Integers and floatingpoint values are the basic building blocks of arithmetic and computation. Builtin representations of such values are called numeric primitives, while representations of integers and floatingpoint numbers as immediate values in code are known as numeric literals. For example, 1
is an integer literal, while 1.0
is a floatingpoint literal; their binary inmemory representations as objects are numeric primitives.
Julia provides a broad range of primitive numeric types, and a full complement of arithmetic and bitwise operators as well as standard mathematical functions are defined over them. These map directly onto numeric types and operations that are natively supported on modern computers, thus allowing Julia to take full advantage of computational resources. Additionally, Julia provides software support for Arbitrary Precision Arithmetic, which can handle operations on numeric values that cannot be represented effectively in native hardware representations, but at the cost of relatively slower performance.
The following are Julia's primitive numeric types:
Integer types:
Type  Signed?  Number of bits  Smallest value  Largest value 

Int8 
✓  8  2^7  2^7  1 
UInt8 
8  0  2^8  1  
Int16 
✓  16  2^15  2^15  1 
UInt16 
16  0  2^16  1  
Int32 
✓  32  2^31  2^31  1 
UInt32 
32  0  2^32  1  
Int64 
✓  64  2^63  2^63  1 
UInt64 
64  0  2^64  1  
Int128 
✓  128  2^127  2^127  1 
UInt128 
128  0  2^128  1  
Bool 
N/A  8 
false (0) 
true (1) 
Floatingpoint types:
Type  Precision  Number of bits 

Float16 
half  16 
Float32 
single  32 
Float64 
double  64 
Additionally, full support for Complex and Rational Numbers is built on top of these primitive numeric types. All numeric types interoperate naturally without explicit casting, thanks to a flexible, userextensible type promotion system.
Literal integers are represented in the standard manner:
julia> 1 1 julia> 1234 1234
The default type for an integer literal depends on whether the target system has a 32bit architecture or a 64bit architecture:
# 32bit system: julia> typeof(1) Int32 # 64bit system: julia> typeof(1) Int64
The Julia internal variable Sys.WORD_SIZE
indicates whether the target system is 32bit or 64bit:
# 32bit system: julia> Sys.WORD_SIZE 32 # 64bit system: julia> Sys.WORD_SIZE 64
Julia also defines the types Int
and UInt
, which are aliases for the system's signed and unsigned native integer types respectively:
# 32bit system: julia> Int Int32 julia> UInt UInt32 # 64bit system: julia> Int Int64 julia> UInt UInt64
Larger integer literals that cannot be represented using only 32 bits but can be represented in 64 bits always create 64bit integers, regardless of the system type:
# 32bit or 64bit system: julia> typeof(3000000000) Int64
Unsigned integers are input and output using the 0x
prefix and hexadecimal (base 16) digits 09af
(the capitalized digits AF
also work for input). The size of the unsigned value is determined by the number of hex digits used:
julia> 0x1 0x01 julia> typeof(ans) UInt8 julia> 0x123 0x0123 julia> typeof(ans) UInt16 julia> 0x1234567 0x01234567 julia> typeof(ans) UInt32 julia> 0x123456789abcdef 0x0123456789abcdef julia> typeof(ans) UInt64
This behavior is based on the observation that when one uses unsigned hex literals for integer values, one typically is using them to represent a fixed numeric byte sequence, rather than just an integer value.
Recall that the variable ans
is set to the value of the last expression evaluated in an interactive session. This does not occur when Julia code is run in other ways.
Binary and octal literals are also supported:
julia> 0b10 0x02 julia> typeof(ans) UInt8 julia> 0o10 0x08 julia> typeof(ans) UInt8
The minimum and maximum representable values of primitive numeric types such as integers are given by the typemin()
and typemax()
functions:
julia> (typemin(Int32), typemax(Int32)) (2147483648, 2147483647) julia> for T in [Int8,Int16,Int32,Int64,Int128,UInt8,UInt16,UInt32,UInt64,UInt128] println("$(lpad(T,7)): [$(typemin(T)),$(typemax(T))]") end Int8: [128,127] Int16: [32768,32767] Int32: [2147483648,2147483647] Int64: [9223372036854775808,9223372036854775807] Int128: [170141183460469231731687303715884105728,170141183460469231731687303715884105727] UInt8: [0,255] UInt16: [0,65535] UInt32: [0,4294967295] UInt64: [0,18446744073709551615] UInt128: [0,340282366920938463463374607431768211455]
The values returned by typemin()
and typemax()
are always of the given argument type. (The above expression uses several features we have yet to introduce, including for loops, Strings, and Interpolation, but should be easy enough to understand for users with some existing programming experience.)
In Julia, exceeding the maximum representable value of a given type results in a wraparound behavior:
julia> x = typemax(Int64) 9223372036854775807 julia> x + 1 9223372036854775808 julia> x + 1 == typemin(Int64) true
Thus, arithmetic with Julia integers is actually a form of modular arithmetic. This reflects the characteristics of the underlying arithmetic of integers as implemented on modern computers. In applications where overflow is possible, explicit checking for wraparound produced by overflow is essential; otherwise, the BigInt
type in Arbitrary Precision Arithmetic is recommended instead.
Integer division (the div
function) has two exceptional cases: dividing by zero, and dividing the lowest negative number (typemin()
) by 1. Both of these cases throw a DivideError
. The remainder and modulus functions (rem
and mod
) throw a DivideError
when their second argument is zero.
Literal floatingpoint numbers are represented in the standard formats:
julia> 1.0 1.0 julia> 1. 1.0 julia> 0.5 0.5 julia> .5 0.5 julia> 1.23 1.23 julia> 1e10 1.0e10 julia> 2.5e4 0.00025
The above results are all Float64
values. Literal Float32
values can be entered by writing an f
in place of e
:
julia> 0.5f0 0.5f0 julia> typeof(ans) Float32 julia> 2.5f4 0.00025f0
Values can be converted to Float32
easily:
julia> Float32(1.5) 1.5f0 julia> typeof(ans) Float32
Hexadecimal floatingpoint literals are also valid, but only as Float64
values:
julia> 0x1p0 1.0 julia> 0x1.8p3 12.0 julia> 0x.4p1 0.125 julia> typeof(ans) Float64
Halfprecision floatingpoint numbers are also supported (Float16
), but they are implemented in software and use Float32
for calculations.
julia> sizeof(Float16(4.)) 2 julia> 2*Float16(4.) Float16(8.0)
The underscore _
can be used as digit separator:
julia> 10_000, 0.000_000_005, 0xdead_beef, 0b1011_0010 (10000, 5.0e9, 0xdeadbeef, 0xb2)
Floatingpoint numbers have two zeros, positive zero and negative zero. They are equal to each other but have different binary representations, as can be seen using the bits
function: :
julia> 0.0 == 0.0 true julia> bits(0.0) "0000000000000000000000000000000000000000000000000000000000000000" julia> bits(0.0) "1000000000000000000000000000000000000000000000000000000000000000"
There are three specified standard floatingpoint values that do not correspond to any point on the real number line:
Float16 
Float32 
Float64 
Name  Description 

Inf16 
Inf32 
Inf 
positive infinity  a value greater than all finite floatingpoint values 
Inf16 
Inf32 
Inf 
negative infinity  a value less than all finite floatingpoint values 
NaN16 
NaN32 
NaN 
not a number  a value not == to any floatingpoint value (including itself) 
For further discussion of how these nonfinite floatingpoint values are ordered with respect to each other and other floats, see Numeric Comparisons. By the IEEE 754 standard, these floatingpoint values are the results of certain arithmetic operations:
julia> 1/Inf 0.0 julia> 1/0 Inf julia> 5/0 Inf julia> 0.000001/0 Inf julia> 0/0 NaN julia> 500 + Inf Inf julia> 500  Inf Inf julia> Inf + Inf Inf julia> Inf  Inf NaN julia> Inf * Inf Inf julia> Inf / Inf NaN julia> 0 * Inf NaN
The typemin()
and typemax()
functions also apply to floatingpoint types:
julia> (typemin(Float16),typemax(Float16)) (Inf16, Inf16) julia> (typemin(Float32),typemax(Float32)) (Inf32, Inf32) julia> (typemin(Float64),typemax(Float64)) (Inf, Inf)
Most real numbers cannot be represented exactly with floatingpoint numbers, and so for many purposes it is important to know the distance between two adjacent representable floatingpoint numbers, which is often known as machine epsilon.
Julia provides eps()
, which gives the distance between 1.0
and the next larger representable floatingpoint value:
julia> eps(Float32) 1.1920929f7 julia> eps(Float64) 2.220446049250313e16 julia> eps() # same as eps(Float64) 2.220446049250313e16
These values are 2.0^23
and 2.0^52
as Float32
and Float64
values, respectively. The eps()
function can also take a floatingpoint value as an argument, and gives the absolute difference between that value and the next representable floating point value. That is, eps(x)
yields a value of the same type as x
such that x + eps(x)
is the next representable floatingpoint value larger than x
:
julia> eps(1.0) 2.220446049250313e16 julia> eps(1000.) 1.1368683772161603e13 julia> eps(1e27) 1.793662034335766e43 julia> eps(0.0) 5.0e324
The distance between two adjacent representable floatingpoint numbers is not constant, but is smaller for smaller values and larger for larger values. In other words, the representable floatingpoint numbers are densest in the real number line near zero, and grow sparser exponentially as one moves farther away from zero. By definition, eps(1.0)
is the same as eps(Float64)
since 1.0
is a 64bit floatingpoint value.
Julia also provides the nextfloat()
and prevfloat()
functions which return the next largest or smallest representable floatingpoint number to the argument respectively:
julia> x = 1.25f0 1.25f0 julia> nextfloat(x) 1.2500001f0 julia> prevfloat(x) 1.2499999f0 julia> bits(prevfloat(x)) "00111111100111111111111111111111" julia> bits(x) "00111111101000000000000000000000" julia> bits(nextfloat(x)) "00111111101000000000000000000001"
This example highlights the general principle that the adjacent representable floatingpoint numbers also have adjacent binary integer representations.
If a number doesn't have an exact floatingpoint representation, it must be rounded to an appropriate representable value, however, if wanted, the manner in which this rounding is done can be changed according to the rounding modes presented in the IEEE 754 standard.
julia> x = 1.1; y = 0.1; julia> x + y 1.2000000000000002 julia> setrounding(Float64,RoundDown) do x + y end 1.2
The default mode used is always RoundNearest
, which rounds to the nearest representable value, with ties rounded towards the nearest value with an even least significant bit.
Floatingpoint arithmetic entails many subtleties which can be surprising to users who are unfamiliar with the lowlevel implementation details. However, these subtleties are described in detail in most books on scientific computation, and also in the following references:
The definitive guide to floating point arithmetic is the IEEE 7542008 Standard; however, it is not available for free online.
For a brief but lucid presentation of how floatingpoint numbers are represented, see John D. Cook's article on the subject as well as his introduction to some of the issues arising from how this representation differs in behavior from the idealized abstraction of real numbers.
Also recommended is Bruce Dawson's series of blog posts on floatingpoint numbers.
For an excellent, indepth discussion of floatingpoint numbers and issues of numerical accuracy encountered when computing with them, see David Goldberg's paper What Every Computer Scientist Should Know About FloatingPoint Arithmetic.
For even more extensive documentation of the history of, rationale for, and issues with floatingpoint numbers, as well as discussion of many other topics in numerical computing, see the collected writings of William Kahan, commonly known as the "Father of FloatingPoint". Of particular interest may be An Interview with the Old Man of FloatingPoint.
To allow computations with arbitraryprecision integers and floating point numbers, Julia wraps the GNU Multiple Precision Arithmetic Library (GMP) and the GNU MPFR Library, respectively. The BigInt
and BigFloat
types are available in Julia for arbitrary precision integer and floating point numbers respectively.
Constructors exist to create these types from primitive numerical types, and parse()
can be used to construct them from AbstractString
s. Once created, they participate in arithmetic with all other numeric types thanks to Julia's type promotion and conversion mechanism:
julia> BigInt(typemax(Int64)) + 1 9223372036854775808 julia> parse(BigInt, "123456789012345678901234567890") + 1 123456789012345678901234567891 julia> parse(BigFloat, "1.23456789012345678901") 1.234567890123456789010000000000000000000000000000000000000000000000000000000004 julia> BigFloat(2.0^66) / 3 2.459565876494606882133333333333333333333333333333333333333333333333333333333344e+19 julia> factorial(BigInt(40)) 815915283247897734345611269596115894272000000000
However, type promotion between the primitive types above and BigInt
/BigFloat
is not automatic and must be explicitly stated.
julia> x = typemin(Int64) 9223372036854775808 julia> x = x  1 9223372036854775807 julia> typeof(x) Int64 julia> y = BigInt(typemin(Int64)) 9223372036854775808 julia> y = y  1 9223372036854775809 julia> typeof(y) BigInt
The default precision (in number of bits of the significand) and rounding mode of BigFloat
operations can be changed globally by calling setprecision()
and setrounding()
, and all further calculations will take these changes in account. Alternatively, the precision or the rounding can be changed only within the execution of a particular block of code by using the same functions with a do
block:
julia> setrounding(BigFloat, RoundUp) do BigFloat(1) + parse(BigFloat, "0.1") end 1.100000000000000000000000000000000000000000000000000000000000000000000000000003 julia> setrounding(BigFloat, RoundDown) do BigFloat(1) + parse(BigFloat, "0.1") end 1.099999999999999999999999999999999999999999999999999999999999999999999999999986 julia> setprecision(40) do BigFloat(1) + parse(BigFloat, "0.1") end 1.1000000000004
To make common numeric formulas and expressions clearer, Julia allows variables to be immediately preceded by a numeric literal, implying multiplication. This makes writing polynomial expressions much cleaner:
julia> x = 3 3 julia> 2x^2  3x + 1 10 julia> 1.5x^2  .5x + 1 13.0
It also makes writing exponential functions more elegant:
julia> 2^2x 64
The precedence of numeric literal coefficients is the same as that of unary operators such as negation. So 2^3x
is parsed as 2^(3x)
, and 2x^3
is parsed as 2*(x^3)
.
Numeric literals also work as coefficients to parenthesized expressions:
julia> 2(x1)^2  3(x1) + 1 3
Additionally, parenthesized expressions can be used as coefficients to variables, implying multiplication of the expression by the variable:
julia> (x1)x 6
Neither juxtaposition of two parenthesized expressions, nor placing a variable before a parenthesized expression, however, can be used to imply multiplication:
julia> (x1)(x+1) ERROR: MethodError: objects of type Int64 are not callable julia> x(x+1) ERROR: MethodError: objects of type Int64 are not callable
Both expressions are interpreted as function application: any expression that is not a numeric literal, when immediately followed by a parenthetical, is interpreted as a function applied to the values in parentheses (see Functions for more about functions). Thus, in both of these cases, an error occurs since the lefthand value is not a function.
The above syntactic enhancements significantly reduce the visual noise incurred when writing common mathematical formulae. Note that no whitespace may come between a numeric literal coefficient and the identifier or parenthesized expression which it multiplies.
Juxtaposed literal coefficient syntax may conflict with two numeric literal syntaxes: hexadecimal integer literals and engineering notation for floatingpoint literals. Here are some situations where syntactic conflicts arise:
The hexadecimal integer literal expression 0xff
could be interpreted as the numeric literal 0
multiplied by the variable xff
.
The floatingpoint literal expression 1e10
could be interpreted as the numeric literal 1
multiplied by the variable e10
, and similarly with the equivalent E
form.
In both cases, we resolve the ambiguity in favor of interpretation as a numeric literals:
Expressions starting with 0x
are always hexadecimal literals.
Expressions starting with a numeric literal followed by e
or E
are always floatingpoint literals.
Julia provides functions which return literal 0 and 1 corresponding to a specified type or the type of a given variable.
Function  Description 

zero(x) 
Literal zero of type x or type of variable x

one(x) 
Literal one of type x or type of variable x

These functions are useful in Numeric Comparisons to avoid overhead from unnecessary type conversion.
Examples:
julia> zero(Float32) 0.0f0 julia> zero(1.0) 0.0 julia> one(Int32) 1 julia> one(BigFloat) 1.000000000000000000000000000000000000000000000000000000000000000000000000000000
© 2009–2016 Jeff Bezanson, Stefan Karpinski, Viral B. Shah, and other contributors
Licensed under the MIT License.
https://docs.julialang.org/en/release0.6/manual/integersandfloatingpointnumbers/