BigNumber(235.4325)
    z = new BigNumber('0.0098074')

    // Serialize an array of three BigNumbers
    str = JSON.stringify( [x, y, z] )
    // "["1.777e+459","235.4325","0.0098074"]"

    // Return an array of three BigNumbers
    JSON.parse(str, function (key, val) {
        return key === '' ? val : new BigNumber(val)
    })

##### toNumber`.toNumber() ⇒ number`

Returns the value of this BigNumber as a JavaScript number primitive.

Type coercion with, for example, the unary plus operator will also work, except that a BigNumber with the value minus zero will be converted to positive zero.

    x = new BigNumber(456.789)
    x.toNumber()                    // 456.789
    +x                              // 456.789

    y = new BigNumber('45987349857634085409857349856430985')
    y.toNumber()                    // 4.598734985763409e+34

    z = new BigNumber(-0)
    1 / +z                          // Infinity
    1 / z.toNumber()                // -Infinity

##### toPower`.pow(n [, m]) ⇒ BigNumber`

`n`: number: integer, `-9007199254740991` to `9007199254740991` inclusive
`m`: number|string|BigNumber

Returns a BigNumber whose value is the value of this BigNumber raised to the power `n`, and optionally modulo a modulus `m`.

If `n` is negative the result is rounded according to the current `DECIMAL_PLACES` and `ROUNDING_MODE` configuration.

If `n` is not an integer or is out of range:

If `ERRORS` is `true` a BigNumber Error is thrown,
else if `n` is greater than `9007199254740991`, it is interpreted as `Infinity`;
else if `n` is less than `-9007199254740991`, it is interpreted as `-Infinity`;
else if `n` is otherwise a number, it is truncated to an integer;
else it is interpreted as `NaN`.

As the number of digits of the result of the power operation can grow so large so quickly, e.g. 123.45610000 has over `50000` digits, the number of significant digits calculated is limited to the value of the `POW_PRECISION` setting (default value: `100`) unless a modulus `m` is specified.

Set `POW_PRECISION` to `0` for an unlimited number of significant digits to be calculated (this will cause the method to slow dramatically for larger exponents).

Negative exponents will be calculated to the number of decimal places specified by `DECIMAL_PLACES` (but not to more than `POW_PRECISION` significant digits).

If `m` is specified and the value of `m`, `n` and this BigNumber are positive integers, then a fast modular exponentiation algorithm is used, otherwise if any of the values is not a positive integer the operation will simply be performed as `x.toPower(n).modulo(m)` with a `POW_PRECISION` of `0`.

    Math.pow(0.7, 2)                // 0.48999999999999994
    x = new BigNumber(0.7)
    x.toPower(2)                    // '0.49'
    BigNumber(3).pow(-2)            // '0.11111111111111111111'

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

`sd`: number: integer, `1` to `1e+9` inclusive
`rm`: number: integer, `0` to `8` inclusive

Returns a string representing the value of this BigNumber rounded to `sd` significant digits using rounding mode `rm`.

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

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

See Errors for the treatment of other non-integer or out of range `sd` or `rm` values.

    x = 45.6
    y = new BigNumber(x)
    x.toPrecision()                 // '45.6'
    y.toPrecision()                 // '45.6'
    x.toPrecision(1)                // '5e+1'
    y.toPrecision(1)                // '5e+1'
    y.toPrecision(2, 0)             // '4.6e+1'  (ROUND_UP)
    y.toPrecision(2, 1)             // '4.5e+1'  (ROUND_DOWN)
    x.toPrecision(5)                // '45.600'
    y.toPrecision(5)                // '45.600'

##### toString`.toString([base]) ⇒ string`

`base`: number: integer, `2` to `64` inclusive

Returns a string representing the value of this BigNumber in the specified base, or base `10` if `base` is omitted or is `null` or `undefined`.

For bases above `10`, values from `10` to `35` are represented by `a-z` (as with `Number.prototype.toString`), `36` to `61` by `A-Z`, and `62` and `63` by `$` and `_` respectively.

If a base is specified the value is rounded according to the current `DECIMAL_PLACES` and `ROUNDING_MODE` configuration.

If a base is not specified, and this BigNumber has a positive exponent that is equal to or greater than the positive component of the current `EXPONENTIAL_AT` setting, or a negative exponent equal to or less than the negative component of the setting, then exponential notation is returned.

If `base` is `null` or `undefined` it is ignored.

See Errors for the treatment of other non-integer or out of range `base` values.

    x = new BigNumber(750000)
    x.toString()                    // '750000'
    BigNumber.config({ EXPONENTIAL_AT: 5 })
    x.toString()                    // '7.5e+5'

    y = new BigNumber(362.875)
    y.toString(2)                   // '101101010.111'
    y.toString(9)                   // '442.77777777777777777778'
    y.toString(32)                  // 'ba.s'

    BigNumber.config({ DECIMAL_PLACES: 4 });
    z = new BigNumber('1.23456789')
    z.toString()                    // '1.23456789'
    z.toString(10)                  // '1.2346'

##### truncated`.trunc() ⇒ BigNumber`

Returns a BigNumber whose value is the value of this BigNumber truncated to a whole number.

    x = new BigNumber(123.456)
    x.truncated()                   // '123'
    y = new BigNumber(-12.3)
    y.trunc()                       // '-12'

##### valueOf`.valueOf() ⇒ string`

As `toString`, but does not accept a base argument and includes the minus sign for negative zero.

    x = new BigNumber('-0')
    x.toString()                    // '0'
    x.valueOf()                     // '-0'
    y = new BigNumber('1.777e+457')
    y.valueOf()                     // '1.777e+457'

#### Properties

A BigNumber is an object with three properties:

Property | Description | Type | Value
--- | --- | --- | ---
c | coefficient* | number`[]` | Array of base `1e14` numbers
e | exponent | number | Integer, `-1000000000` to `1000000000` inclusive
s | sign | number | `-1` or `1`

*significand

The value of any of the three properties may also be `null`.

From v2.0.0 of this library, the value of the coefficient of a BigNumber is stored in a normalised base `100000000000000` floating point format, as opposed to the base `10` format used in v1.x.x

This change means the properties of a BigNumber are now best considered to be read-only. Previously it was acceptable to change the exponent of a BigNumber by writing to its exponent property directly, but this is no longer recommended as the number of digits in the first element of the coefficient array is dependent on the exponent, so the coefficient would also need to be altered.

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

    x = new BigNumber(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 BigNumber('-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

#### Zero, NaN and Infinity

The table below shows how ±`0`, `NaN` and ±`Infinity` are stored.

| c | e | s
--- | --- | --- | ---
±0 | `[0]` | `0` | `±1`
NaN | `null` | `null` | `null`
±Infinity | `null` | `null` | `±1`

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

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

#### Errors

The errors that are thrown are generic `Error` objects with `name` BigNumber Error.

The table below shows the errors that may be thrown if `ERRORS` is `true`, and the action taken if `ERRORS` is `false`.

Method(s) | ERRORS: true Throw BigNumber Error | ERRORS: false Action on invalid argument
--- | --- | ---
`BigNumber comparedTo dividedBy dividedToIntegerBy equals greaterThan greaterThanOrEqualTo lessThan lessThanOrEqualTo minus modulo plus times` | number type has more than 15 significant digits | Accept.
not a base... number | Substitute `NaN`.
base not an integer | Truncate to integer. Ignore if not a number.
base out of range | Ignore.
not a number* | Substitute `NaN`.
`another` | not an object | Ignore.
`config` | `DECIMAL_PLACES` not an integer | Truncate to integer. Ignore if not a number.
`DECIMAL_PLACES` out of range | Ignore.
`ROUNDING_MODE` not an integer | Truncate to integer. Ignore if not a number.
`ROUNDING_MODE` out of range | Ignore.
`EXPONENTIAL_AT` not an integer or not [integer, integer] | Truncate to integer(s). Ignore if not number(s).
`EXPONENTIAL_AT` out of range or not [negative, positive] | Ignore.
`RANGE` not an integer or not [integer, integer] | Truncate to integer(s). Ignore if not number(s).
`RANGE` cannot be zero | Ignore.
`RANGE` out of range or not [negative, positive] | Ignore.
`ERRORS` not a boolean or binary digit | Ignore.
`CRYPTO` not a boolean or binary digit | Ignore.
`CRYPTO` crypto unavailable | Ignore.
`MODULO_MODE` not an integer | Truncate to integer. Ignore if not a number.
`MODULO_MODE` out of range | Ignore.
`POW_PRECISION` not an integer | Truncate to integer. Ignore if not a number.
`POW_PRECISION` out of range | Ignore.
`FORMAT` not an object | Ignore.
`precision` | argument not a boolean or binary digit | Ignore.
`round` | decimal places not an integer | Truncate to integer. Ignore if not a number.
decimal places out of range | Ignore.
rounding mode not an integer | Truncate to integer. Ignore if not a number.
rounding mode out of range | Ignore.
`shift` | argument not an integer | Truncate to integer. Ignore if not a number.
argument out of range | Substitute ±`Infinity`.
`toExponential` `toFixed` `toFormat` | decimal places not an integer | Truncate to integer. Ignore if not a number.
decimal places out of range | Ignore.
rounding mode not an integer | Truncate to integer. Ignore if not a number.
rounding mode out of range | Ignore.
`toFraction` | max denominator not an integer | Truncate to integer. Ignore if not a number.
max denominator out of range | Ignore.
`toDigits` `toPrecision` | precision not an integer | Truncate to integer. Ignore if not a number.
precision out of range | Ignore.
rounding mode not an integer | Truncate to integer. Ignore if not a number.
rounding mode out of range | Ignore.
`toPower` | exponent not an integer | Truncate to integer. Substitute `NaN` if not a number.
exponent out of range | Substitute ±`Infinity`.
`toString` | base not an integer | Truncate to integer. Ignore if not a number.
base out of range | Ignore.

*No error is thrown if the value is `NaN` or 'NaN'.

The message of a BigNumber Error will also contain the name of the method from which the error originated.

To determine if an exception is a BigNumber Error:

    try {
        // ...
    } catch (e) {
        if ( e instanceof Error && e.name == 'BigNumber Error' ) {
            // ...
        }
    }

#### FAQ

###### Why are trailing fractional zeros removed from BigNumbers?

Some 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 it 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 have, `0.95 + 1.09995 = 2.04995`, and if we add the two highest values we have, `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 may be 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 may be 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.