Integers and Floating-Point Numbers

Integers and floating-point values are the basic building blocks of arithmetic and computation. Built-in representations of such values are called numeric primitives, while representations of integers and floating-point numbers as immediate values in code are known as numeric literals. For example, 1 is an integer literal, while 1.0 is a floating-point literal; their binary in-memory 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:
TypeSigned?Number of bitsSmallest valueLargest value
Int88-2^72^7 - 1
UInt8802^8 - 1
Int1616-2^152^15 - 1
UInt161602^16 - 1
Int3232-2^312^31 - 1
UInt323202^32 - 1
Int6464-2^632^63 - 1
UInt646402^64 - 1
Int128128-2^1272^127 - 1
UInt12812802^128 - 1
BoolN/A8false (0)true (1)
  • Floating-point types:
TypePrecisionNumber of bits

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, user-extensible type promotion system.


Literal integers are represented in the standard manner:

julia> 1

julia> 1234

The default type for an integer literal depends on whether the target system has a 32-bit architecture or a 64-bit architecture:

# 32-bit system:
julia> typeof(1)

# 64-bit system:
julia> typeof(1)

The Julia internal variable Sys.WORD_SIZE indicates whether the target system is 32-bit or 64-bit:

# 32-bit system:
julia> Sys.WORD_SIZE

# 64-bit system:
julia> Sys.WORD_SIZE

Julia also defines the types Int and UInt, which are aliases for the system's signed and unsigned native integer types respectively:

# 32-bit system:
julia> Int
julia> UInt

# 64-bit system:
julia> Int
julia> UInt

Larger integer literals that cannot be represented using only 32 bits but can be represented in 64 bits always create 64-bit integers, regardless of the system type:

# 32-bit or 64-bit system:
julia> typeof(3000000000)

Unsigned integers are input and output using the 0x prefix and hexadecimal (base 16) digits 0-9a-f (the capitalized digits A-F also work for input). The size of the unsigned value is determined by the number of hex digits used:

julia> x = 0x1

julia> typeof(x)

julia> x = 0x123

julia> typeof(x)

julia> x = 0x1234567

julia> typeof(x)

julia> x = 0x123456789abcdef

julia> typeof(x)

julia> x = 0x11112222333344445555666677778888

julia> typeof(x)

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.

Binary and octal literals are also supported:

julia> x = 0b10

julia> typeof(x)

julia> x = 0o010

julia> typeof(x)

julia> x = 0x00000000000000001111222233334444

julia> typeof(x)

As for hexadecimal literals, binary and octal literals produce unsigned integer types. The size of the binary data item is the minimal needed size, if the leading digit of the literal is not 0. In the case of leading zeros, the size is determined by the minimal needed size for a literal, which has the same length but leading digit 1. It means that:

  • 0x1 and 0x12 are UInt8 literals,
  • 0x123 and 0x1234 are UInt16 literals,
  • 0x12345 and 0x12345678 are UInt32 literals,
  • 0x123456789 and 0x1234567890adcdef are UInt64 literals, etc.

Even if there are leading zero digits which don’t contribute to the value, they count for determining storage size of a literal. So 0x01 is a UInt8 while 0x0001 is a UInt16.

That allows the user to control the size.

Values which cannot be stored in UInt128 cannot be written as such literals.

Binary, octal, and hexadecimal literals may be signed by a - immediately preceding the unsigned literal. They produce an unsigned integer of the same size as the unsigned literal would do, with the two's complement of the value:

julia> -0x2

julia> -0x0002

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))]")
   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 that have yet to be introduced, including for loops, Strings, and Interpolation, but should be easy enough to understand for users with some existing programming experience.)

Overflow behavior

In Julia, exceeding the maximum representable value of a given type results in a wraparound behavior:

julia> x = typemax(Int64)

julia> x + 1

julia> x + 1 == typemin(Int64)

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.

An example of overflow behavior and how to potentially resolve it is as follows:

julia> 10^19

julia> big(10)^19

Division errors

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.

Floating-Point Numbers

Literal floating-point numbers are represented in the standard formats, using E-notation when necessary:

julia> 1.0

julia> 1.

julia> 0.5

julia> .5

julia> -1.23

julia> 1e10

julia> 2.5e-4

The above results are all Float64 values. Literal Float32 values can be entered by writing an f in place of e:

julia> x = 0.5f0

julia> typeof(x)

julia> 2.5f-4

Values can be converted to Float32 easily:

julia> x = Float32(-1.5)

julia> typeof(x)

Hexadecimal floating-point literals are also valid, but only as Float64 values, with p preceding the base-2 exponent:

julia> 0x1p0

julia> 0x1.8p3

julia> x = 0x.4p-1

julia> typeof(x)

Half-precision floating-point numbers are also supported (Float16), but they are implemented in software and use Float32 for calculations.

julia> sizeof(Float16(4.))

julia> 2*Float16(4.)

The underscore _ can be used as digit separator:

julia> 10_000, 0.000_000_005, 0xdead_beef, 0b1011_0010
(10000, 5.0e-9, 0xdeadbeef, 0xb2)

Floating-point zero

Floating-point 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 bitstring function:

julia> 0.0 == -0.0

julia> bitstring(0.0)

julia> bitstring(-0.0)

Special floating-point values

There are three specified standard floating-point values that do not correspond to any point on the real number line:

Inf16Inf32Infpositive infinitya value greater than all finite floating-point values
-Inf16-Inf32-Infnegative infinitya value less than all finite floating-point values
NaN16NaN32NaNnot a numbera value not == to any floating-point value (including itself)

For further discussion of how these non-finite floating-point values are ordered with respect to each other and other floats, see Numeric Comparisons. By the IEEE 754 standard, these floating-point values are the results of certain arithmetic operations:

julia> 1/Inf

julia> 1/0

julia> -5/0

julia> 0.000001/0

julia> 0/0

julia> 500 + Inf

julia> 500 - Inf

julia> Inf + Inf

julia> Inf - Inf

julia> Inf * Inf

julia> Inf / Inf

julia> 0 * Inf

julia> NaN == NaN

julia> NaN != NaN

julia> NaN < NaN

julia> NaN > NaN

The typemin and typemax functions also apply to floating-point types:

julia> (typemin(Float16),typemax(Float16))
(-Inf16, Inf16)

julia> (typemin(Float32),typemax(Float32))
(-Inf32, Inf32)

julia> (typemin(Float64),typemax(Float64))
(-Inf, Inf)

Machine epsilon

Most real numbers cannot be represented exactly with floating-point numbers, and so for many purposes it is important to know the distance between two adjacent representable floating-point numbers, which is often known as machine epsilon.

Julia provides eps, which gives the distance between 1.0 and the next larger representable floating-point value:

julia> eps(Float32)

julia> eps(Float64)

julia> eps() # same as eps(Float64)

These values are 2.0^-23 and 2.0^-52 as Float32 and Float64 values, respectively. The eps function can also take a floating-point 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 floating-point value larger than x:

julia> eps(1.0)

julia> eps(1000.)

julia> eps(1e-27)

julia> eps(0.0)

The distance between two adjacent representable floating-point numbers is not constant, but is smaller for smaller values and larger for larger values. In other words, the representable floating-point 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 64-bit floating-point value.

Julia also provides the nextfloat and prevfloat functions which return the next largest or smallest representable floating-point number to the argument respectively:

julia> x = 1.25f0

julia> nextfloat(x)

julia> prevfloat(x)

julia> bitstring(prevfloat(x))

julia> bitstring(x)

julia> bitstring(nextfloat(x))

This example highlights the general principle that the adjacent representable floating-point numbers also have adjacent binary integer representations.

Rounding modes

If a number doesn't have an exact floating-point representation, it must be rounded to an appropriate representable value. However, the manner in which this rounding is done can be changed if required according to the rounding modes presented in the IEEE 754 standard.

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.

Background and References

Floating-point arithmetic entails many subtleties which can be surprising to users who are unfamiliar with the low-level implementation details. However, these subtleties are described in detail in most books on scientific computation, and also in the following references:

Arbitrary Precision Arithmetic

To allow computations with arbitrary-precision 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 the string literal @big_str or parse can be used to construct them from AbstractStrings. BigInts can also be input as integer literals when they are too big for other built-in integer types. Note that as there is no unsigned arbitrary-precision integer type in Base (BigInt is sufficient in most cases), hexadecimal, octal and binary literals can be used (in addition to decimal literals).

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

julia> big"123456789012345678901234567890" + 1

julia> parse(BigInt, "123456789012345678901234567890") + 1

julia> string(big"2"^200, base=16)

julia> 0x100000000000000000000000000000000-1 == typemax(UInt128)

julia> 0x000000000000000000000000000000000

julia> typeof(ans)

julia> big"1.23456789012345678901"

julia> parse(BigFloat, "1.23456789012345678901")

julia> BigFloat(2.0^66) / 3

julia> factorial(BigInt(40))

However, type promotion between the primitive types above and BigInt/BigFloat is not automatic and must be explicitly stated.

julia> x = typemin(Int64)

julia> x = x - 1

julia> typeof(x)

julia> y = BigInt(typemin(Int64))

julia> y = y - 1

julia> typeof(y)

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")

julia> setrounding(BigFloat, RoundDown) do
           BigFloat(1) + parse(BigFloat, "0.1")

julia> setprecision(40) do
           BigFloat(1) + parse(BigFloat, "0.1")

Numeric Literal Coefficients

To make common numeric formulae 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

julia> 2x^2 - 3x + 1

julia> 1.5x^2 - .5x + 1

It also makes writing exponential functions more elegant:

julia> 2^2x

The precedence of numeric literal coefficients is slightly lower than that of unary operators such as negation. So -2x is parsed as (-2) * x and √2x is parsed as (√2) * x. However, numeric literal coefficients parse similarly to unary operators when combined with exponentiation. For example 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(x-1)^2 - 3(x-1) + 1

The precedence of numeric literal coefficients used for implicit multiplication is higher than other binary operators such as multiplication (*), and division (/, \, and //). This means, for example, that 1 / 2im equals -0.5im and 6 // 2(2 + 1) equals 1 // 1.

Additionally, parenthesized expressions can be used as coefficients to variables, implying multiplication of the expression by the variable:

julia> (x-1)x

Neither juxtaposition of two parenthesized expressions, nor placing a variable before a parenthesized expression, however, can be used to imply multiplication:

julia> (x-1)(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 left-hand 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.

Syntax Conflicts

Juxtaposed literal coefficient syntax may conflict with some numeric literal syntaxes: hexadecimal, octal and binary integer literals and engineering notation for floating-point 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. Similar ambiguities arise with octal and binary literals like 0o777 or 0b01001010.
  • The floating-point literal expression 1e10 could be interpreted as the numeric literal 1 multiplied by the variable e10, and similarly with the equivalent E form.
  • The 32-bit floating-point literal expression 1.5f22 could be interpreted as the numeric literal 1.5 multiplied by the variable f22.

In all cases the ambiguity is resolved in favor of interpretation as numeric literals:

  • Expressions starting with 0x/0o/0b are always hexadecimal/octal/binary literals.
  • Expressions starting with a numeric literal followed by e or E are always floating-point literals.
  • Expressions starting with a numeric literal followed by f are always 32-bit floating-point literals.

Unlike E, which is equivalent to e in numeric literals for historical reasons, F is just another letter and does not behave like f in numeric literals. Hence, expressions starting with a numeric literal followed by F are interpreted as the numerical literal multiplied by a variable, which means that, for example, 1.5F22 is equal to 1.5 * F22.

Literal zero and one

Julia provides functions which return literal 0 and 1 corresponding to a specified type or the type of a given variable.

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.


julia> zero(Float32)

julia> zero(1.0)

julia> one(Int32)

julia> one(BigFloat)