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BigNumbers

For calculations with an arbitrary precision, math.js supports a BigNumber datatype. BigNumber support is powered by decimal.js.

Usage

A BigNumber can be created using the function bignumber:

math.bignumber('2.3e+500') // BigNumber, 2.3e+500

Most functions can determine the type of output from the type of input: a number as input will return a number as output, a BigNumber as input returns a BigNumber as output. Functions which cannot determine the type of output from the input (for example math.evaluate) use the default number type number, which can be configured when instantiating math.js. To configure the use of BigNumbers instead of numbers by default, configure math.js like:

math.config({
  number: 'BigNumber',      // Default type of number:
                            // 'number' (default), 'BigNumber', or 'Fraction'
  precision: 64,            // Number of significant digits for BigNumbers
  relTol: 1e-60,
  absTol: 1e-63
})

// use math
math.evaluate('0.1 + 0.2')  // BigNumber, 0.3

The default precision for BigNumber is 64 digits, and can be configured with the option precision.

Note that we also change the configuration of relTol and absTol to be close to the precision limit of our BigNumbers. relTol and absTol are used for example in relational and rounding functions (equal, larger, smaller, round, floor, etc) to determine when a value is nearly equal, see Equality. If we would leave relTol and absTol unchanged, having the default value of 1e-12 and 1e-15 respectively, we could get inaccurate and misleading results since we're now working with a higher precision.

Support

Most functions in math.js support BigNumbers, but not all of them. For example the function random doesn't support BigNumbers.

Round-off errors

Calculations with BigNumber are much slower than calculations with Number, but they can be executed with an arbitrary precision. By using a higher precision, it is less likely that round-off errors occur:

// round-off errors with numbers
math.add(0.1, 0.2)                                     // Number, 0.30000000000000004
math.divide(0.3, 0.2)                                  // Number, 1.4999999999999998

// no round-off errors with BigNumbers :)
math.add(math.bignumber(0.1), math.bignumber(0.2))     // BigNumber, 0.3
math.divide(math.bignumber(0.3), math.bignumber(0.2))  // BigNumber, 1.5

Limitations

It's important to realize that BigNumbers do not solve all problems related to precision and round-off errors. Numbers with an infinite number of digits cannot be represented with a regular number nor a BigNumber. Though a BigNumber can store a much larger number of digits, the amount of digits remains limited if only to keep calculations fast enough to remain practical.

const one = math.bignumber(1)
const three = math.bignumber(3)
const third = math.divide(one, three)
console.log(third.toString())
// outputs 0.3333333333333333333333333333333333333333333333333333333333333333

const ans = math.multiply(third, three)
console.log(ans.toString())
// outputs 0.9999999999999999999999999999999999999999999999999999999999999999
// this should be 1 again, but `third` is rounded to a limited number of digits 3

Conversion

BigNumbers can be converted to numbers and vice versa using the functions number and bignumber. When converting a BigNumber to a number, the high precision of the BigNumber will be lost. When a BigNumber is too large to be represented as Number, it will be initialized as Infinity.

// converting numbers and BigNumbers
const a = math.number(0.3)                         // number, 0.3
const b = math.bignumber(a)                        // BigNumber, 0.3
const c = math.number(b)                           // number, 0.3

// exceeding the maximum of a number
const d = math.bignumber('1.2e500')                // BigNumber, 1.2e+500
const e = math.number(d)                           // number, Infinity

// loosing precision when converting to number
const f = math.bignumber('0.2222222222222222222')  // BigNumber, 0.2222222222222222222
const g = math.number(f)                           // number,    0.2222222222222222