Math

Numeric types

The two most common number types are Int32 and Float64. The number in the name denotes the size in bits: Int32 is a 32-bit integer type), Float64 is a 64-bit floating point number.

  • An integer literal is written as a series of one or more base-10 digits (0-9) without leading zeros. The default type is Int32.
  • A floating-point literal is written as a series of two or more base-10 digits (0-9) with a point (.) somewhere in the middle, indicating the decimal point. The default type is Float64.

All numeric types allow underscores at any place in the middle. This is useful to write large numbers in a more readable way: 100000 can be written as 100_000.

  1. p! 1, typeof(1)
  2. p! 1.0, typeof(1.0)
  3. p! 100_000, typeof(100_000)
  4. p! 100_000.0, typeof(100_000.0)

Float values print with a decimal point. Integer values don’t.

!!! info There are quite a few more numeric types, but most of them are intended only for special use cases such as binary protocols, specific numeric algorithms, and performance optimization. You probably don’t need them for everyday programs.

  1. See [Integer literal reference](../../syntax_and_semantics/literals/integers.md) and [Float literal reference](../../syntax_and_semantics/literals/floats.md)
  2. for a full reference on all primitive number types and alternative representations.

Arithmetic

Equality and Comparison

Numbers of the same numerical value are considered equal regarding the equality operator ==, independent of their type.

  1. p! 1 == 1,
  2.   1 == 2,
  3.   1.0 == 1,
  4.   -2000.0 == -2000

Besides the equality operator, there are also comparison operators. They determine the relationship between two values. As with equality, comparability is also independent of type.

  1. p! 2 > 1,
  2.   1 >= 1,
  3.   1 < 2,
  4.   1 <= 2

The universal comparison operator is <=>, also called Spaceship operator for its appearance. It compares its operands and returns a value that is either zero (both operands are equal), a positive value (the first operand is bigger), or a negative value (the second operand is bigger). It combines the behaviour of all other comparison operators.

  1. p! 1 <=> 1,
  2.   2 <=> 1,
  3.   1 <=> 2

Operators

Basic arithmetic operations can be performed with operators. Most operators are binary (i.e. two operands), and written in infix notation (i.e. between the operands). Some operators are unary (i.e. one operand), and written in prefix notation (i.e. before the operand). The value of the expression is the result of the operation.

  1. p! 1 + 1, # addition
  2.   1 - 1,  # subtraction
  3.   2 * 3,  # multiplication
  4.   2 ** 4, # exponentiation
  5.   2 / 3,  # division
  6.   2 // 3, # floor division
  7.   3 % 2,  # modulus
  8.   -1      # negation (unary)

As you can see, the result of most of these operations between integer operands is also an integer value. The division operator (/) is an exception. It always returns a float value. The floor division operator (//) however returns an integer value, but it’s obviously reduced to integer precision. An operation between integer and float operands always returns a float value. Otherwise, the return type is usually the type of the first operand.

!!! info A full list of operators is available in the Operator reference.

Precedence

When several operators are combined, the question arises in which order are they executed. In math, there are several rules, like multiplication and division taking precedence over addition and subtraction. Crystal operators implement these precedence rules.

A tool to structure operations are parentheses. An operator expression in parentheses always takes precedence over external operators.

  1. p! 4 + 5 * 2,
  2.   (4 + 5) * 2

!!! info All the precedence rules are detailed in the the Operator reference.

Number Methods

Some less common math operations are not operators, but named methods.

  1. p! -5.abs,   # absolute value
  2.   4.3.round, # round to nearest integer
  3.   5.even?,   # odd/even check
  4.   10.gcd(16) # greatest common divisor

!!! info A full list of numerical methods is available in the Number API docs (also check subtypes).

Math Methods

Some arithmetic methods are not defined on the number types directly but in the Math namespace.

  1. p! Math.cos(1),     # cosine
  2.   Math.sin(1),      # sine
  3.   Math.tan(1),      # tangent
  4.   Math.log(42),     # natural logarithm
  5.   Math.log10(312),  # logarithm to base 10
  6.   Math.log(312, 5), # logarithm to base 5
  7.   Math.sqrt(9)      # square root

!!! info A full list of math methods is available in the Math API docs.

Constants

Some mathematical constants are available as constants of the Math module.

  1. p! Math::E,  # Euler's number
  2.   Math::TAU, # Full circle constant (2 * PI)
  3.   Math::PI   # Archimedes' constant (TAU / 2)