Integers and Floating-Point Numbers

# 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
--- | --- | --- | --- | ---
`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)

- Floating-point 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, user-extensible type promotion system.

## Integers

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 `Int` and `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

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.)

### Overflow behavior

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.

### 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:

    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 `e`:

    julia> 0.5f0
    0.5f0

    julia> typeof(ans)
    Float32

    julia> 2.5f-4
    0.00025f0

Values can be converted to `Float32` easily:

    julia> Float32(-1.5)
    -1.5f0

    julia> typeof(ans)
    Float32

Hexadecimal floating-point literals are also valid, but only as `Float64` values:

    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)

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 `bits` function: :

    julia> 0.0 == -0.0
    true

    julia> bits(0.0)
    "0000000000000000000000000000000000000000000000000000000000000000"

    julia> bits(-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:

`Float16` | `Float32` | `Float64` | Name | Description
--- | --- | --- | --- | ---
`Inf16` | `Inf32` | `Inf` | positive infinity | a value greater than all finite floating-point values
`-Inf16` | `-Inf32` | `-Inf` | negative infinity | a value less than all finite floating-point values
`NaN16` | `NaN32` | `NaN` | not a number | a 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
    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 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)
    1.1920929f-7

    julia> eps(Float64)
    2.220446049250313e-16

    julia> eps() # same as eps(Float64)
    2.220446049250313e-16

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)
    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 `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
    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 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, 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.

Warning

Rounding is generally only correct for basic arithmetic functions (`+()`, `-()`, `*()`, `/()` and `sqrt()`) and type conversion operations. Many other functions assume the default `RoundNearest` mode is set, and can give erroneous results when operating under other rounding modes.

### 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 `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

## Numeric Literal Coefficients

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(x-1)^2 - 3(x-1) + 1
    3

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.

### Syntax Conflicts

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 `0xff` could be interpreted as the numeric literal `0` multiplied by the variable `xff`.

- 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.

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 floating-point literals.

## 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.

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