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:
|Type||Signed?||Number of bits||Smallest value||Largest value|
|✓||8||-2^7||2^7 - 1|
|8||0||2^8 - 1|
|✓||16||-2^15||2^15 - 1|
|16||0||2^16 - 1|
|✓||32||-2^31||2^31 - 1|
|32||0||2^32 - 1|
|✓||64||-2^63||2^63 - 1|
|64||0||2^64 - 1|
|✓||128||-2^127||2^127 - 1|
|128||0||2^128 - 1|
- Floating-point types:
|Type||Precision||Number 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 1 julia> 1234 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) Int32 # 64-bit system: julia> typeof(1) Int64
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 32 # 64-bit system: julia> Sys.WORD_SIZE 64
Julia also defines the types
UInt, which are aliases for the system's signed and unsigned native integer types respectively:
# 32-bit system: julia> Int Int32 julia> UInt UInt32 # 64-bit 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 64-bit integers, regardless of the system type:
# 32-bit or 64-bit system: julia> typeof(3000000000) Int64
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> 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 julia> 0x11112222333344445555666677778888 0x11112222333344445555666677778888 julia> typeof(ans) UInt128
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> 0o010 0x08 julia> typeof(ans) UInt8 julia> 0x00000000000000001111222233334444 0x00000000000000001111222233334444 julia> typeof(ans) UInt128
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. 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 0xfe julia> -0x0002 0xfffe
The minimum and maximum representable values of primitive numeric types such as integers are given by the
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
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 (
mod) throw a
DivideError when their second argument is zero.
Literal floating-point numbers are represented in the standard formats, using E-notation when necessary:
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> 2.5e-4 0.00025
The above results are all
Float64 values. Literal
Float32 values can be entered by writing an
f in place of
julia> 0.5f0 0.5f0 julia> typeof(ans) Float32 julia> 2.5f-4 0.00025f0
Values can be converted to
julia> Float32(-1.5) -1.5f0 julia> typeof(ans) Float32
Hexadecimal floating-point literals are also valid, but only as
Float64 values, with
p preceding the base-2 exponent:
julia> 0x1p0 1.0 julia> 0x1.8p3 12.0 julia> 0x.4p-1 0.125 julia> typeof(ans) Float64
Half-precision floating-point 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)
_ 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 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
julia> 0.0 == -0.0 true julia> bitstring(0.0) "0000000000000000000000000000000000000000000000000000000000000000" julia> bitstring(-0.0) "1000000000000000000000000000000000000000000000000000000000000000"
Special floating-point values
There are three specified standard floating-point values that do not correspond to any point on the real number line:
|positive infinity||a value greater than all finite floating-point values|
|negative infinity||a value less than all finite floating-point values|
|not a number||a value not |
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 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
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)
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.
eps, which gives the distance between
1.0 and the next larger representable floating-point value:
julia> eps(Float32) 1.1920929f-7 julia> eps(Float64) 2.220446049250313e-16 julia> eps() # same as eps(Float64) 2.220446049250313e-16
These values are
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
julia> eps(1.0) 2.220446049250313e-16 julia> eps(1000.) 1.1368683772161603e-13 julia> eps(1e-27) 1.793662034335766e-43 julia> eps(0.0) 5.0e-324
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
1.0 is a 64-bit floating-point value.
Julia also provides the
prevfloat functions which return the next largest or smallest representable floating-point number to the argument respectively:
julia> x = 1.25f0 1.25f0 julia> nextfloat(x) 1.2500001f0 julia> prevfloat(x) 1.2499999f0 julia> bitstring(prevfloat(x)) "00111111100111111111111111111111" julia> bitstring(x) "00111111101000000000000000000000" julia> bitstring(nextfloat(x)) "00111111101000000000000000000001"
This example highlights the general principle that the adjacent representable floating-point numbers also have adjacent binary integer representations.
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:
- The definitive guide to floating point arithmetic is the IEEE 754-2008 Standard; however, it is not available for free online.
- For a brief but lucid presentation of how floating-point 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 floating-point numbers.
- For an excellent, in-depth discussion of floating-point numbers and issues of numerical accuracy encountered when computing with them, see David Goldberg's paper What Every Computer Scientist Should Know About Floating-Point Arithmetic.
- For even more extensive documentation of the history of, rationale for, and issues with floating-point 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 Floating-Point". Of particular interest may be An Interview with the Old Man of Floating-Point.
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
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
AbstractStrings. 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
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
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
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
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 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 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
2x^3 is parsed as
Numeric literals also work as coefficients to parenthesized expressions:
julia> 2(x-1)^2 - 3(x-1) + 1 3
The precedence of numeric literal coefficients used for implicit multiplication is higher than other binary operators such as multiplication (
*), and division (
//). This means, for example, that
1 / 2im equals
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 6
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.
Juxtaposed literal coefficient syntax may conflict with two numeric literal syntaxes: hexadecimal integer literals and engineering notation for floating-point literals. Here are some situations where syntactic conflicts arise:
- The hexadecimal integer literal expression
0xffcould be interpreted as the numeric literal
0multiplied by the variable
- The floating-point literal expression
1e10could be interpreted as the numeric literal
1multiplied by the variable
e10, and similarly with the equivalent
- The 32-bit floating-point literal expression
1.5f22could be interpreted as the numeric literal
1.5multiplied by the variable
In all cases, we resolve the ambiguity in favor of interpretation as numeric literals:
- Expressions starting with
0xare always hexadecimal literals.
- Expressions starting with a numeric literal followed by
Eare always floating-point literals.
- Expressions starting with a numeric literal followed by
fare always 32-bit floating-point literals.
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.
|Literal zero of type |
|Literal one of type |
These functions are useful in Numeric Comparisons to avoid overhead from unnecessary type conversion.
julia> zero(Float32) 0.0f0 julia> zero(1.0) 0.0 julia> one(Int32) 1 julia> one(BigFloat) 1.0