# 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](bignumbers.md).
- Fraction, which stores numbers in terms of a numerator and denominators,
described on the page [Fractions](fractions.md).
## 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.eval`) use the default number type, which
can be configured when instantiating math.js:
```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:
```js
// a round-off error
0.1 + 0.2; // 0.30000000000000004
math.add(0.1, 0.2); // 0.30000000000000004
```
In most cases, round-off errors don'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:
```js
// prevent round-off errors showing up in output
var ans = math.add(0.1, 0.2); // 0.30000000000000004
math.format(ans, {precision: 14}); // '0.3'
```
## 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`.
```js
// 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 difference between the compared values is smaller than the configured
option `epsilon`. In pseudo code (without exceptions for 0, Infinity and NaN):
diff = abs(x - y)
nearlyEqual = (diff <= max(abs(x), abs(y)) * EPSILON) OR (diff < DBL_EPSILON)
where:
- `EPSILON` is the relative difference between x and y. Epsilon is configurable
and is `1e-14` by default. See [Configuration](../configuration.md).
- `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:
```js
// 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`.