Can anyone please elaborate Number precision and overflow in JavaScript?

Question

I was looking at Number.isInteger() on MDN. I came upon this example -

Number.isInteger(5.000000000000001); // false

I changed the value to 5.0000000000000001 and if I execute the same method on the console, the result is:

Number.isInteger(5.0000000000000001); // true

Also 5.0000000000000001 === 5 comes out to be true.

Can someone please explain what's happening here?

Answer 1

JavaScript uses the IEEE-754 basic 64-bit binary floating-point format. In this format, every finite number is expressed as an integer M times a power of 2, 2^E. The magnitude of M must be less than 2⁵³, and E may be from −1074 to 971. (The more common description scales M and E differently, so that M has fraction bits instead of being an integer, but this is equivalently mathematically.)

For the number 5.000000000000001, the most precisely it can be represented is to use E = −50, because this makes M as large as will fit, about 5•2⁵⁰ = 1.25•2⁵². If we made E one smaller, M would have to be about 1.25•2⁵³, which exceeds the 2⁵³ limit.

So, given E = −50, we want 5.000000000000001 = M•2⁻⁵⁰, which means M = 5.000000000000001•2⁵⁰ = 5•2⁵⁰ + .000000000000001•2⁵⁰. Obviously the first term, 5•2⁵⁰, is an integer. The second term, .000000000000001•2⁵⁰, is about 1.1259. So, when converting 5.000000000000001 to JavaScript’s Number format, we need to make M an integer, so we have to round 1.1259 to 1. The result is M = 5•2⁵⁰ + 1, so the number is (5•2⁵⁰ + 1)•2⁻⁵⁰ = 5.00000000000000088817841970012523233890533447265625.

On the other hand, when we have 5.0000000000000001, with one more zero, we have 5.0000000000000001•2⁵⁰ = 5•2⁵⁰ + .0000000000000001•2⁵⁰. In this case, the second term, .0000000000000001•2⁵⁰, is about .11259. So, when converting 5.0000000000000001 to JavaScript’s Number format, we have to round .11259 to 0, so the number is ((5•2⁵⁰)•2⁻⁵⁰ = 5.