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I work in Javascript with integer numbers only (mainly adding numbers and shifting them). I wonder how big they can be without loosing any bits.
For example, how big X can be such that 1 << X will represent 2^X ?
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The JavaScript Number type is a double-precision 64-bit binary format IEEE 754 value, like double in Java or C#.
Number type. The Number type is a double-precision 64-bit binary format IEEE 754 value.
32-bit integer has range from -2147483648 ( -2^31 ) to 2147483647 ( 2^31 − 1 )
Since JavaScript does not natively support 64-bit integers, 64-bit Ids are instead represented as strings. Strings containing 64-bit Ids are distinguished from ordinary strings through use of the Id64String type alias.
All numbers in JavaScript are actually IEEE-754 compliant floating-point doubles. These have a 53-bit mantissa which should mean that any integer value with a magnitude of approximately 9 quadrillion or less -- more specifically, 9,007,199,254,740,991 -- will be represented accurately.
NOTICE: in 2018 main browsers and NodeJS are working also with the new Javascript's primitive-type, BigInt, solving the problems with integer value magnitude.
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All answers are partially wrong - Maybe due the new ES6/ES7 specs - , read why:
First of all, in JavaScript, the representation of the number is 2^53 - 1 that is true for @Luke answer, we can prove that by running Number.MAX_SAFE_INTEGER that will show a big number, then we do log2 to confirm that the number of bits is the same :
Number.MAX_SAFE_INTEGER 9007199254740991 Math.log2(9007199254740991) 53 
However, Bitwise operation are calculated on 32 bits ( 4 bytes ), meaning if you exceed 32bits shifts you will start loosing bits.
Welcome to Javascript!
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