How javascript treat large integers (more than 52 bits)?

Question

Consider this code (node v5.0.0)

const a = Math.pow(2, 53)
const b = Math.pow(2, 53) + 1
const c = Math.pow(2, 53) + 2

console.log(a === b) // true
console.         


        

Answer 1

To supplement to other answers here, it's worth mentioning that BigInt exists. This allows JavaScript to handle arbitrarily large integers.

Use the n suffix on your numbers and use regular operators like 2n ** 53n + 2n. Important to point out that a BigInt is not a Number, but you can do range-limited interoperation with Number via explicit conversions.

Some examples at the Node.js REPL:

> 999999999999999999999999999999n + 1n
1000000000000000000000000000000n
> 2n ** 53n
9007199254740992n
> 2n ** 53n + 1n
9007199254740993n
> 2n ** 53n == 2n ** 53n + 1n
false
> typeof 1n
'bigint'
> 3 * 4n
TypeError: Cannot mix BigInt and other types, use explicit conversions
> BigInt(3) * 4n
12n
> 3 * Number(4n)
12
> Number(2n ** 53n) == Number(2n ** 53n + 1n)
true

Answer 2

How javascript treat large integers?

JS does not have integers. JS numbers are 64 bit floats. They are stored as a mantissa and an exponent.

The precision is given by the mantissa, the magnitude by the exponent.

If your number needs more precision than what can be stored in the mantissa, the least significant bits will be truncated.

9007199254740992; // 9007199254740992
(9007199254740992).toString(2);
// "100000000000000000000000000000000000000000000000000000"
//  \        \                    ...                   /\
//   1        10                                      53  54
// The 54-th is not stored, but is not a problem because it's 0

9007199254740993; // 9007199254740992
(9007199254740993).toString(2);
// "100000000000000000000000000000000000000000000000000000"
//  \        \                    ...                   /\
//   1        10                                      53  54
// The 54-th bit should be 1, but the mantissa only has 53 bits!

9007199254740994; // 9007199254740994
(9007199254740994).toString(2);
// "100000000000000000000000000000000000000000000000000010"
//  \        \                    ...                   /\
//   1        10                                      53  54
// The 54-th is not stored, but is not a problem because it's 0

Then, you can store all these integers:

-9007199254740992, -9007199254740991, ..., 9007199254740991, 9007199254740992

The second one is called the minimum safe integer:

The value of Number.MIN_SAFE_INTEGER is the smallest integer n such that n and n − 1 are both exactly representable as a Number value.

The value of Number.MIN_SAFE_INTEGER is −9007199254740991 (−(2⁵³−1)).

The second last one is called the maximum safe integer:

The value of Number.MAX_SAFE_INTEGER is the largest integer n such that n and n + 1 are both exactly representable as a Number value.

The value of Number.MAX_SAFE_INTEGER is 9007199254740991 (2⁵³−1).

Answer 3

Answering your second question, here is your maximum safe integer in JavaScript:

console.log( Number.MAX_SAFE_INTEGER );

All the rest is written in MDN:

The MAX_SAFE_INTEGER constant has a value of 9007199254740991. The reasoning behind that number is that JavaScript uses double-precision floating-point format numbers as specified in IEEE 754 and can only safely represent numbers between -(2 ** 53 - 1) and 2 ** 53 - 1.

Safe in this context refers to the ability to represent integers exactly and to correctly compare them. For example, Number.MAX_SAFE_INTEGER + 1 === Number.MAX_SAFE_INTEGER + 2 will evaluate to true, which is mathematically incorrect. See Number.isSafeInteger() for more information.

Answer 4

.:: JavaScript only supports 53 bit integers ::.

All numbers in JavaScript are floating point which means that integers are always represented as

sign × mantissa × 2exponent

The mantissa has 53 bits. You can use the exponent to get higher integers, but then they won’t be contiguous, any more. For example, you generally need to multiply the mantissa by two (exponent 1) in order to reach the 54th bit.

However, if you multiply by two, you will only be able to represent every second integer:

Math.pow(2, 53)      // 54 bits 9007199254740992
Math.pow(2, 53) + 1  // 9007199254740992
Math.pow(2, 53) + 2  //9007199254740994
Math.pow(2, 53) + 3  //9007199254740996
Math.pow(2, 53) + 4  //9007199254740996

Rounding effects during the addition make things unpredictable for odd increments (+1 versus +3). The actual representation is a bit more complicated but this explanation should help you understand the basic problem.

You can safely use strint library to encode large integers in strings and perform arithmetic operations on them too.

Here is the full article.

Answer 5

Number.MAX_VALUE will tell you the largest floating-point value representable in your JS implementation. The answer will likely be: 1.7976931348623157e+308. But that doesn't mean that every integer up to 10^308 can be represented exactly. As your example code shows, beyond 2^53 only even numbers can be represented, and as you go farther out on the number line the gaps get much wider.

If you need exact integers larger than 2^53, you probably want to work with a bignum package, which allows for arbitrarily large integers (within the bounds of available memory). Two packages that I happen to know are:

BigInt by Leemon

and

Crunch