A JavaScript library for arbitrary-precision arithmetic.
+ + ++ 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.
+
BigNumber(n [, base]) ⇒ BigNumber
+
+ n: number|string|BigNumber
+ base: number: integer, 2 to 36 inclusive. (See
+ ALPHABET to extend this range).
+
+ Returns a new instance of a BigNumber object with value n, where n
+ is a numeric value in the specified base, or base 10 if
+ base is omitted or is null or undefined.
+
+ Note that the BigNnumber constructor accepts an n of type number purely
+ as a convenience so that string quotes don't have to be typed when entering literal values,
+ and that it is the toString value of n that is used rather than its
+ underlying binary floating point value converted to decimal.
+
+x = new BigNumber(123.4567) // '123.4567' +// 'new' is optional +y = BigNumber(x) // '123.4567'+
+ If n is a base 10 value it can be in normal or exponential notation.
+ Values in other bases must be in normal notation. Values in any base can have fraction digits,
+ i.e. digits after the decimal point.
+
+new BigNumber(43210) // '43210'
+new BigNumber('4.321e+4') // '43210'
+new BigNumber('-735.0918e-430') // '-7.350918e-428'
+new BigNumber('123412421.234324', 5) // '607236.557696'
+
+ Signed 0, signed Infinity and NaN are supported.
+
+new BigNumber('-Infinity') // '-Infinity'
+new BigNumber(NaN) // 'NaN'
+new BigNumber(-0) // '0'
+new BigNumber('.5') // '0.5'
+new BigNumber('+2') // '2'
+
+ String values in hexadecimal literal form, e.g. '0xff' or '0xFF'
+ (but not '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.
+
+new BigNumber(-10110100.1, 2) // '-180.5'
+new BigNumber('-0b10110100.1') // '-180.5'
+new BigNumber('ff.8', 16) // '255.5'
+new BigNumber('0xff.8') // '255.5'
+
+ If a base is specified, n is rounded according to the current
+ DECIMAL_PLACES and
+ ROUNDING_MODE settings. This includes base
+ 10 so don't include a base parameter for decimal values unless
+ this behaviour is wanted.
+
BigNumber.config({ DECIMAL_PLACES: 5 })
+new BigNumber(1.23456789) // '1.23456789'
+new BigNumber(1.23456789, 10) // '1.23457'
+ An error is thrown if base is invalid. See Errors.
+ There is no limit to the number of digits of a value of type string (other than
+ that of JavaScript's maximum array size). See RANGE to set
+ the maximum and minimum possible exponent value of a BigNumber.
+
+new BigNumber('5032485723458348569331745.33434346346912144534543')
+new BigNumber('4.321e10000000')
+ BigNumber NaN is returned if n is invalid
+ (unless BigNumber.DEBUG is true, see below).
+new BigNumber('.1*') // 'NaN'
+new BigNumber('blurgh') // 'NaN'
+new BigNumber(9, 2) // 'NaN'
+
+ To aid in debugging, if BigNumber.DEBUG is true then an error will
+ be thrown on an invalid n. An error will also be thrown if n is of
+ type number and has more than 15 significant digits, as calling
+ toString or valueOf on
+ these numbers may not result in the intended value.
+
+console.log(823456789123456.3) // 823456789123456.2 +new BigNumber(823456789123456.3) // '823456789123456.2' +BigNumber.DEBUG = true +// '[BigNumber Error] Number primitive has more than 15 significant digits' +new BigNumber(823456789123456.3) +// '[BigNumber Error] Not a base 2 number' +new BigNumber(9, 2)+
+ A BigNumber can also be created from an object literal.
+ Use isBigNumber to check that it is well-formed.
+
new BigNumber({ s: 1, e: 2, c: [ 777, 12300000000000 ], _isBigNumber: true }) // '777.123'
+
+
+
+
+ The static methods of a BigNumber constructor.
+ + + + +.clone([object]) ⇒ BigNumber constructor
+ object: object
+ Returns a new independent BigNumber constructor with configuration as described by
+ object (see config), or with the default
+ configuration if object is null or undefined.
+
+ Throws if object is not an object. See Errors.
+
BigNumber.config({ DECIMAL_PLACES: 5 })
+BN = BigNumber.clone({ DECIMAL_PLACES: 9 })
+
+x = new BigNumber(1)
+y = new BN(1)
+
+x.div(3) // 0.33333
+y.div(3) // 0.333333333
+
+// BN = BigNumber.clone({ DECIMAL_PLACES: 9 }) is equivalent to:
+BN = BigNumber.clone()
+BN.config({ DECIMAL_PLACES: 9 })
+
+
+
+ set([object]) ⇒ object
+ object: object: an object that contains some or all of the following
+ properties.
+
Configures the settings for this particular BigNumber constructor.
+ +DECIMAL_PLACES0 to 1e+9 inclusive20
+ BigNumber.config({ DECIMAL_PLACES: 5 })
+BigNumber.set({ DECIMAL_PLACES: 5 }) // equivalent
+ ROUNDING_MODE0 to 8 inclusive4 (ROUND_HALF_UP)
+ decimalPlaces,
+ precision,
+ toExponential,
+ toFixed,
+ toFormat and
+ toPrecision.
+ BigNumber.config({ ROUNDING_MODE: 0 })
+BigNumber.set({ ROUNDING_MODE: BigNumber.ROUND_UP }) // equivalent
+ EXPONENTIAL_AT0 to 1e+9 inclusive, or
+ -1e+9 to 0 inclusive, integer
+ 0 to 1e+9 inclusive ][-7, 20]
+ toString returns exponential notation.
+ [-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 })
+ 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.
+ toString with a base argument, e.g. toString(10), will
+ also always return normal notation.
+ RANGE1 to 1e+9 inclusive, or
+ -1e+9 to -1 inclusive, integer
+ 1 to 1e+9 inclusive ][-1e+9, 1e+9]
+ Infinity and underflow to
+ zero occurs.
+ Infinity and those with a
+ negative exponent of greater magnitude become zero.
+ 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
+ 9.999...e+1000000000.1e-1000000000.
+ CRYPTOtrue or false.false
+ 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 Node.js.
+ CRYPTO to true will fail and an exception will be thrown.
+ CRYPTO is false then the source of randomness used will be
+ Math.random (which is assumed to generate at least 30 bits of
+ randomness).
+ random.
+// Node.js
+const crypto = require('crypto'); // CommonJS
+import * as crypto from 'crypto'; // ES module
+
+global.crypto = crypto;
+
+BigNumber.config({ CRYPTO: true })
+BigNumber.config().CRYPTO // true
+BigNumber.random() // 0.54340758610486147524
+ MODULO_MODE0 to 9 inclusive1 (ROUND_DOWN)
+ a mod n.q = a / n, is calculated according to the
+ ROUNDING_MODE that corresponds to the chosen
+ MODULO_MODE.
+ r, is calculated as: r = a - n * q.| 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))
+ |
+
modulo.BigNumber.config({ MODULO_MODE: BigNumber.EUCLID })
+BigNumber.config({ MODULO_MODE: 9 }) // equivalent
+ POW_PRECISION0 to 1e+9 inclusive.0
+ 0, the number of significant digits will not be limited.exponentiatedBy.BigNumber.config({ POW_PRECISION: 100 })FORMATFORMAT object configures the format of the string returned by the
+ toFormat method.
+ FORMAT object that are
+ recognised, and their default values.
+ FORMAT object will not be checked for validity. The existing
+ FORMAT object will simply be replaced by the object that is passed in.
+ The object can include any number of the properties shown below.
+ toFormat for examples of usage.
+BigNumber.config({
+ FORMAT: {
+ // string to prepend
+ prefix: '',
+ // decimal separator
+ decimalSeparator: '.',
+ // grouping separator of the integer part
+ groupSeparator: ',',
+ // primary grouping size of the integer part
+ groupSize: 3,
+ // secondary grouping size of the integer part
+ secondaryGroupSize: 0,
+ // grouping separator of the fraction part
+ fractionGroupSeparator: ' ',
+ // grouping size of the fraction part
+ fractionGroupSize: 0,
+ // string to append
+ suffix: ''
+ }
+});
+ ALPHABET'0123456789abcdefghijklmnopqrstuvwxyz'
+ BigNumber constructor or
+ toString.
+ '+' and '-', or the decimal separator '.'.
+ // duodecimal (base 12)
+BigNumber.config({ ALPHABET: '0123456789TE' })
+x = new BigNumber('T', 12)
+x.toString() // '10'
+x.toString(12) // 'T'
+ Returns an object with the above properties and their current values.
+
+ Throws if object is not an object, or if an invalid value is assigned to
+ one or more of the above properties. See Errors.
+
+BigNumber.config({
+ DECIMAL_PLACES: 40,
+ ROUNDING_MODE: BigNumber.ROUND_HALF_CEIL,
+ EXPONENTIAL_AT: [-10, 20],
+ RANGE: [-500, 500],
+ CRYPTO: true,
+ MODULO_MODE: BigNumber.ROUND_FLOOR,
+ POW_PRECISION: 80,
+ FORMAT: {
+ groupSize: 3,
+ groupSeparator: ' ',
+ decimalSeparator: ','
+ },
+ ALPHABET: '0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ$_'
+});
+
+obj = BigNumber.config();
+obj.DECIMAL_PLACES // 40
+obj.RANGE // [-500, 500]
+
+
+
+ .isBigNumber(value) ⇒ boolean
+ value: any
+ Returns true if value is a BigNumber instance, otherwise returns
+ false.
+
x = 42 +y = new BigNumber(x) + +BigNumber.isBigNumber(x) // false +y instanceof BigNumber // true +BigNumber.isBigNumber(y) // true + +BN = BigNumber.clone(); +z = new BN(x) +z instanceof BigNumber // false +BigNumber.isBigNumber(z) // true+
+ If value is a BigNumber instance and BigNumber.DEBUG is true,
+ then this method will also check if value is well-formed, and throw if it is not.
+ See Errors.
+
+ The check can be useful if creating a BigNumber from an object literal. + See BigNumber. +
++x = new BigNumber(10) + +// Change x.c to an illegitimate value. +x.c = NaN + +BigNumber.DEBUG = false + +// No error. +BigNumber.isBigNumber(x) // true + +BigNumber.DEBUG = true + +// Error. +BigNumber.isBigNumber(x) // '[BigNumber Error] Invalid BigNumber'+ + + +
.max(n...) ⇒ BigNumber
+ n: number|string|BigNumber
+ See BigNumber for further parameter details.
+
+ Returns a BigNumber whose value is the maximum of the arguments. +
+The return value is always exact and unrounded.
+x = new BigNumber('3257869345.0378653')
+BigNumber.maximum(4e9, x, '123456789.9') // '4000000000'
+
+arr = [12, '13', new BigNumber(14)]
+BigNumber.max.apply(null, arr) // '14'
+
+
+
+ .min(n...) ⇒ BigNumber
+ n: number|string|BigNumber
+ See BigNumber for further parameter details.
+
+ Returns a BigNumber whose value is the minimum of the arguments. +
+The return value is always exact and unrounded.
+x = new BigNumber('3257869345.0378653')
+BigNumber.minimum(4e9, x, '123456789.9') // '123456789.9'
+
+arr = [2, new BigNumber(-14), '-15.9999', -12]
+BigNumber.min.apply(null, arr) // '-15.9999'
+
+
+
+ .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).
+
+ To be able to set CRYPTO to true when using
+ Node.js, the crypto object must be available globally:
+
// Node.js
+const crypto = require('crypto'); // CommonJS
+import * as crypto from 'crypto'; // ES module
+global.crypto = crypto;
+
+ 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.
+
+ Throws if dp is invalid. See Errors.
+
BigNumber.config({ DECIMAL_PLACES: 10 })
+BigNumber.random() // '0.4117936847'
+BigNumber.random(20) // '0.78193327636914089009'
+
+
+
+ .sum(n...) ⇒ BigNumber
+ n: number|string|BigNumber
+ See BigNumber for further parameter details.
+
Returns a BigNumber whose value is the sum of the arguments.
+The return value is always exact and unrounded.
+x = new BigNumber('3257869345.0378653')
+BigNumber.sum(4e9, x, '123456789.9') // '7381326134.9378653'
+
+arr = [2, new BigNumber(14), '15.9999', 12]
+BigNumber.sum.apply(null, arr) // '43.9999'
+
+
+
+
+ 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
+
+ undefined|false|true
+
+ If BigNumber.DEBUG is set true then an error will be thrown
+ if this BigNumber constructor receives an invalid value, such as
+ a value of type number with more than 15 significant digits.
+ See BigNumber.
+
+ An error will also be thrown if the isBigNumber
+ method receives a BigNumber that is not well-formed.
+ See isBigNumber.
+
BigNumber.DEBUG = true+ + +
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.
+ + + +.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'+ + + +
.comparedTo(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.comparedTo(NaN) // null
+y.comparedTo('110', 2) // -1
+
+
+
+ .dp([dp [, rm]]) ⇒ BigNumber|number
+
+ dp: number: integer, 0 to 1e+9 inclusive
+ rm: number: integer, 0 to 8 inclusive
+
+ If dp is a number, 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 the number of decimal places of the value of this BigNumber, or null if
+ the value of this BigNumber is ±Infinity or NaN.
+
+ If rm is omitted, or is null or undefined,
+ ROUNDING_MODE is used.
+
+ Throws if dp or rm is invalid. See Errors.
+
+x = new BigNumber(1234.56)
+x.decimalPlaces(1) // '1234.6'
+x.dp() // 2
+x.decimalPlaces(2) // '1234.56'
+x.dp(10) // '1234.56'
+x.decimalPlaces(0, 1) // '1234'
+x.dp(0, 6) // '1235'
+x.decimalPlaces(1, 1) // '1234.5'
+x.dp(1, BigNumber.ROUND_HALF_EVEN) // '1234.6'
+x // '1234.56'
+y = new BigNumber('9.9e-101')
+y.dp() // 102
+
+
+
+ .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 settings.
+
+x = new BigNumber(355) +y = new BigNumber(113) +x.dividedBy(y) // '3.14159292035398230088' +x.div(5) // '71' +x.div(47, 16) // '5'+ + + +
.idiv(n [, base]) ⇒
+ BigNumber
+
+ n: number|string|BigNumber
+ base: number
+ See BigNumber for further parameter details.
+
+ Returns 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.idiv(0.7) // '7'
+x.idiv('0.f', 16) // '5'
+
+
+
+ .pow(n [, m]) ⇒ BigNumber
+
+ n: number|string|BigNumber: integer
+ m: number|string|BigNumber
+
+ Returns a BigNumber whose value is the value of this BigNumber exponentiated by
+ n, i.e. raised to the power n, and optionally modulo a modulus
+ m.
+
+ Throws if n is not an integer. See Errors.
+
+ If n is negative the result is rounded according to the current
+ DECIMAL_PLACES and
+ ROUNDING_MODE settings.
+
+ 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 (unless a modulus
+ m is specified).
+
+ By default POW_PRECISION is set to 0.
+ This means that an unlimited number of significant digits will be calculated, and that the
+ method's performance will decrease dramatically for larger exponents.
+
+ If m is specified and the value of m, n and this
+ BigNumber are integers, and n is positive, then a fast modular exponentiation
+ algorithm is used, otherwise the operation will be performed as
+ x.exponentiatedBy(n).modulo(m) with a
+ POW_PRECISION of 0.
+
+Math.pow(0.7, 2) // 0.48999999999999994 +x = new BigNumber(0.7) +x.exponentiatedBy(2) // '0.49' +BigNumber(3).pow(-2) // '0.11111111111111111111'+ + + +
.integerValue([rm]) ⇒ BigNumber
+
+ rm: number: integer, 0 to 8 inclusive
+
+ Returns a BigNumber whose value is the value of this BigNumber rounded to an integer using
+ rounding mode rm.
+
+ If rm is omitted, or is null or undefined,
+ ROUNDING_MODE is used.
+
+ Throws if rm is invalid. See Errors.
+
+x = new BigNumber(123.456) +x.integerValue() // '123' +x.integerValue(BigNumber.ROUND_CEIL) // '124' +y = new BigNumber(-12.7) +y.integerValue() // '-13' +y.integerValue(BigNumber.ROUND_DOWN) // '-12'+
+ The following is an example of how to add a prototype method that emulates JavaScript's
+ Math.round function. Math.ceil, Math.floor and
+ Math.trunc can be emulated in the same way with
+ BigNumber.ROUND_CEIL, BigNumber.ROUND_FLOOR and
+ BigNumber.ROUND_DOWN respectively.
+
+BigNumber.prototype.round = function () {
+ return this.integerValue(BigNumber.ROUND_HALF_CEIL);
+};
+x.round() // '123'
+
+
+
+ .eq(n [, base]) ⇒ boolean
+ n: number|string|BigNumber
+ base: number
+ See BigNumber for further parameter details.
+
+ Returns true if the value of this BigNumber is equal to 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.isEqualTo('1e-324') // false
+BigNumber(-0).eq(x) // true ( -0 === 0 )
+BigNumber(255).eq('ff', 16) // true
+
+y = new BigNumber(NaN)
+y.isEqualTo(NaN) // false
+
+
+
+ .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.
+
.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.isGreaterThan(BigNumber(0.3).minus(0.2)) // false +BigNumber(0).gt(x) // false +BigNumber(11, 3).gt(11.1, 2) // true+ + + +
.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.isGreaterThanOrEqualTo(0.1) // true
+BigNumber(1).gte(x) // true
+BigNumber(10, 18).gte('i', 36) // true
+
+
+
+ .isInteger() ⇒ boolean
+ Returns true if the value of this BigNumber is an integer, otherwise returns
+ false.
+
+x = new BigNumber(1) +x.isInteger() // true +y = new BigNumber(123.456) +y.isInteger() // false+ + + +
.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.isLessThan(0.1) // false +BigNumber(0).lt(x) // true +BigNumber(11.1, 2).lt(11, 3) // true+ + + +
.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.isLessThanOrEqualTo(BigNumber(0.3).minus(0.2)) // true
+BigNumber(-1).lte(x) // true
+BigNumber(10, 18).lte('i', 36) // true
+
+
+
+ .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() ⇒ boolean
+ Returns true if the sign of this BigNumber is negative, otherwise returns
+ false.
+
+x = new BigNumber(-0) +x.isNegative() // true +y = new BigNumber(2) +y.isNegative() // false+
Note: n < 0 can be used if n <= -Number.MIN_VALUE.
.isPositive() ⇒ boolean
+ Returns true if the sign of this BigNumber is positive, otherwise returns
+ false.
+
+x = new BigNumber(-0) +x.isPositive() // false +y = new BigNumber(2) +y.isPositive() // true+ + + +
.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.isNegative() // true +y = new BigNumber(Infinity) +y.isZero() // false+
Note: n == 0 can be used if n >= Number.MIN_VALUE.
.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'+ + + +
.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'
+
+
+
+ .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 multiplied by n.
+
The return value is always exact and unrounded.
+
+0.6 * 3 // 1.7999999999999998
+x = new BigNumber(0.6)
+y = x.multipliedBy(3) // '1.8'
+BigNumber('7e+500').times(y) // '1.26e+501'
+x.multipliedBy('-a', 16) // '-6'
+
+
+
+ .negated() ⇒ 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.negated() // '1.3'+ + + +
.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) // '1.1'
+x.plus('0.1', 8) // '0.225'
+
+
+
+ .sd([d [, rm]]) ⇒ BigNumber|number
+
+ d: number|boolean: integer, 1 to 1e+9
+ inclusive, or true or false
+ rm: number: integer, 0 to 8 inclusive.
+
+ If d is a number, returns a BigNumber whose value is the value of this BigNumber
+ rounded to a precision of d significant digits using rounding mode
+ rm.
+
+ If d is omitted or is null or undefined, the return
+ value is the number of significant digits of the value of this BigNumber, or null
+ if the value of this BigNumber is ±Infinity or NaN.
+
+ If d is true then any trailing zeros of the integer
+ part of a number are counted as significant digits, otherwise they are not.
+
+ If rm is omitted or is null or undefined,
+ ROUNDING_MODE will be used.
+
+ Throws if d or rm is invalid. See Errors.
+
+x = new BigNumber(9876.54321) +x.precision(6) // '9876.54' +x.sd() // 9 +x.precision(6, BigNumber.ROUND_UP) // '9876.55' +x.sd(2) // '9900' +x.precision(2, 1) // '9800' +x // '9876.54321' +y = new BigNumber(987000) +y.precision() // 3 +y.sd(true) // 6+ + + +
.shiftedBy(n) ⇒ BigNumber
+ n: number: integer,
+ -9007199254740991 to 9007199254740991 inclusive
+
+ Returns a BigNumber whose value is the value of this BigNumber shifted by 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.
+
+ Throws if n is invalid. See Errors.
+
+x = new BigNumber(1.23) +x.shiftedBy(3) // '1230' +x.shiftedBy(-3) // '0.00123'+ + + +
.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 settings.
+
+ 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'+ + + +
.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.
+
+ Throws if dp or rm is invalid. See Errors.
+
+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([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.
+
+ Throws if dp or rm is invalid. See Errors.
+
+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([dp [, rm[, format]]]) ⇒ string
+
+ dp: number: integer, 0 to 1e+9 inclusive
+ rm: number: integer, 0 to 8 inclusive
+ format: object: see FORMAT
+
+
+ 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 FORMAT and the examples below for the properties of the
+ format object, their types, and their usage. A formatting object may contain
+ some or all of the recognised properties.
+
+ 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.
+ If format is omitted or is null or undefined, the
+ FORMAT object is used.
+
+ Throws if dp, rm or format is invalid. See
+ Errors.
+
+fmt = {
+ prefix: '',
+ decimalSeparator: '.',
+ groupSeparator: ',',
+ groupSize: 3,
+ secondaryGroupSize: 0,
+ fractionGroupSeparator: ' ',
+ fractionGroupSize: 0,
+ suffix: ''
+}
+
+x = new BigNumber('123456789.123456789')
+
+// Set the global formatting options
+BigNumber.config({ FORMAT: fmt })
+
+x.toFormat() // '123,456,789.123456789'
+x.toFormat(3) // '123,456,789.123'
+
+// If a reference to the object assigned to FORMAT has been retained,
+// the format properties can be changed directly
+fmt.groupSeparator = ' '
+fmt.fractionGroupSize = 5
+x.toFormat() // '123 456 789.12345 6789'
+
+// Alternatively, pass the formatting options as an argument
+fmt = {
+ prefix: '=> ',
+ decimalSeparator: ',',
+ groupSeparator: '.',
+ groupSize: 3,
+ secondaryGroupSize: 2
+}
+
+x.toFormat() // '123 456 789.12345 6789'
+x.toFormat(fmt) // '=> 12.34.56.789,123456789'
+x.toFormat(2, fmt) // '=> 12.34.56.789,12'
+x.toFormat(3, BigNumber.ROUND_UP, fmt) // '=> 12.34.56.789,124'
+
+
+
+ .toFraction([maximum_denominator])
+ ⇒ [BigNumber, BigNumber]
+
+ maximum_denominator:
+ number|string|BigNumber: integer >= 1 and <=
+ Infinity
+
+ Returns an array of two BigNumbers 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 maximum_denominator.
+
+ If a maximum_denominator is not specified, or is null or
+ undefined, the denominator will be the lowest value necessary to represent the
+ number exactly.
+
+ Throws if maximum_denominator is invalid. See Errors.
+
+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() ⇒ stringAs 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() ⇒ numberReturns the value of this BigNumber as a JavaScript number primitive.
++ This method is identical to using type coercion with the unary plus operator. +
+
+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.toNumber() // -Infinity
+1 / +z // -Infinity
+
+
+
+ .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.
+
+ Throws if sd or rm is invalid. See Errors.
+
+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([base]) ⇒ string
+ base: number: integer, 2 to ALPHABET.length
+ inclusive (see ALPHABET).
+
+ 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, and using the default base conversion alphabet
+ (see ALPHABET), values from 10 to
+ 35 are represented by a-z
+ (as with Number.prototype.toString).
+
+ If a base is specified the value is rounded according to the current
+ DECIMAL_PLACES
+ and ROUNDING_MODE settings.
+
+ 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.
+ Throws if base is invalid. See Errors.
+
+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'
+
+
+
+ .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'
+
+
+
+ The properties of a BigNumber instance:
+| 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 c, e and s properties may also
+ be null.
+
+ The above properties are best considered to be read-only. In early versions of this library it + was okay to change the exponent of a BigNumber by writing to its exponent property directly, + but this is no longer reliable as the value of the first element of the coefficient array is + now dependent on the exponent. +
++ 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
+
+
+
+
+ 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+ + + +
The table below shows the errors that are thrown.
+
+ The errors are generic Error objects whose message begins
+ '[BigNumber Error]'.
+
| Method | +Throws | +
|---|---|
+ BigNumber+ comparedTo+ dividedBy+ dividedToIntegerBy+ isEqualTo+ isGreaterThan+ isGreaterThanOrEqualTo+ isLessThan+ isLessThanOrEqualTo+ minus+ modulo+ plus+ multipliedBy
+ |
+ Base not a primitive number | +
| Base not an integer | +|
| Base out of range | +|
| Number primitive has more than 15 significant digits* | +|
| Not a base... number* | +|
| Not a number* | +|
clone |
+ Object expected | +
config |
+ Object expected | +
DECIMAL_PLACES not a primitive number |
+ |
DECIMAL_PLACES not an integer |
+ |
DECIMAL_PLACES out of range |
+ |
ROUNDING_MODE not a primitive number |
+ |
ROUNDING_MODE not an integer |
+ |
ROUNDING_MODE out of range |
+ |
EXPONENTIAL_AT not a primitive number |
+ |
EXPONENTIAL_AT not an integer |
+ |
EXPONENTIAL_AT out of range |
+ |
RANGE not a primitive number |
+ |
RANGE not an integer |
+ |
RANGE cannot be zero |
+ |
RANGE cannot be zero |
+ |
CRYPTO not true or false |
+ |
crypto unavailable |
+ |
MODULO_MODE not a primitive number |
+ |
MODULO_MODE not an integer |
+ |
MODULO_MODE out of range |
+ |
POW_PRECISION not a primitive number |
+ |
POW_PRECISION not an integer |
+ |
POW_PRECISION out of range |
+ |
FORMAT not an object |
+ |
ALPHABET invalid |
+ |
+ decimalPlaces+ precision+ random+ shiftedBy+ toExponential+ toFixed+ toFormat+ toPrecision
+ |
+ Argument not a primitive number | +
| Argument not an integer | +|
| Argument out of range | +|
+ decimalPlaces+ precision
+ |
+ Argument not true or false | +
exponentiatedBy |
+ Argument not an integer | +
isBigNumber |
+ Invalid BigNumber* | +
+ minimum+ maximum
+ |
+ Not a number* | +
+ random
+ |
+ crypto unavailable | +
+ toFormat
+ |
+ Argument not an object | +
toFraction |
+ Argument not an integer | +
| Argument out of range | +|
toString |
+ Base not a primitive number | +
| Base not an integer | +|
| Base out of range | +
*Only thrown if BigNumber.DEBUG is true.
To determine if an exception is a BigNumber Error:
+
+try {
+ // ...
+} catch (e) {
+ if (e instanceof Error && e.message.indexOf('[BigNumber Error]') === 0) {
+ // ...
+ }
+}
+
+
+
+
+ To prevent the accidental use of a BigNumber in primitive number operations, or the
+ accidental addition of a BigNumber to a string, the valueOf method can be safely
+ overwritten as shown below.
+
+ The valueOf method is the same as the
+ toJSON method, and both are the same as the
+ toString method except they do not take a base
+ argument and they include the minus sign for negative zero.
+
+BigNumber.prototype.valueOf = function () {
+ throw Error('valueOf called!')
+}
+
+x = new BigNumber(1)
+x / 2 // '[BigNumber Error] valueOf called!'
+x + 'abc' // '[BigNumber Error] valueOf called!'
+
+
+
+
+ + 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.
+