Chapter 11. Numbers
JavaScript has a single type for all numbers: it treats all of them as floatingpoint numbers. However, the dot is not displayed if there are no digits after the decimal point:
> 5.000 5
Internally, most JavaScript engines optimize and do distinguish between floatingpoint numbers and integers (details: Integers in JavaScript). But that is something that programmers don’t see.
JavaScript numbers are double
(64bit) values, based on the IEEE Standard for FloatingPoint Arithmetic (IEEE 754). That standard is used by many programming languages.
Number Literals
A number literal can be an integer, floating point, or (integer) hexadecimal:
> 35 // integer 35 > 3.141 // floating point 3.141 > 0xFF // hexadecimal 255
Exponent
An exponent, eX
, is an abbreviation for “multiply with 10^{X}”:
> 5e2 500 > 5e2 0.05 > 0.5e2 50
Invoking Methods on Literals
With number literals, the dot for accessing a property must be distinguished from the decimal dot. This leaves you with the following options if you want to invoke toString()
on the number literal 123
:
123
..
toString
()
123
.
toString
()
// space before the dot
123.0
.
toString
()
(
123
).
toString
()
Converting to Number
Values are converted to numbers as follows:
Value  Result 




A boolean 

 
A number  Same as input (nothing to convert) 
A string  Parse the number in the string (ignoring leading and trailing whitespace); the empty string is converted to 0.
Example: 
An object  Call 
When converting the empty string to a number, NaN
would arguably be a better result. The result 0 was chosen to help with empty numeric input fields, in line with what other programming languages did in the mid1990s.^{[12]}
Manually Converting to Number
The two most common ways to convert any value to a number are:
 (Invoked as a function, not as a constructor) 

I prefer Number()
, because it is more descriptive. Here are some examples:
> Number('') 0 > Number('123') 123 > Number('\t\v\r12.34\n ') // ignores leading and trailing whitespace 12.34 > Number(false) 0 > Number(true) 1
parseFloat()
The global function parseFloat()
provides another way to convert values to numbers. However, Number()
is usually a better choice, as we shall see in a moment. This code:
parseFloat
(
str
)
converts str
to string, trims leading whitespace, and then parses the longest prefix that is a floatingpoint number. If no such prefix exists (e.g., in an empty string), NaN
is returned.
Comparing parseFloat()
and Number()
:
Applying
parseFloat()
to a nonstring is less efficient, because it coerces its argument to a string before parsing it. As a consequence, many values thatNumber()
converts to actual numbers are converted toNaN
byparseFloat()
:> parseFloat(true) // same as parseFloat('true') NaN > Number(true) 1 > parseFloat(null) // same as parseFloat('null') NaN > Number(null) 0
parseFloat()
parses the empty string asNaN
:> parseFloat('') NaN > Number('') 0
parseFloat()
parses until the last legal character, meaning you get a result where you may not want one:> parseFloat('123.45#') 123.45 > Number('123.45#') NaN
parseFloat()
ignores leading whitespace and stops before illegal characters (which include whitespace):> parseFloat('\t\v\r12.34\n ') 12.34
Number()
ignores both leading and trailing whitespace (but other illegal characters lead toNaN
).
Special Number Values
JavaScript has several special number values:

Two error values,
NaN
andInfinity
. 
Two values for zero,
+0
and0
. JavaScript has two zeros, a positive zero and a negative zero, because the sign and the magnitude of a number are stored separately. In most of this book, I pretend that there is only a single zero, and you almost never see in JavaScript that there are two of them.
NaN
The error value NaN
(an abbreviation for “not a number”) is, ironically, a number value:
> typeof NaN 'number'
It is produced by errors such as the following:
A number could not be parsed:
> Number('xyz') NaN > Number(undefined) NaN
An operation failed:
> Math.acos(2) NaN > Math.log(1) NaN > Math.sqrt(1) NaN
One of the operands is
NaN
(this ensures that, if an error occurs during a longer computation, you can see it in the final result):> NaN + 3 NaN > 25 / NaN NaN
Pitfall: checking whether a value is NaN
NaN
is the only value that is not equal to itself:
> NaN === NaN false
Strict equality (===
) is also used by Array.prototype.indexOf
. You therefore can’t search for NaN
in an array via that method:
> [ NaN ].indexOf(NaN) 1
If you want to check whether a value is NaN
, you have to use the global function isNaN()
:
> isNaN(NaN) true > isNaN(33) false
However, isNaN
does not work properly with nonnumbers, because it first converts those to numbers. That conversion can produce NaN
and then the function incorrectly returns true
:
> isNaN('xyz') true
Thus, it is best to combine isNaN
with a type check:
function
myIsNaN
(
value
)
{
return
typeof
value
===
'number'
&&
isNaN
(
value
);
}
Alternatively, you can check whether the value is unequal to itself (as NaN
is the only value with this trait). But that is less selfexplanatory:
function
myIsNaN
(
value
)
{
return
value
!==
value
;
}
Note that this behavior is dictated by IEEE 754. As noted in Section 7.11, “Details of comparison predicates”:^{[13]}
Every NaN shall compare unordered with everything, including itself.
Infinity
Infinity
is an error value indicating one of two problems: a number can’t be represented because its magnitude is too large, or a division by zero has happened.
Infinity
is larger than any other number (except NaN
). Similarly, Infinity
is smaller than any other number (except NaN
). That makes them useful as default values—for example, when you are looking for a minimum or maximum.
Error: a number’s magnitude is too large
How large a number’s magnitude can become is determined by its internal representation (as discussed in The Internal Representation of Numbers), which is the arithmetic product of:
 A mantissa (a binary number 1.f_{1}f_{2}...)
 2 to the power of an exponent
The exponent must be between (and excluding) −1023 and 1024. If the exponent is too small, the number becomes 0. If the exponent is too large, it becomes Infinity
. 2^{1023} can still be represented, but 2^{1024} can’t:
> Math.pow(2, 1023) 8.98846567431158e+307 > Math.pow(2, 1024) Infinity
Error: division by zero
Dividing by zero produces Infinity
as an error value:
> 3 / 0 Infinity > 3 / 0 Infinity
Computing with Infinity
You get the error result NaN
if you try to “neutralize” one Infinity
with another one:
> Infinity  Infinity NaN > Infinity / Infinity NaN
If you try to go beyond Infinity
, you still get Infinity
:
> Infinity + Infinity Infinity > Infinity * Infinity Infinity
Checking for Infinity
Strict and lenient equality work fine for Infinity
:
> var x = Infinity; > x === Infinity true
Additionally, the global function isFinite()
allows you to check whether a value is an actual number (neither infinite nor NaN
):
> isFinite(5) true > isFinite(Infinity) false > isFinite(NaN) false
Two Zeros
Because JavaScript’s numbers keep magnitude and sign separate, each nonnegative number has a negative, including 0
.
The rationale for this is that whenever you represent a number digitally, it can become so small that it is indistinguishable from 0, because the encoding is not precise enough to represent the difference. Then a signed zero allows you to record “from which direction” you approached zero; that is, what sign the number had before it was considered zero. Wikipedia nicely sums up the pros and cons of signed zeros:
It is claimed that the inclusion of signed zero in IEEE 754 makes it much easier to achieve numerical accuracy in some critical problems, in particular when computing with complex elementary functions. On the other hand, the concept of signed zero runs contrary to the general assumption made in most mathematical fields (and in most mathematics courses) that negative zero is the same thing as zero. Representations that allow negative zero can be a source of errors in programs, as software developers do not realize (or may forget) that, while the two zero representations behave as equal under numeric comparisons, they are different bit patterns and yield different results in some operations.
Best practice: pretend there’s only one zero
JavaScript goes to great lengths to hide the fact that there are two zeros. Given that it normally doesn’t matter that they are different, it is recommended that you play along with the illusion of the single zero. Let’s examine how that illusion is maintained.
In JavaScript, you normally write 0
, which means +0
. But 0
is also displayed as simply 0
. This is what you see when you use a browser command line or the Node.js REPL:
> 0 0
That is because the standard toString()
method converts both zeros to the same '0'
:
> (0).toString() '0' > (+0).toString() '0'
Equality doesn’t distinguish zeros, either. Not even ===
:
> +0 === 0 true
Array.prototype.indexOf
uses ===
to search for elements, maintaining the illusion:
> [ 0, +0 ].indexOf(+0) 0 > [ +0, 0 ].indexOf(0) 0
The ordering operators also consider the zeros to be equal:
> 0 < +0 false > +0 < 0 false
Distinguishing the two zeros
How can you actually observe that the two zeros are different? You can divide by zero (Infinity
and +Infinity
can be distinguished by ===
):
> 3 / 0 Infinity > 3 / +0 Infinity
Another way to perform the division by zero is via Math.pow()
(see Numerical Functions):
> Math.pow(0, 1) Infinity > Math.pow(+0, 1) Infinity
Math.atan2()
(see Trigonometric Functions) also reveals that the zeros are different:
> Math.atan2(0, 1) 3.141592653589793 > Math.atan2(+0, 1) 3.141592653589793
The canonical way of telling the two zeros apart is the division by zero. Therefore, a function for detecting negative zeros would look like this:
function
isNegativeZero
(
x
)
{
return
x
===
0
&&
(
1
/
x
<
0
);
}
Here is the function in use:
> isNegativeZero(0) false > isNegativeZero(0) true > isNegativeZero(33) false
The Internal Representation of Numbers
JavaScript numbers have 64bit precision, which is also called double precision (type double
in some programming languages). The internal representation is based on the IEEE 754 standard. The 64 bits are distributed between a number’s sign, exponent, and fraction as follows:
Sign  Exponent ∈ [−1023, 1024]  Fraction 
1 bit  11 bits  52 bits 
Bit 63  Bits 62–52  Bits 51–0 
The value of a number is computed by the following formula:
(–1)^{sign} × %1.fraction × 2^{exponent}
The prefixed percentage sign (%
) means that the number in the middle is written in binary notation: a 1, followed by a binary point, followed by a binary fraction—namely the binary digits of the fraction (a natural number). Here are some examples of this representation:
+0  (sign = 0, fraction = 0, exponent = −1023)  
–0  (sign = 1, fraction = 0, exponent = −1023)  
1  = (−1)^{0} × %1.0 × 2^{0}  (sign = 0, fraction = 0, exponent = 0) 
2  = (−1)^{0} × %1.0 × 2^{1}  
3  = (−1)^{0} × %1.1 × 2^{1}  (sign = 0, fraction = 2^{51}, exponent = 0) 
0.5  = (−1)^{0} × %1.0 × 2^{−1}  
−1  = (−1)^{1} × %1.0 × 2^{0} 
The encodings of +0, −0, and 3 can be explained as follows:
 ±0: Given that the fraction is always prefixed by a 1, it’s impossible to represent 0 with it. Hence, JavaScript encodes a zero via the fraction 0 and the special exponent −1023. The sign can be either positive or negative, meaning that JavaScript has two zeros (see Two Zeros).
 3: Bit 51 is the most significant (highest) bit of the fraction. That bit is 1.
Special Exponents
The previously mentioned representation of numbers is called normalized. In that case, the exponent e is in the range −1023 < e < 1024 (excluding lower and upper bounds). −1023 and 1024 are special exponents:

1024 is used for error values such as
NaN
andInfinity
. −1023 is used for:
 Zero (if the fraction is 0, as just explained)
 Small numbers close to zero (if the fraction is not 0).
To enable both applications, a different, socalled denormalized, representation is used:
(–1)^{sign} × %0.fraction × 2^{–1022}
To compare, the smallest (as in “closest to zero”) numbers in normalized representation are:
(–1)^{sign} × %1.fraction × 2^{–1022}
Denormalized numbers are smaller, because there is no leading digit 1.
Handling Rounding Errors
JavaScript’s numbers are usually entered as decimal floatingpoint numbers, but they are internally represented as binary floatingpoint numbers. That leads to imprecision. To understand why, let’s forget JavaScript’s internal storage format and take a general look at what fractions can be well represented by decimal floatingpoint numbers and by binary floatingpoint numbers. In the decimal system, all fractions are a mantissa m divided by a power of 10:
So, in the denominator, there are only tens. That’s why cannot be expressed precisely as a decimal floatingpoint number—there is no way to get a 3 into the denominator. Binary floatingpoint numbers only have twos in the denominator. Let’s examine which decimal floatingpoint numbers can be represented well as binary and which can’t. If there are only twos in the denominator, the decimal number can be represented:
 0.5_{dec} = = = 0.1_{bin}
 0.75_{dec} = = = 0.11_{bin}
 0.125_{dec} = = = 0.001_{bin}
Other fractions cannot be represented precisely, because they have numbers other than 2 in the denominator (after prime factorization):
 0.1_{dec} = =
 0.2_{dec} = =
You can’t normally see that JavaScript doesn’t store exactly 0.1 internally. But you can make it visible by multiplying it with a high enough power of 10:
> 0.1 * Math.pow(10, 24) 1.0000000000000001e+23
And if you add two imprecisely represented numbers, the result is sometimes imprecise enough that the imprecision becomes visible:
> 0.1 + 0.2 0.30000000000000004
Another example:
> 0.1 + 1  1 0.10000000000000009
Due to rounding errors, as a best practice you should not compare nonintegers directly. Instead, take an upper bound for rounding errors into consideration. Such an upper bound is called a machine epsilon. The standard epsilon value for double precision is 2^{−53}:
var
EPSILON
=
Math
.
pow
(
2
,

53
);
function
epsEqu
(
x
,
y
)
{
return
Math
.
abs
(
x

y
)
<
EPSILON
;
}
epsEqu()
ensures correct results where a normal comparison would be inadequate:
> 0.1 + 0.2 === 0.3 false > epsEqu(0.1+0.2, 0.3) true
Integers in JavaScript
As mentioned before, JavaScript has only floatingpoint numbers. Integers appear internally in two ways. First, most JavaScript engines store a small enough number without a decimal fraction as an integer (with, for example, 31 bits) and maintain that representation as long as possible. They have to switch back to a floatingpoint representation if a number’s magnitude grows too large or if a decimal fraction appears.
Second, the ECMAScript specification has integer operators: namely, all of the bitwise operators. Those operators convert their operands to 32bit integers and return 32bit integers. For the specification, integer only means that the numbers don’t have a decimal fraction, and 32bit means that they are within a certain range. For engines, 32bit integer means that an actual integer (nonfloatingpoint) representation can usually be introduced or maintained.
Ranges of Integers
Internally, the following ranges of integers are important in JavaScript:
Safe integers (see Safe Integers), the largest practically usable range of integers that JavaScript supports:
 53 bits plus a sign, range (−2^{53}, 2^{53})
Array indices (see Array Indices):
 32 bits, unsigned
 Maximum length: 2^{32}−1
 Range of indices: [0, 2^{32}−1) (excluding the maximum length!)
Bitwise operands (see Bitwise Operators):

Unsigned right shift operator (
>>>
): 32 bits, unsigned, range [0, 2^{32})  All other bitwise operators: 32 bits, including a sign, range [−2^{31}, 2^{31})

Unsigned right shift operator (
“Char codes,” UTF16 code units as numbers:

Accepted by
String.fromCharCode()
(see String Constructor Method) 
Returned by
String.prototype.charCodeAt()
(see Extract Substrings)  16 bits, unsigned

Accepted by
Representing Integers as FloatingPoint Numbers
JavaScript can only handle integer values up to a magnitude of 53 bits (the 52 bits of the fraction plus 1 indirect bit, via the exponent; see The Internal Representation of Numbers for details).
The following table explains how JavaScript represents 53bit integers as floatingpoint numbers:
Bits  Range  Encoding 
1 bit  0  
1 bit  1  %1 × 2^{0} 
2 bits  2–3  %1.f_{51} × 2^{1} 
3 bits  4–7 = 2^{2}–(2^{3}−1)  %1.f_{51}f_{50} × 2^{2} 
4 bits  2^{3}–(2^{4}−1)  %1.f_{51}f_{50}f_{49} × 2^{3} 
⋯  ⋯  ⋯ 
53 bits  2^{52}–(2^{53}−1)  %1.f_{51}⋯f_{0} × 2^{52} 
There is no fixed sequence of bits that represents the integer. Instead, the mantissa %1.f is shifted by the exponent, so that the leading digit 1 is in the right place. In a way, the exponent counts the number of digits of the fraction that are in active use (the remaining digits are 0). That means that for 2 bits, we use one digit of the fraction and for 53 bits, we use all digits of the fraction. Additionally, we can represent 2^{53} as %1.0 × 2^{53}, but we get problems with higher numbers:
Bits  Range  Encoding 
54 bits  2^{53}–(2^{54}−1)  %1.f_{51}⋯f_{0}0 × 2^{53} 
55 bits  2^{54}–(2^{55}−1)  %1.f_{51}⋯f_{0}00 × 2^{54} 
⋯ 
For 54 bits, the least significant digit is always 0, for 55 bits the two least significant digits are always 0, and so on. That means that for 54 bits, we can only represent every second number, for 55 bits only every fourth number, and so on. For example:
> Math.pow(2, 53)  1 // OK 9007199254740991 > Math.pow(2, 53) // OK 9007199254740992 > Math.pow(2, 53) + 1 // can't be represented 9007199254740992 > Math.pow(2, 53) + 2 // OK 9007199254740994
Best practice
If you work with integers of up to 53 bits magnitude, you are fine. Unfortunately, you’ll often encounter 64bit unsigned integers in programming (Twitter IDs, databases, etc.). These must be stored in strings in JavaScript. If you want to perform arithmetic with such integers, you need special libraries. There are plans to bring larger integers to JavaScript, but that will take a while.
Safe Integers
JavaScript can only safely represent integers i in the range −2^{53} < i < 2^{53}. This section examines what that means and what the consequences are. It is based on an email by Mark S. Miller to the esdiscuss mailing list.
The idea of a safe integer centers on how mathematical integers are represented in JavaScript. In the range (−2^{53}, 2^{53}) (excluding the lower and upper bounds), JavaScript integers are safe: there is a onetoone mapping between mathematical integers and their representations in JavaScript.
Beyond this range, JavaScript integers are unsafe: two or more mathematical integers are represented as the same JavaScript integer. For example, starting at 2^{53}, JavaScript can represent only every second mathematical integer (the previous section explains why). Therefore, a safe JavaScript integer is one that unambiguously represents a single mathematical integer.
Definitions in ECMAScript 6
ECMAScript 6 will provide the following constants:
Number
.
MAX_SAFE_INTEGER
=
Math
.
pow
(
2
,
53
)

1
;
Number
.
MIN_SAFE_INTEGER
=

Number
.
MAX_SAFE_INTEGER
;
It will also provide a function for determining whether an integer is safe:
Number
.
isSafeInteger
=
function
(
n
)
{
return
(
typeof
n
===
'number'
&&
Math
.
round
(
n
)
===
n
&&
Number
.
MIN_SAFE_INTEGER
<=
n
&&
n
<=
Number
.
MAX_SAFE_INTEGER
);
}
For a given value n
, this function first checks whether n
is a number and an integer. If both checks succeed, n
is safe if it is greater than or equal to MIN_SAFE_INTEGER
and less than or equal to MAX_SAFE_INTEGER
.
Safe results of arithmetic computations
How can we make sure that results of arithmetic computations are correct? For example, the following result is clearly not correct:
> 9007199254740990 + 3 9007199254740992
We have two safe operands, but an unsafe result:
> Number.isSafeInteger(9007199254740990) true > Number.isSafeInteger(3) true > Number.isSafeInteger(9007199254740992) false
The following result is also incorrect:
> 9007199254740995  10 9007199254740986
This time, the result is safe, but one of the operands isn’t:
> Number.isSafeInteger(9007199254740995) false > Number.isSafeInteger(10) true > Number.isSafeInteger(9007199254740986) true
Therefore, the result of applying an integer operator op
is guaranteed to be correct only if all operands and the result are safe. More formally:
isSafeInteger(a) && isSafeInteger(b) && isSafeInteger(a op b)
implies that a op b
is a correct result.
Converting to Integer
In JavaScript, all numbers are floating point. Integers are floatingpoint numbers without a fraction. Converting a number n
to an integer means finding the integer that is “closest” to n
(where the meaning of “closest” depends on how you convert). You have several options for performing this conversion:

The
Math
functionsMath.floor()
,Math.ceil()
, andMath.round()
(see Integers via Math.floor(), Math.ceil(), and Math.round()) 
The custom function
ToInteger()
(see Integers via the Custom Function ToInteger())  Binary bitwise operators (see 32bit Integers via Bitwise Operators)

The global function
parseInt()
(see Integers via parseInt())
Spoiler: #1 is usually the best choice, #2 and #3 have niche applications, and #4 is OK for parsing strings, but not for converting numbers to integers.
Integers via Math.floor(), Math.ceil(), and Math.round()
The following three functions are usually the best way of converting a number to an integer:
Math.floor()
converts its argument to the closest lower integer:> Math.floor(3.8) 3 > Math.floor(3.8) 4
Math.ceil()
converts its argument to the closest higher integer:> Math.ceil(3.2) 4 > Math.ceil(3.2) 3
Math.round()
converts its argument to the closest integer:> Math.round(3.2) 3 > Math.round(3.5) 4 > Math.round(3.8) 4
The result of rounding
3.5
may be surprising:> Math.round(3.2) 3 > Math.round(3.5) 3 > Math.round(3.8) 4
Therefore,
Math.round(x)
is the same as:Math
.
ceil
(
x
+
0.5
)
Integers via the Custom Function ToInteger()
Another good option for converting any value to an integer is the internal ECMAScript operation ToInteger()
, which removes the fraction of a floatingpoint number. If it was accessible in JavaScript, it would work like this:
> ToInteger(3.2) 3 > ToInteger(3.5) 3 > ToInteger(3.8) 3 > ToInteger(3.2) 3 > ToInteger(3.5) 3 > ToInteger(3.8) 3
The ECMAScript specification defines the result of ToInteger(number)
as:
sign(number) × floor(abs(number))
For what it does, this formula is relatively complicated because floor
seeks the closest larger integer; if you want to remove the fraction of a negative integer, you have to seek the closest smaller integer.
The following code implements the operation in JavaScript. We avoid the sign
operation by using ceil
if the number is negative:
function
ToInteger
(
x
)
{
x
=
Number
(
x
);
return
x
<
0
?
Math
.
ceil
(
x
)
:
Math
.
floor
(
x
);
}
Binary bitwise operators (see Binary Bitwise Operators) convert (at least) one of their operands to a 32bit integer that is then manipulated to produce a result that is also a 32bit integer. Therefore, if you choose the other operand appropriately, you get a fast way to convert an arbitrary number to a 32bit integer (that is either signed or unsigned).
Bitwise Or ()
If the mask, the second operand, is 0, you don’t change any bits and the result is the first operand, coerced to a signed 32bit integer. This is the canonical way to execute this kind of coercion and is used, for example, by asm.js (refer back to Is JavaScript Fast Enough?):
// Convert x to a signed 32bit integer
function
ToInt32
(
x
)
{
return
x

0
;
}
ToInt32()
removes the fraction and applies modulo 2^{32}:
> ToInt32(1.001) 1 > ToInt32(1.999) 1 > ToInt32(1) 1 > ToInt32(1) 1 > ToInt32(Math.pow(2, 32)+1) 1 > ToInt32(Math.pow(2, 32)1) 1
Shift operators
The same trick that worked for bitwise Or also works for shift operators: if you shift by zero bits, the result of a shift operation is the first operand, coerced to a 32bit integer. Here are some examples of implementing operations of the ECMAScript specification via shift operators:
// Convert x to a signed 32bit integer
function
ToInt32
(
x
)
{
return
x
<<
0
;
}
// Convert x to a signed 32bit integer
function
ToInt32
(
x
)
{
return
x
>>
0
;
}
// Convert x to an unsigned 32bit integer
function
ToUint32
(
x
)
{
return
x
>>>
0
;
}
> ToUint32(1) 4294967295 > ToUint32(Math.pow(2, 32)1) 4294967295 > ToUint32(Math.pow(2, 32)) 0
Should I use bitwise operators to coerce to integer?
You have to decide for yourself if the slight increase in efficiency is worth your code being harder to understand. Also note that bitwise operators artificially limit themselves to 32 bits, which is often neither necessary nor useful. Using one of the Math
functions, possibly in addition to Math.abs()
, is a more selfexplanatory and arguably better choice.
Integers via parseInt()
parseInt
(
str
,
radix
?
)
parses the string str
(nonstrings are coerced) as an integer. The function ignores leading whitespace and considers as many consecutive legal digits as it can find.
The radix
The range of the radix is 2 ≤ radix
≤ 36.
It determines the base of the number to be parsed. If the radix is greater than 10, letters are used as digits (caseinsensitively), in addition to 0–9.
If radix
is missing, then it is assumed to be 10, except if str
begins with “0x” or “0X,” in which case radix
is set to 16 (hexadecimal):
> parseInt('0xA') 10
If radix
is already 16, then the hexadecimal prefix is optional:
> parseInt('0xA', 16) 10 > parseInt('A', 16) 10
So far I have described the behavior of parseInt()
according to the ECMAScript specification. Additionally, some engines set the radix to 8 if str
starts with a zero:
> parseInt('010') 8 > parseInt('0109') // ignores digits ≥ 8 8
Thus, it is best to always explicitly state the radix, to always call parseInt()
with two arguments.
Here are a few examples:
> parseInt('') NaN > parseInt('zz', 36) 1295 > parseInt(' 81', 10) 81 > parseInt('12**', 10) 12 > parseInt('12.34', 10) 12 > parseInt(12.34, 10) 12
Don’t use parseInt()
to convert a number to an integer. The last example gives us hope that we might be able to use parseInt()
for converting numbers to integers. Alas, here is an example where the conversion is incorrect:
> parseInt(1000000000000000000000.5, 10) 1
Explanation
The argument is first converted to a string:
> String(1000000000000000000000.5) '1e+21'
parseInt
doesn’t consider “e” to be an integer digit and thus stops parsing after the 1. Here’s another example:
> parseInt(0.0000008, 10) 8 > String(0.0000008) '8e7'
Summary
parseInt()
shouldn’t be used to convert numbers to integers: coercion to string is an unnecessary detour and even then, the result is not always correct.
parseInt()
is useful for parsing strings, but you have to be aware that it stops at the first illegal digit. Parsing strings via Number()
(see The Function Number) is less forgiving, but may produce nonintegers.
Arithmetic Operators
The following operators are available for numbers:

number1 + number2
Numerical addition, unless either of the operands is a string. Then both operands are converted to strings and concatenated (see The Plus Operator (+)):
> 3.1 + 4.3 7.4 > 4 + ' messages' '4 messages'

number1  number2
 Subtraction.

number1 * number2
 Multiplication.

number1 / number2
 Division.

number1 % number2
Remainder:
> 9 % 7 2 > 9 % 7 2
Warning
This operation is not modulo. It returns a value whose sign is the same as the first operand (more details in a moment).

number
 Negates its argument.

+number
 Leaves its argument as is; nonnumbers are converted to a number.

++variable
,variable
Returns the current value of the variable after incrementing (or decrementing) it by 1:
> var x = 3; > ++x 4 > x 4

variable++
,variable
Increments (or decrements) the value of the variable by 1 and returns it:
> var x = 3; > x++ 3 > x 4
Mnemonic: increment (++) and decrement () operators
The position of the operand can help you remember whether it is returned before or after incrementing (or decrementing) it. If the operand comes before the increment operator, it is returned before incrementing it. If the operand comes after the operator, it is incremented and then returned. (The decrement operator works similarly.)
Bitwise Operators
JavaScript has several bitwise operators that work with 32bit integers. That is, they convert their operands to 32bit integers and produce a result that is a 32bit integer. Use cases for these operators include processing binary protocols, special algorithms, etc.
Background Knowledge
This section explains a few concepts that will help you understand bitwise operators.
Binary complements
Two common ways of computing a binary complement (or inverse) of a binary number are:
 Ones’ complement
You compute the ones’ complement
~x
of a numberx
by inverting each of the 32 digits. Let’s illustrate the ones’ complement via fourdigit numbers. The ones’ complement of1100
is0011
. Adding a number to its ones’ complement results in a number whose digits are all 1:1 + ~1 = 0001 + 1110 = 1111
 Twos’ complement
The twos’ complement
x
of a numberx
is the ones’ complement plus one. Adding a number to its twos’ complement results in0
(ignoring overflow beyond the most significant digit). Here’s an example using fourdigit numbers:1 + 1 = 0001 + 1111 = 0000
Signed 32bit integers
32bit integers don’t have an explicit sign, but you can still encode negative numbers. For example, −1 can be encoded as the twos’ complement of 1: adding 1 to the result yields 0 (within 32 bits). The boundary between positive and negative numbers is fluid; 4294967295 (2^{32}−1) and −1 are the same integer here. But you have to decide on a sign when you convert such an integer from or to a JavaScript number, which has an explicit sign as opposed to an implicit one. Therefore, signed 32bit integers are partitioned into two groups:
 Highest bit is 0: number is zero or positive.
 Highest bit is 1: number is negative.
The highest bit is often called the sign bit. Accordingly, 4294967295, interpreted as a signed 32bit integer, becomes −1 when converted to a JavaScript number:
> ToInt32(4294967295) 1
ToInt32()
is explained in 32bit Integers via Bitwise Operators.
Note
Only the unsigned right shift operator (>>>
) works with unsigned 32bit integers; all other bitwise operators work with signed 32bit integers.
Inputting and outputting binary numbers
In the following examples, we work with binary numbers via the following two operations:
parseInt(str, 2)
(see Integers via parseInt()) parses a stringstr
in binary notation (base 2). For example:> parseInt('110', 2) 6
num.toString(2)
(see Number.prototype.toString(radix?)) converts the numbernum
to a string in binary notation. For example:> 6..toString(2) '110'
Bitwise Not Operator
~number
computes the ones’ complement of number
:
> (~parseInt('11111111111111111111111111111111', 2)).toString(2) '0'
Binary Bitwise Operators
JavaScript has three binary bitwise operators:
number1 & number2
(bitwise And):> (parseInt('11001010', 2) & parseInt('1111', 2)).toString(2) '1010'
number1  number2
(bitwise Or):> (parseInt('11001010', 2)  parseInt('1111', 2)).toString(2) '11001111'
number1 ^ number2
(bitwise Xor; eXclusive Or):> (parseInt('11001010', 2) ^ parseInt('1111', 2)).toString(2) '11000101'
There are two ways to intuitively understand binary bitwise operators:
 One boolean operation per bit
In the following formulas,
n_{i}
means biti
of numbern
interpreted as a boolean (0 isfalse
, 1 istrue
). For example,2_{0}
isfalse
;2_{1}
istrue
:
And:
result_{i} = number1_{i} && number2_{i}

Or:
result_{i} = number1_{i}  number2_{i}
Xor:
result_{i} = number1_{i} ^^ number2_{i}
The operator
^^
does not exist. If it did, it would work like this (the result istrue
if exactly one of the operands istrue
):x
^^
y
===
(
x
&&
!
y
)

(
!
x
&&
y
)

And:

Changing bits of
number1
vianumber2

And: Keeps only those bits of
number1
that are set innumber2
. This operation is also called masking, withnumber2
being the mask. 
Or: Sets all bits of
number1
that are set innumber2
and keeps all other bits unchanged. 
Xor: Inverts all bits of
number1
that are set innumber2
and keeps all other bits unchanged.

And: Keeps only those bits of
Bitwise Shift Operators
JavaScript has three bitwise shift operators:
number << digitCount
(left shift):> (parseInt('1', 2) << 1).toString(2) '10'
number >> digitCount
(signed right shift):The 32bit binary number is interpreted as signed (see the preceding section). When shifting right, the sign is preserved:
> (parseInt('11111111111111111111111111111110', 2) >> 1).toString(2) '1'
We have rightshifted –2. The result, –1, is equivalent to a 32bit integer whose digits are all 1 (the twos’ complement of 1). In other words, a signed right shift by one digit divides both negative and positive integers by two.
number >>> digitCount` (unsigned right shift):
> (parseInt('11100', 2) >>> 1).toString(2) '1110'
As you can see, this operator shifts in zeros from the left.
The Function Number
The function Number
can be invoked in two ways:

Number(value)
As a normal function, it converts
value
to a primitive number (see Converting to Number):> Number('123') 123 > typeof Number(3) // no change 'number'

new Number(num)
As a constructor, it creates a new instance of
Number
(see Wrapper Objects for Primitives), an object that wrapsnum
(after converting it to a number). For example:> typeof new Number(3) 'object'
The former invocation is the common one.
Number Constructor Properties
The object Number
has the following properties:

Number.MAX_VALUE
The largest positive number that can be represented. Internally, all digits of its fraction are ones and the exponent is maximal, at 1023. If you try to increment the exponent by multiplying it by two, the result is the error value
Infinity
(see Infinity):> Number.MAX_VALUE 1.7976931348623157e+308 > Number.MAX_VALUE * 2 Infinity

Number.MIN_VALUE
The smallest representable positive number (greater than zero, a tiny fraction):
> Number.MIN_VALUE 5e324

Number.NaN

The same value as the global
NaN
. 
Number.NEGATIVE_INFINITY
> Number.NEGATIVE_INFINITY === Infinity true

Number.POSITIVE_INFINITY
> Number.POSITIVE_INFINITY === Infinity true
Number Prototype Methods
All methods of primitive numbers are stored in Number.prototype
(see Primitives Borrow Their Methods from Wrappers).
Number.prototype.toFixed(fractionDigits?)
Number.prototype.toFixed(fractionDigits?)
returns an exponentfree representation of the number, rounded to fractionDigits
digits. If the parameter is omitted, the value 0 is used:
> 0.0000003.toFixed(10) '0.0000003000' > 0.0000003.toString() '3e7'
If the number is greater than or equal to 10^{21}, then this method works the same as toString()
. You get a number in exponential notation:
> 1234567890123456789012..toFixed() '1.2345678901234568e+21' > 1234567890123456789012..toString() '1.2345678901234568e+21'
Number.prototype.toPrecision(precision?)
Number.prototype.toPrecision(precision?)
prunes the mantissa to precision
digits before using a conversion algorithm similar to toString()
. If no precision is given, toString()
is used directly:
> 1234..toPrecision(3) '1.23e+3' > 1234..toPrecision(4) '1234' > 1234..toPrecision(5) '1234.0' > 1.234.toPrecision(3) '1.23'
You need the exponential notation to display 1234 with a precision of three digits.
Number.prototype.toString(radix?)
For Number.prototype.toString(radix?)
, the parameter radix
indicates the base of the system in which the number is to be displayed. The most common radices are 10 (decimal), 2 (binary), and 16 (hexadecimal):
> 15..toString(2) '1111' > 65535..toString(16) 'ffff'
The radix must be at least 2 and at most 36. Any radix greater than 10 leads to alphabetical characters being used as digits, which explains the maximum 36, as the Latin alphabet has 26 characters:
> 1234567890..toString(36) 'kf12oi'
The global function parseInt
(see Integers via parseInt()) allows you to convert such notations back to a number:
> parseInt('kf12oi', 36) 1234567890
Decimal exponential notation
For the radix 10, toString()
uses exponential notation (with a single digit before the decimal point) in two cases. First, if there are more than 21 digits before the decimal point of a number:
> 1234567890123456789012 1.2345678901234568e+21 > 123456789012345678901 123456789012345680000
Second, if a number starts with 0.
followed by more than five zeros and a nonzero digit:
> 0.0000003 3e7 > 0.000003 0.000003
In all other cases, a fixed notation is used.
Number.prototype.toExponential(fractionDigits?)
Number.prototype.toExponential(fractionDigits?)
forces a number to be expressed in exponential notation. fractionDigits
is a number between 0 and 20 that determines how many digits should be shown after the decimal point. If it is omitted, then as many significant digits are included as necessary to uniquely specify the number.
In this example, we force more precision when toString()
would also use exponential notation. Results are mixed, because we reach the limits of the precision that can be achieved when converting binary numbers to a decimal notation:
> 1234567890123456789012..toString() '1.2345678901234568e+21' > 1234567890123456789012..toExponential(20) '1.23456789012345677414e+21'
In this example, the magnitude of the number is not large enough for an exponent being displayed by toString()
. However, toExponential()
does display an exponent:
> 1234..toString() '1234' > 1234..toExponential(5) '1.23400e+3' > 1234..toExponential() '1.234e+3'
In this example, we get exponential notation when the fraction is not small enough:
> 0.003.toString() '0.003' > 0.003.toExponential(4) '3.0000e3' > 0.003.toExponential() '3e3'
Functions for Numbers
The following functions operate on numbers:

isFinite(number)

Checks whether
number
is an actual number (neitherInfinity
norNaN
). For details, see Checking for Infinity. 
isNaN(number)

Returns
true
ifnumber
isNaN
. For details, see Pitfall: checking whether a value is NaN. 
parseFloat(str)

Turns
str
into a floatingpoint number. For details, see parseFloat(). 
parseInt(str, radix?)

Parses
str
as an integer whose base isradix
(2–36). For details, see Integers via parseInt().
Sources for This Chapter
I referred to the following sources while writing this chapter:
 “IEEE Standard 754 Floating Point Numbers” by Steve Hollasch
 “Data Types and Scaling (FixedPoint Blockset)” in the MATLAB documentation
 “IEEE floating point” on Wikipedia
^{[12] }Source: Brendan Eich, http://bit.ly/1lKzQeC.
^{[13] }Béla Varga (@netzzwerg) pointed out that IEEE 754 specifies NaN as not equal to itself.
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