奇怪的JavaScript编号行为

Question

I found the following weird behavior while working on JavaScript numbers.

var baseNum = Math.pow(2, 53);
console.log(baseNum); //prints 9007199254740992

console.log(baseNum + 1); //prints 9007199254740992 again!

console.log(baseNum + 2); //prints 9007199254740994, 2 more than +1

console.log(baseNum + 3) // prints 9007199254740996, 2 more than +2
console.log(baseNum + 4) // prints 9007199254740996, same as +3

What is happening here? I understand that JavaScript can only represent numbers upto 2^53 (they are internally 'double'?), but why this behavior?

If 2^53 is the practical max, then why do we have Number.MAX_VALUE ( 1.7976931348623157e+308 )?.

Answer 1

The number is really a double. The mantissa has 52-bits ( source and extra information on doubles ). Therefore, storing 2^53 chops off the one bit.

The number is stored using 3 pieces a sign bit (fairly straight forward) and two other pieces, the mantissa M and the exponent E. The number is calculated as:

(1 + M/2^53) * 2^(E-1023)

I might have some of the specifics there a little off, but the basic idea is there. So when the number is 2^53, 2^(E-1023) = 2^53 and since there are only 52 bits in M, you can no longer represent the lowest bit.

Answer 2

The answer that @CrazyCasta gave is good.

The only thing to add is to your second question:

If 2^53 is the practical max, then why do we have Number.MAX_VALUE ( 1.7976931348623157e+308 )?

As you've demonstrated, it can store numbers larger than 2^53 , but with precision worse than 2^0 . As the numbers grow ever larger, they lose more and more precision.

So max value in Number.MAX_VALUE means the "largest" value it can represent; but it doesn't mean that accuracy is the same as a value near 2^1 or 2^53 .

A corollary to this is that Number.MIN_VALUE is the smallest value that Number can contain; not the most negative . That is, it is the closest non-zero number to zero: 5.00E-324 (notice that's a positive number!).

Answer 3

The maximum value storable in a long is much larger than the maximum value storable with exact precision in a long. Floating-point numbers have a fixed maximum number of significant digits, but the magnitude of the number can get much larger.

There are a certain number of bits allocated for an exponent (power of two), and that's implicitly multiplied by the mantissa stored in the rest of the bits. Beyond a certain point, you lose precision, but you can keep incrementing the exponent to represent larger and larger magnitudes.