All numbers in JavaScript are stored in 64bit format IEEE754 also known as “double precision”.
Let’s recap what we know about them and add a little bit more.
More ways to write a number
Imagine, we need to write a billion. The obvious way is:
let billion = 1000000000;
But in real life we usually dislike writing many zeroes. It’s easy to mistype. Also we are lazy. We we usually write something like "1bn"
for a billion or "7.3bn"
for 7 billions 300 millions. The similar is true for other big numbers.
In JavaScript, we can do almost the same by appending the letter "e"
to the number and specifying the zeroes count:
let billion = 1e9; // 1 billion, literally: 1 and 9 zeroes
alert( 7.3e9 ); // 7.3 billions (7,300,000,000)
In other words, "e"
multiplies the number by 1
with the given zeroes count.
1e3 = 1 * 1000
1.23e6 = 1.23 * 1000000
Now let’s write something very small. Say, 1 microsecond (one millionth of a second):
let ms = 0.000001;
Also the same "e"
can help. If we’d like not to write down the zeroes explicitly, the same number is:
let ms = 1e6; // six zeroes to the left from 1
If we count the zeroes in 0.000001
, there are 6 of them. So naturally it’s 1e6
.
In other words, a negative number after "e"
means a division by 1 with the given number of zeries:
// 3 divides by 1 with 3 zeroes
1e3 = 1 / 1000 (=0.001)
// 6 divides by 1 with 6 zeroes
1.23e6 = 1.23 / 1000000 (=0.00000123)
Hex, binary and octal numbers
Hexadecimal numbers are widely used in JavaScript: to represent colors, encode characters and for many other things. So there exists a short way to write them: 0x
and then the number.
For instance:
alert( 0xff ); // 255
alert( 0xFF ); // 255 (the same, case doesn't matter)
Binary and octal numeral systems are rarely used, but also supported using 0b
and 0o
prefixes:
let a = 0b11111111; // binary form of 255
let b = 0o377; // octal form of 255
alert( a == b ); // true, the same number 255 at both sides
There are only 3 numeral systems with such support. For other numeral systems we should use function parseInt
(later in this chapter).
toString(base)
The method num.toString(base)
returns a string representation of num
in the numeral system with the given base
.
For example:
let num = 255;
alert( num.toString(16) ); // ff
alert( num.toString(2) ); // 11111111
The base
can vary from 2
to 36
. By default it’s 10
.
Most often use cases are:

base=16 is used for hex colors, character encodings etc, digits can be
0..9
orA..F
. 
base=2 is mostly for debugging bitwise operations, digits can be
0
or1
. 
base=36 is the maximum, digits can be
0..9
orA..Z
. The whole latin alphabet is used to represent a number. A funny, but useful case for36
is when we need to turn a long numeric identifier into something shorter, for example to make a short url. Can simply represent it in the numeral system with base36
:alert( 123456..toString(36) ); // 2n9c
Please note that two dots in 123456..toString(36)
is not a typo. If we want to call a method directly on a number, like toString
in the example above, then we need to place two dots ..
after it.
If we placed a single dot: 123456.toString(36)
, then there would be an error, because JavaScript syntax implies the decimal part after the first dot. And if we place one more dot, then JavaScript knows that the decimal part is empty and now goes the method.
Also could write (123456).toString(36)
.
Rounding
One of most often operations with numbers is the rounding.
There are following builtin functions for rounding:
Math.floor
 Rounds down:
3.1
becomes3
, and1.1
becomes2
. Math.ceil
 Rounds up:
3.1
becomes4
, and1.1
becomes1
. Math.round
 Rounds to the nearest integer:
3.1
becomes3
,3.6
becomes4
and1.1
becomes1
. Math.trunc
(not supported by Internet Explorer) Removes the decimal part:
3.1
becomes3
,1.1
becomes1
.
Here’s the table to summarize the differences between them:
Math.floor 
Math.ceil 
Math.round 
Math.trunc 


3.1 
3 
4 
3 
3 
3.6 
3 
4 
4 
3 
1.1 
2 
1 
1 
1 
1.6 
2 
1 
2 
1 
These functions cover all possible ways to deal with the decimal part as a whole. But what if we’d like to round the number to nth
digit after the point?
For instance, we have 1.2345
and want to round it to 2 digits, getting only 1.23
.
There are two ways to do so.

Multiplyanddivide.
For instance, to round the number to the 2nd digit after the point, we can multiply the number by
100
, call the rounding function and then divide back.let num = 1.23456; alert( Math.floor(num * 100) / 100 ); // 1.23456 > 123.456 > 123 > 1.23

The method toFixed(n) rounds the number to
n
digits after the point and returns a string representation of the result.let num = 12.34; alert( num.toFixed(1) ); // "12.3"
The rounding goes to the nearest value, similar to
Math.round
:let num = 12.36; alert( num.toFixed(1) ); // "12.4"
Please note that result of
toFixed
is a string. If the decimal part is shorter than required, zeroes are appended to its end:let num = 12.34; alert( num.toFixed(5) ); // "12.34000", added zeroes to make exactly 5 digits
We can convert it to a number using the unary plus or a
Number()
call:+num.toFixed(5)
.
Imprecise calculations
Internally, a number is represented in 64bit format IEEE754. So, there are exactly 64 bits to store a number: 52 of them are used to store the digits, 11 of them store the position of the decimal point (they are zero for integer numbers) and 1 bit for the sign.
If a number is too big, it would overflow the 64bit storage, potentially giving an infinity:
alert( 1e500 ); // Infinity
But what may be a little bit more obvious, but happens much often is the loss of precision.
Consider this (falsy!) test:
alert( 0.1 + 0.2 == 0.3 ); // false
Yes, indeed, if we check whether the sum of 0.1
and 0.2
is 0.3
, we get false
.
Strange! What is it then if not 0.3
?
alert( 0.1 + 0.2 ); // 0.30000000000000004
Ouch! There are more consequences than an incorrect comparison here. Imagine you’re making an eshopping site and the visitor puts $0.10
and $0.20
goods into his chart. The order total will be $0.30000000000000004
. That would surprise anyone.
Why does it work like that?
A number is stored in memory in it’s binary form, as a sequence of ones and zeroes. But fractions like 0.1
, 0.2
that look simple in the decimal numeric system are actually unending fractions in their binary form.
In other words, what is 0.1
? It is one divided by ten 1/10
, onetenth. In decimal numeral system such numbers are easily representable. Compare it to onethird: 1/3
. It becomes an endless fraction 0.33333(3)
.
So, division by powers 10
is guaranteed to look well in the decimal system, but the division by 3
is not. For the same reason, in the binary numeral system, the division by powers of 2
is guaranteed to look good, but 1/10
becomes an endless binary fraction.
There’s just no way to store exactly 0.1 or exactly 0.2 in the binary system, just like there is no way to store onethird as a decimal fraction.
The numeric format IEEE754 solves that by storing the nearest possible number. There are rounding rules that normally don’t allow us to see that “tiny precision loss”, so the number shows up as 0.3
. But the loss still exists.
We can see it like this:
alert( 0.1.toFixed(20) ); // 0.10000000000000000555
And when we sum two numbers, then their “precision losses” sum up.
That’s why 0.1 + 0.2
is not exactly 0.3
.
The same issue exists in many other programming languages.
PHP, Java, C, Perl, Ruby give exactly the same result, because they are based on the same numeric format.
Can we work around the problem? Sure, there’s a number of ways:

We can round the result with the help of a method toFixed(n):
let sum = 0.1 + 0.2; alert( sum.toFixed(2) ); // 0.30
Please note that
toFixed
always returns a string. It ensures that it has 2 digits after the decimal point. That’s actually convenient if we have an eshopping and need to show$0.30
. For other cases we can use the unary plus to coerce it into a number:let sum = 0.1 + 0.2; alert( +sum.toFixed(2) ); // 0.3

We can temporarily turn numbers into integers for the maths and then go back. That would looks like this:
alert( (0.1*10 + 0.2*10) / 10 ); // 0.3
It works, because when we get
0.1*10 = 1
and0.2 * 10 = 2
then both numbers are integers, there’s no precision loss for them. 
If it’s a shop, then the most radical solution would be to store all prices in cents. No fractions at all. But what if we apply a discount of 30%? In practice, totally evading fractions is rarely feasible, so the solutions listed above are here to help.
Try running this:
// Hello! I'm a selfincreasing number!
alert( 9999999999999999 ); // shows 10000000000000000
The reason is the same: loss of precision. There are 64 bits for the number, 52 of them can be used to store digits, and that’s not enough. So the least significant digits disappear.
JavaScript doesn’t trigger an error in such case. It does the best to fit the number into the format. Unfortunately, the format is not big enough.
Another funny consequence of the internal representation is the existance of two zeroes: 0
and 0
.
That’s because a sign is represented by a single bit, so every number can be positive or negative, including the zero.
In most cases the distinction is unnoticeable, because operators are suited to treat them as the same.
Tests: isFinite and isNaN
Remember the two special numeric values?
Infinite
(andInfinite
) is a special numeric value that is greater (less) than anything.NaN
represends an error.
They belong to the type number
, but are not “normal” numbers, so there are special functions to check for them:

isNaN(value)
converts its argument to a number and then tests if for beingNaN
:alert( isNaN(NaN) ); // true alert( isNaN("str") ); // true
But do we need the function? Can we just use the comparison
=== NaN
? Sorry, but no. The valueNaN
is unique in that it does not equal anything including itself:alert( NaN === NaN ); // false

isFinite(value)
converts its argument to a number and returnstrue
if it’s a regular number, notNaN/Infinity/Infinity
:alert( isFinite("15") ); // true alert( isFinite("str") ); // false, because a special value: NaN alert( isFinite(Infinity) ); // false, because a special value: Infinity
Sometimes isFinite
is used to validate the string value for being a regular number:
let num = +prompt("Enter a number", '');
// will be true unless you enter Infinity, Infinity or not a number
alert( isFinite(num) );
Please note that an empty or a spaceonly string is treated as 0
in all numeric functions including isFinite
.
Object.is
There is a special builtin method Object.is that compares values like ===
, but is more reliable for two edge cases:
 It works with
NaN
:Object.is(NaN, NaN) === true
, that’s a good thing.  Values
0
and0
are different:Object.is(0, 0) === false
, it rarely matters, but these values technically are different.
In all other cases, Object.is(a, b)
is the same as a === b
.
This way of comparison is often used in JavaScript specification. When an internal algorithm needs to compare two values for being exactly the same, it uses Object.is
(internally called SameValue).
parseInt and parseFloat
The numeric conversion using a plus +
or Number()
is strict. If a value is not exactly a number, it fails:
alert( +"100px" ); // NaN
The sole exception is spaces before and after the line, they are ignored.
But in real life we often have values in units, like "100px"
or "12pt"
in CSS. Also in many countries the currency symbol goes after the amount, so we have "19€"
and would like to extract a numeric value out of that.
That’s what parseInt
and parseFloat
are for.
They “read” a number from a string until they can. In case of an error, the gathered number is returned. Function parseInt
reads an integer number, parseFloat
reads any number:
alert( parseInt('100px') ); // 100
alert( parseFloat('12.5em') ); // 12.5
alert( parseInt('12.3') ); // 12, only integer part
alert( parseFloat('12.3.4') ); // 12.3, the second point stops the reading
Of course, there are situations when parseInt/parseFloat
return NaN
. It happens when no digits could be read:
alert( parseInt('a123') ); // NaN, the first symbol stops he process
parseInt(str, radix)
The parseInt()
function has an optional second parameter. It specifies the base of the numeral system, so parseInt
can also parse strings of hex numbers, binary numbers and so on:
alert( parseInt('0xff', 16) ); // 255
alert( parseInt('ff', 16) ); // 255, without 0x also works
alert( parseInt('2n9c', 36) ); // 123456
Other math functions
JavaScript has a builtin Math object which contains a small library of mathematical functions and constants.
A few examples:
Math.random()

Returns a random number from 0 to 1 (not including 1)
alert( Math.random() ); // 0.1234567894322 alert( Math.random() ); // 0.5435252343232 alert( Math.random() ); // ... (any random numbers)
Math.max(a, b, c...)
/Math.min(a, b, c...)

Return the greatest/smallest from the arbitrary number of arguments.
alert( Math.max(3, 5, 10, 0, 1) ); // 5 alert( Math.min(1, 2 ); // 1
Math.pow(n, power)

Returns
n
raised the given poweralert( Math.pow(2, 10) ); // 2 in power 10 = 1024
There are more functions and constants in Math
, including trigonometry, you can find them in the docs for the Math object.
Summary
To write big numbers:
 Append
"e"
with the zeroes count to the number. Like:123e6
is123
with 6 zeroes.  A negative number after
"e"
causes the number to be divided by 1 with given zeroes. That’s for onemillionth or such.
For different numeral systems:
 Can write numbers directly in hex (
0x
), octal (0o
) and binary (0b
) systems parseInt(str, base)
parses an integer from any numeral system with base:2 ≤ base ≤ 36
.num.toString(base)
converts a number to a string in the numeral system with the givenbase
.
For converting values like 12pt
and 100px
to a number:
 Use
parseInt/parseFloat
for the “soft” conversion, which reads a number from a string until it can.
For fractions:
 Round using
Math.floor
,Math.ceil
,Math.trunc
,Math.round
ornum.toFixed(precision)
.  Remember about the loss of precision when working with fractions.
More mathematical functions:
 See the Math object when you need them. The library is very small, but can cover basic needs.
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