# scala.math

package scala.math

The package object `scala.math` contains methods for performing basic numeric operations such as elementary exponential, logarithmic, root and trigonometric functions.

All methods forward to java.lang.Math unless otherwise noted.

### Mathematical Constants

final val E: 2.718281828459045d

The `Double` value that is closer than any other to `e`, the base of the natural logarithms.

The `Double` value that is closer than any other to `e`, the base of the natural logarithms.

Source:
package.scala
final val Pi: 3.141592653589793d

The `Double` value that is closer than any other to `pi`, the ratio of the circumference of a circle to its diameter.

The `Double` value that is closer than any other to `pi`, the ratio of the circumference of a circle to its diameter.

Source:
package.scala

### Minimum and Maximum

Find the min or max of two numbers. Note: scala.collection.IterableOnceOps has min and max methods which determine the min or max of a collection.

def max(x: Int, y: Int): Int
def max(x: Long, y: Long): Long
def max(x: Float, y: Float): Float
def max(x: Double, y: Double): Double
def min(x: Int, y: Int): Int
def min(x: Long, y: Long): Long
def min(x: Float, y: Float): Float
def min(x: Double, y: Double): Double

### Rounding

def rint(x: Double): Double

Returns the `Double` value that is closest in value to the argument and is equal to a mathematical integer.

Returns the `Double` value that is closest in value to the argument and is equal to a mathematical integer.

Value parameters:
x

a `Double` value

Returns:

the closest floating-point value to a that is equal to a mathematical integer.

Source:
package.scala
@deprecated("This is an integer type; there is no reason to round it. Perhaps you meant to call this with a floating-point value?", "2.11.0")
def round(x: Long): Long

There is no reason to round a `Long`, but this method prevents unintended conversion to `Float` followed by rounding to `Int`.

There is no reason to round a `Long`, but this method prevents unintended conversion to `Float` followed by rounding to `Int`.

Note:

Does not forward to java.lang.Math

Deprecated
Source:
package.scala
def round(x: Float): Int

Returns the closest `Int` to the argument.

Returns the closest `Int` to the argument.

Value parameters:
x

a floating-point value to be rounded to a `Int`.

Returns:

the value of the argument rounded to the nearest `Int` value.

Source:
package.scala
def round(x: Double): Long

Returns the closest `Long` to the argument.

Returns the closest `Long` to the argument.

Value parameters:
x

a floating-point value to be rounded to a `Long`.

Returns:

the value of the argument rounded to the nearest`long` value.

Source:
package.scala

### Scaling

Scaling with rounding guarantees

def scalb(d: Double, scaleFactor: Int): Double
def scalb(f: Float, scaleFactor: Int): Float

### Exponential and Logarithmic

def exp(x: Double): Double

Returns Euler's number `e` raised to the power of a `Double` value.

Returns Euler's number `e` raised to the power of a `Double` value.

Value parameters:
x

the exponent to raise `e` to.

Returns:

the value `ea`, where `e` is the base of the natural logarithms.

Source:
package.scala

Returns `exp(x) - 1`.

Returns `exp(x) - 1`.

Source:
package.scala
def log(x: Double): Double

Returns the natural logarithm of a `Double` value.

Returns the natural logarithm of a `Double` value.

Value parameters:
x

the number to take the natural logarithm of

Returns:

the value `logâ‚‘(x)` where `e` is Eulers number

Source:
package.scala

Returns the base 10 logarithm of the given `Double` value.

Returns the base 10 logarithm of the given `Double` value.

Source:
package.scala

Returns the natural logarithm of the sum of the given `Double` value and 1.

Returns the natural logarithm of the sum of the given `Double` value and 1.

Source:
package.scala
def pow(x: Double, y: Double): Double

Returns the value of the first argument raised to the power of the second argument.

Returns the value of the first argument raised to the power of the second argument.

Value parameters:
x

the base.

y

the exponent.

Returns:

the value `xy`.

Source:
package.scala

### Angular Measurement Conversion

Converts an angle measured in radians to an approximately equivalent angle measured in degrees.

Converts an angle measured in radians to an approximately equivalent angle measured in degrees.

Value parameters:
x

Returns:

the measurement of the angle `x` in degrees.

Source:
package.scala

Converts an angle measured in degrees to an approximately equivalent angle measured in radians.

Converts an angle measured in degrees to an approximately equivalent angle measured in radians.

Value parameters:
x

an angle, in degrees

Returns:

the measurement of the angle `x` in radians.

Source:
package.scala

### Hyperbolic

def cosh(x: Double): Double

Returns the hyperbolic cosine of the given `Double` value.

Returns the hyperbolic cosine of the given `Double` value.

Source:
package.scala
def sinh(x: Double): Double

Returns the hyperbolic sine of the given `Double` value.

Returns the hyperbolic sine of the given `Double` value.

Source:
package.scala
def tanh(x: Double): Double

Returns the hyperbolic tangent of the given `Double` value.

Returns the hyperbolic tangent of the given `Double` value.

Source:
package.scala

### Absolute Values

Determine the magnitude of a value by discarding the sign. Results are >= 0.

def abs(x: Int): Int
def abs(x: Long): Long
def abs(x: Float): Float

### Signs

For `signum` extract the sign of a value. Results are -1, 0 or 1. Note the `signum` methods are not pure forwarders to the Java versions. In particular, the return type of `java.lang.Long.signum` is `Int`, but here it is widened to `Long` so that each overloaded variant will return the same numeric type it is passed.

def copySign(magnitude: Double, sign: Double): Double
def copySign(magnitude: Float, sign: Float): Float
def signum(x: Int): Int
Note:

Forwards to java.lang.Integer

Source:
package.scala
def signum(x: Long): Long
Note:

Forwards to java.lang.Long

Source:
package.scala

### Root Extraction

def cbrt(x: Double): Double

Returns the cube root of the given `Double` value.

Returns the cube root of the given `Double` value.

Value parameters:
x

the number to take the cube root of

Returns:

the value âˆ›x

Source:
package.scala
def sqrt(x: Double): Double

Returns the square root of a `Double` value.

Returns the square root of a `Double` value.

Value parameters:
x

the number to take the square root of

Returns:

the value âˆšx

Source:
package.scala

### Polar Coordinates

def atan2(y: Double, x: Double): Double

Converts rectangular coordinates `(x, y)` to polar `(r, theta)`.

Converts rectangular coordinates `(x, y)` to polar `(r, theta)`.

Value parameters:
x

the ordinate coordinate

y

the abscissa coordinate

Returns:

the theta component of the point `(r, theta)` in polar coordinates that corresponds to the point `(x, y)` in Cartesian coordinates.

Source:
package.scala
def hypot(x: Double, y: Double): Double

Returns the square root of the sum of the squares of both given `Double` values without intermediate underflow or overflow.

Returns the square root of the sum of the squares of both given `Double` values without intermediate underflow or overflow.

The r component of the point `(r, theta)` in polar coordinates that corresponds to the point `(x, y)` in Cartesian coordinates.

Source:
package.scala

### Unit of Least Precision

def ulp(x: Double): Double

Returns the size of an ulp of the given `Double` value.

Returns the size of an ulp of the given `Double` value.

Source:
package.scala
def ulp(x: Float): Float

Returns the size of an ulp of the given `Float` value.

Returns the size of an ulp of the given `Float` value.

Source:
package.scala

### Pseudo Random Number Generation

def random(): Double

Returns a `Double` value with a positive sign, greater than or equal to `0.0` and less than `1.0`.

Returns a `Double` value with a positive sign, greater than or equal to `0.0` and less than `1.0`.

Source:
package.scala

### Exact Arithmetic

Integral addition, multiplication, stepping and conversion throwing ArithmeticException instead of underflowing or overflowing

def addExact(x: Int, y: Int): Int
def addExact(x: Long, y: Long): Long

### Modulus and Quotient

Calculate quotient values by rounding to negative infinity

def floorDiv(x: Int, y: Int): Int
def floorDiv(x: Long, y: Long): Long
def floorMod(x: Int, y: Int): Int
def floorMod(x: Long, y: Long): Long

def nextAfter(start: Double, direction: Double): Double
def nextAfter(start: Float, direction: Double): Float

## Type members

### Classlikes

object BigDecimal
Companion:
class
Source:
BigDecimal.scala
final class BigDecimal(val bigDecimal: BigDecimal, val mc: MathContext) extends ScalaNumber with ScalaNumericConversions with Serializable with Ordered[BigDecimal]

`BigDecimal` represents decimal floating-point numbers of arbitrary precision.

`BigDecimal` represents decimal floating-point numbers of arbitrary precision. By default, the precision approximately matches that of IEEE 128-bit floating point numbers (34 decimal digits, `HALF_EVEN` rounding mode). Within the range of IEEE binary128 numbers, `BigDecimal` will agree with `BigInt` for both equality and hash codes (and will agree with primitive types as well). Beyond that range--numbers with more than 4934 digits when written out in full--the `hashCode` of `BigInt` and `BigDecimal` is allowed to diverge due to difficulty in efficiently computing both the decimal representation in `BigDecimal` and the binary representation in `BigInt`.

When creating a `BigDecimal` from a `Double` or `Float`, care must be taken as the binary fraction representation of `Double` and `Float` does not easily convert into a decimal representation. Three explicit schemes are available for conversion. `BigDecimal.decimal` will convert the floating-point number to a decimal text representation, and build a `BigDecimal` based on that. `BigDecimal.binary` will expand the binary fraction to the requested or default precision. `BigDecimal.exact` will expand the binary fraction to the full number of digits, thus producing the exact decimal value corresponding to the binary fraction of that floating-point number. `BigDecimal` equality matches the decimal expansion of `Double`: `BigDecimal.decimal(0.1) == 0.1`. Note that since `0.1f != 0.1`, the same is not true for `Float`. Instead, `0.1f == BigDecimal.decimal((0.1f).toDouble)`.

To test whether a `BigDecimal` number can be converted to a `Double` or `Float` and then back without loss of information by using one of these methods, test with `isDecimalDouble`, `isBinaryDouble`, or `isExactDouble` or the corresponding `Float` versions. Note that `BigInt`'s `isValidDouble` will agree with `isExactDouble`, not the `isDecimalDouble` used by default.

`BigDecimal` uses the decimal representation of binary floating-point numbers to determine equality and hash codes. This yields different answers than conversion between `Long` and `Double` values, where the exact form is used. As always, since floating-point is a lossy representation, it is advisable to take care when assuming identity will be maintained across multiple conversions.

`BigDecimal` maintains a `MathContext` that determines the rounding that is applied to certain calculations. In most cases, the value of the `BigDecimal` is also rounded to the precision specified by the `MathContext`. To create a `BigDecimal` with a different precision than its `MathContext`, use `new BigDecimal(new java.math.BigDecimal(...), mc)`. Rounding will be applied on those mathematical operations that can dramatically change the number of digits in a full representation, namely multiplication, division, and powers. The left-hand argument's `MathContext` always determines the degree of rounding, if any, and is the one propagated through arithmetic operations that do not apply rounding themselves.

Companion:
object
Source:
BigDecimal.scala
object BigInt
Companion:
class
Source:
BigInt.scala
final class BigInt extends ScalaNumber with ScalaNumericConversions with Serializable with Ordered[BigInt]

A type with efficient encoding of arbitrary integers.

A type with efficient encoding of arbitrary integers.

It wraps `java.math.BigInteger`, with optimization for small values that can be encoded in a `Long`.

Companion:
object
Source:
BigInt.scala
trait Equiv[T] extends Serializable

A trait for representing equivalence relations.

A trait for representing equivalence relations. It is important to distinguish between a type that can be compared for equality or equivalence and a representation of equivalence on some type. This trait is for representing the latter.

An equivalence relation is a binary relation on a type. This relation is exposed as the `equiv` method of the `Equiv` trait. The relation must be:

1. reflexive: `equiv(x, x) == true` for any x of type `T`.

2. symmetric: `equiv(x, y) == equiv(y, x)` for any `x` and `y` of type `T`.

3. transitive: if `equiv(x, y) == true` and `equiv(y, z) == true`, then `equiv(x, z) == true` for any `x`, `y`, and `z` of type `T`.

Companion:
object
Source:
Equiv.scala
object Equiv extends LowPriorityEquiv
Companion:
class
Source:
Equiv.scala
trait Fractional[T] extends Numeric[T]
Companion:
object
Source:
Fractional.scala
object Fractional
Companion:
class
Source:
Fractional.scala
trait Integral[T] extends Numeric[T]
Companion:
object
Source:
Integral.scala
object Integral
Companion:
class
Source:
Integral.scala
object Numeric
Companion:
class
Source:
Numeric.scala
trait Numeric[T] extends Ordering[T]
Companion:
object
Source:
Numeric.scala
trait Ordered[A] extends Comparable[A]

A trait for data that have a single, natural ordering.

A trait for data that have a single, natural ordering. See scala.math.Ordering before using this trait for more information about whether to use scala.math.Ordering instead.

Classes that implement this trait can be sorted with scala.util.Sorting and can be compared with standard comparison operators (e.g. > and <).

Ordered should be used for data with a single, natural ordering (like integers) while Ordering allows for multiple ordering implementations. An Ordering instance will be implicitly created if necessary.

scala.math.Ordering is an alternative to this trait that allows multiple orderings to be defined for the same type.

scala.math.PartiallyOrdered is an alternative to this trait for partially ordered data.

For example, create a simple class that implements `Ordered` and then sort it with scala.util.Sorting:

``````case class OrderedClass(n:Int) extends Ordered[OrderedClass] {
def compare(that: OrderedClass) =  this.n - that.n
}

val x = Array(OrderedClass(1), OrderedClass(5), OrderedClass(3))
scala.util.Sorting.quickSort(x)
x``````

It is important that the `equals` method for an instance of `Ordered[A]` be consistent with the compare method. However, due to limitations inherent in the type erasure semantics, there is no reasonable way to provide a default implementation of equality for instances of `Ordered[A]`. Therefore, if you need to be able to use equality on an instance of `Ordered[A]` you must provide it yourself either when inheriting or instantiating.

It is important that the `hashCode` method for an instance of `Ordered[A]` be consistent with the `compare` method. However, it is not possible to provide a sensible default implementation. Therefore, if you need to be able compute the hash of an instance of `Ordered[A]` you must provide it yourself either when inheriting or instantiating.

Companion:
object
Source:
Ordered.scala
object Ordered
Companion:
class
Source:
Ordered.scala
@implicitNotFound(msg = "No implicit Ordering defined for \${T}.")
trait Ordering[T] extends Comparator[T] with PartialOrdering[T] with Serializable

Ordering is a trait whose instances each represent a strategy for sorting instances of a type.

Ordering is a trait whose instances each represent a strategy for sorting instances of a type.

Ordering's companion object defines many implicit objects to deal with subtypes of AnyVal (e.g. `Int`, `Double`), `String`, and others.

To sort instances by one or more member variables, you can take advantage of these built-in orderings using Ordering.by and Ordering.on:

``````import scala.util.Sorting
val pairs = Array(("a", 5, 2), ("c", 3, 1), ("b", 1, 3))

// sort by 2nd element
Sorting.quickSort(pairs)(Ordering.by[(String, Int, Int), Int](_._2))

// sort by the 3rd element, then 1st
Sorting.quickSort(pairs)(Ordering[(Int, String)].on(x => (x._3, x._1)))``````

An `Ordering[T]` is implemented by specifying the compare method, `compare(a: T, b: T): Int`, which decides how to order two instances `a` and `b`. Instances of `Ordering[T]` can be used by things like `scala.util.Sorting` to sort collections like `Array[T]`.

For example:

``````import scala.util.Sorting

case class Person(name:String, age:Int)
val people = Array(Person("bob", 30), Person("ann", 32), Person("carl", 19))

// sort by age
object AgeOrdering extends Ordering[Person] {
def compare(a:Person, b:Person) = a.age.compare(b.age)
}
Sorting.quickSort(people)(AgeOrdering)``````

This trait and scala.math.Ordered both provide this same functionality, but in different ways. A type `T` can be given a single way to order itself by extending `Ordered`. Using `Ordering`, this same type may be sorted in many other ways. `Ordered` and `Ordering` both provide implicits allowing them to be used interchangeably.

You can `import scala.math.Ordering.Implicits._` to gain access to other implicit orderings.

Companion:
object
Source:
Ordering.scala

This is the companion object for the scala.math.Ordering trait.

This is the companion object for the scala.math.Ordering trait.

It contains many implicit orderings as well as well as methods to construct new orderings.

Companion:
class
Source:
Ordering.scala
trait PartialOrdering[T] extends Equiv[T]

A trait for representing partial orderings.

A trait for representing partial orderings. It is important to distinguish between a type that has a partial order and a representation of partial ordering on some type. This trait is for representing the latter.

A partial ordering is a binary relation on a type `T`, exposed as the `lteq` method of this trait. This relation must be:

- reflexive: `lteq(x, x) == true`, for any `x` of type `T`. - anti-symmetric: if `lteq(x, y) == true` and `lteq(y, x) == true` then `equiv(x, y) == true`, for any `x` and `y` of type `T`. - transitive: if `lteq(x, y) == true` and `lteq(y, z) == true` then `lteq(x, z) == true`, for any `x`, `y`, and `z` of type `T`.

Additionally, a partial ordering induces an equivalence relation on a type `T`: `x` and `y` of type `T` are equivalent if and only if `lteq(x, y) && lteq(y, x) == true`. This equivalence relation is exposed as the `equiv` method, inherited from the Equiv trait.

Companion:
object
Source:
PartialOrdering.scala

A class for partially ordered data.

A class for partially ordered data.

Source:
PartiallyOrdered.scala

Conversions which present a consistent conversion interface across all the numeric types, suitable for use in value classes.

Conversions which present a consistent conversion interface across all the numeric types, suitable for use in value classes.

Source:
ScalaNumericConversions.scala

A slightly more specific conversion trait for classes which extend ScalaNumber (which excludes value classes.)

A slightly more specific conversion trait for classes which extend ScalaNumber (which excludes value classes.)

Source:
ScalaNumericConversions.scala