# Numbers #

Math.js supports three types of numbers:

- Number for fast floating point arithmetic, described on this page.
- BigNumber for arbitrary precision arithmetic, described on the page BigNumbers.
- Fraction, which stores numbers in terms of a numerator and denominators, described on the page Fractions.

## Configuration #

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, which
can be configured when instantiating math.js:

```
math.config({
number: 'number' // Default type of number:
// 'number' (default), 'BigNumber', or 'Fraction'
})
```

## Round-off errors #

Math.js uses the built-in JavaScript Number type. A Number is a floating point
number with a limited precision of 64 bits, about 16 digits. The largest integer
number which can be represented by a JavaScript Number
is `+/- 9007199254740992`

(`+/- 2^53`

). Because of the limited precision of
floating point numbers round-off errors can occur during calculations.
This can be easily demonstrated:

```
// a round-off error
0.1 + 0.2 // 0.30000000000000004
math.add(0.1, 0.2) // 0.30000000000000004
```

In most cases, round-off errors donâ€™t matter: they have no significant impact on the results. However, it looks ugly when displaying output to a user. A solution is to limit the precision just below the actual precision of 16 digits in the displayed output:

```
// prevent round-off errors showing up in output
const ans = math.add(0.1, 0.2) // 0.30000000000000004
math.format(ans, {precision: 14}) // '0.3'
```

Alternatives are to use Fractions which store a number as a numerator and denominator, BigNumbers which store a number with a higher precision, or bigint which can store larger integer numbers.

## Minimum and maximum #

A Number can store values between `5e-324`

and `1.7976931348623157e+308`

.
Values smaller than the minimum are stored as `0`

, and values larger than the
maximum are stored as `+/- Infinity`

.

```
// exceeding the maximum and minimum number
console.log(1e309) // Infinity
console.log(1e-324) // 0
```

## Equality #

Because of rounding errors in calculations, it is unsafe to compare JavaScript
Numbers. For example executing `0.1 + 0.2 == 0.3`

in JavaScript will return
false, as the addition `0.1 + 0.2`

introduces a round-off error and does not
return exactly `0.3`

.

To solve this problem, the relational functions of math.js check whether the
relative and absolute differences between the compared values is smaller than the configured
option `relTol`

and `absTol`

. In pseudo code (without exceptions for 0, Infinity and NaN):

```
abs(a-b) <= max(relTol * max(abs(a), abs(b)), absTol)
```

where:

`relTol`

is the relative tolerance between x and y and`absTol`

the absolute tolerance. Relative tolerance and absolute tolerance are configurable and are`1e-12`

and`1e-15`

respectively by default. See Configuration.`DBL_EPSILON`

is the minimum positive floating point number such that`1.0 + DBL_EPSILON !== 1.0`

. This is a constant with a value of approximately`2.2204460492503130808472633361816e-16`

.

Note that the relational functions cannot be used to compare small values
(`< 2.22e-16`

). These values are all considered equal to zero.

Examples:

```
// compare values having a round-off error
console.log(0.1 + 0.2 === 0.3) // false
console.log(math.equal(0.1 + 0.2, 0.3)) // true
// small values (< 2.22e-16) cannot be compared
console.log(3e-20 === 3.1e-20) // false
console.log(math.equal(3e-20, 3.1e-20)) // true
```

The available relational functions are: `compare`

, `equal`

, `larger`

,
`largerEq`

, `smaller`

, `smallerEq`

, `unequal`

.