Module

The Math module contains module functions for basic trigonometric and transcendental functions. See class Float for a list of constants that define Ruby’s floating point accuracy.

Domains and codomains are given only for real (not complex) numbers.

Constants

PI

Definition of the mathematical constant PI as a Float number.

E

Definition of the mathematical constant E (e) as a Float number.

Class Methods

Computes the arc cosine of x. Returns 0..PI.

Domain: [-1, 1]

Codomain: [0, PI]

Math.acos(0) == Math::PI/2  #=> true

Computes the inverse hyperbolic cosine of x.

Domain: [1, INFINITY)

Codomain: [0, INFINITY)

Math.acosh(1) #=> 0.0

Computes the arc sine of x. Returns -PI/2..PI/2.

Domain: [-1, -1]

Codomain: [-PI/2, PI/2]

Math.asin(1) == Math::PI/2  #=> true

Computes the inverse hyperbolic sine of x.

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.asinh(1) #=> 0.881373587019543

Computes the arc tangent of x. Returns -PI/2..PI/2.

Domain: (-INFINITY, INFINITY)

Codomain: (-PI/2, PI/2)

Math.atan(0) #=> 0.0

Computes the arc tangent given y and x. Returns a Float in the range -PI..PI. Return value is a angle in radians between the positive x-axis of cartesian plane and the point given by the coordinates (x, y) on it.

Domain: (-INFINITY, INFINITY)

Codomain: [-PI, PI]

Math.atan2(-0.0, -1.0) #=> -3.141592653589793
Math.atan2(-1.0, -1.0) #=> -2.356194490192345
Math.atan2(-1.0, 0.0)  #=> -1.5707963267948966
Math.atan2(-1.0, 1.0)  #=> -0.7853981633974483
Math.atan2(-0.0, 1.0)  #=> -0.0
Math.atan2(0.0, 1.0)   #=> 0.0
Math.atan2(1.0, 1.0)   #=> 0.7853981633974483
Math.atan2(1.0, 0.0)   #=> 1.5707963267948966
Math.atan2(1.0, -1.0)  #=> 2.356194490192345
Math.atan2(0.0, -1.0)  #=> 3.141592653589793
Math.atan2(INFINITY, INFINITY)   #=> 0.7853981633974483
Math.atan2(INFINITY, -INFINITY)  #=> 2.356194490192345
Math.atan2(-INFINITY, INFINITY)  #=> -0.7853981633974483
Math.atan2(-INFINITY, -INFINITY) #=> -2.356194490192345

Computes the inverse hyperbolic tangent of x.

Domain: (-1, 1)

Codomain: (-INFINITY, INFINITY)

Math.atanh(1) #=> Infinity

Returns the cube root of x.

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

-9.upto(9) {|x|
  p [x, Math.cbrt(x), Math.cbrt(x)**3]
}
#=> [-9, -2.0800838230519, -9.0]
#   [-8, -2.0, -8.0]
#   [-7, -1.91293118277239, -7.0]
#   [-6, -1.81712059283214, -6.0]
#   [-5, -1.7099759466767, -5.0]
#   [-4, -1.5874010519682, -4.0]
#   [-3, -1.44224957030741, -3.0]
#   [-2, -1.25992104989487, -2.0]
#   [-1, -1.0, -1.0]
#   [0, 0.0, 0.0]
#   [1, 1.0, 1.0]
#   [2, 1.25992104989487, 2.0]
#   [3, 1.44224957030741, 3.0]
#   [4, 1.5874010519682, 4.0]
#   [5, 1.7099759466767, 5.0]
#   [6, 1.81712059283214, 6.0]
#   [7, 1.91293118277239, 7.0]
#   [8, 2.0, 8.0]
#   [9, 2.0800838230519, 9.0]

Computes the cosine of x (expressed in radians). Returns a Float in the range -1.0..1.0.

Domain: (-INFINITY, INFINITY)

Codomain: [-1, 1]

Math.cos(Math::PI) #=> -1.0

Computes the hyperbolic cosine of x (expressed in radians).

Domain: (-INFINITY, INFINITY)

Codomain: [1, INFINITY)

Math.cosh(0) #=> 1.0

Calculates the error function of x.

Domain: (-INFINITY, INFINITY)

Codomain: (-1, 1)

Math.erf(0) #=> 0.0

Calculates the complementary error function of x.

Domain: (-INFINITY, INFINITY)

Codomain: (0, 2)

Math.erfc(0) #=> 1.0

Returns e**x.

Domain: (-INFINITY, INFINITY)

Codomain: (0, INFINITY)

Math.exp(0)       #=> 1.0
Math.exp(1)       #=> 2.718281828459045
Math.exp(1.5)     #=> 4.4816890703380645

Returns a two-element array containing the normalized fraction (a Float) and exponent (an Integer) of x.

fraction, exponent = Math.frexp(1234)   #=> [0.6025390625, 11]
fraction * 2**exponent                  #=> 1234.0

Calculates the gamma function of x.

Note that gamma(n) is same as fact(n-1) for integer n > 0. However gamma(n) returns float and can be an approximation.

def fact(n) (1..n).inject(1) {|r,i| r*i } end
1.upto(26) {|i| p [i, Math.gamma(i), fact(i-1)] }
#=> [1, 1.0, 1]
#   [2, 1.0, 1]
#   [3, 2.0, 2]
#   [4, 6.0, 6]
#   [5, 24.0, 24]
#   [6, 120.0, 120]
#   [7, 720.0, 720]
#   [8, 5040.0, 5040]
#   [9, 40320.0, 40320]
#   [10, 362880.0, 362880]
#   [11, 3628800.0, 3628800]
#   [12, 39916800.0, 39916800]
#   [13, 479001600.0, 479001600]
#   [14, 6227020800.0, 6227020800]
#   [15, 87178291200.0, 87178291200]
#   [16, 1307674368000.0, 1307674368000]
#   [17, 20922789888000.0, 20922789888000]
#   [18, 355687428096000.0, 355687428096000]
#   [19, 6.402373705728e+15, 6402373705728000]
#   [20, 1.21645100408832e+17, 121645100408832000]
#   [21, 2.43290200817664e+18, 2432902008176640000]
#   [22, 5.109094217170944e+19, 51090942171709440000]
#   [23, 1.1240007277776077e+21, 1124000727777607680000]
#   [24, 2.5852016738885062e+22, 25852016738884976640000]
#   [25, 6.204484017332391e+23, 620448401733239439360000]
#   [26, 1.5511210043330954e+25, 15511210043330985984000000]

Returns sqrt(x**2 + y**2), the hypotenuse of a right-angled triangle with sides x and y.

Math.hypot(3, 4)   #=> 5.0

Returns the value of fraction*(2**exponent).

fraction, exponent = Math.frexp(1234)
Math.ldexp(fraction, exponent)   #=> 1234.0

Calculates the logarithmic gamma of x and the sign of gamma of x.

Math.lgamma(x) is same as

[Math.log(Math.gamma(x).abs), Math.gamma(x) < 0 ? -1 : 1]

but avoid overflow by Math.gamma(x) for large x.

Math.lgamma(0) #=> [Infinity, 1]

Returns the logarithm of x. If additional second argument is given, it will be the base of logarithm. Otherwise it is e (for the natural logarithm).

Domain: (0, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.log(0)          #=> -Infinity
Math.log(1)          #=> 0.0
Math.log(Math::E)    #=> 1.0
Math.log(Math::E**3) #=> 3.0
Math.log(12, 3)      #=> 2.2618595071429146

Returns the base 10 logarithm of x.

Domain: (0, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.log10(1)       #=> 0.0
Math.log10(10)      #=> 1.0
Math.log10(10**100) #=> 100.0

Returns the base 2 logarithm of x.

Domain: (0, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.log2(1)      #=> 0.0
Math.log2(2)      #=> 1.0
Math.log2(32768)  #=> 15.0
Math.log2(65536)  #=> 16.0

Computes the sine of x (expressed in radians). Returns a Float in the range -1.0..1.0.

Domain: (-INFINITY, INFINITY)

Codomain: [-1, 1]

Math.sin(Math::PI/2) #=> 1.0

Computes the hyperbolic sine of x (expressed in radians).

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.sinh(0) #=> 0.0

Returns the non-negative square root of x.

Domain: [0, INFINITY)

Codomain:[0, INFINITY)

0.upto(10) {|x|
  p [x, Math.sqrt(x), Math.sqrt(x)**2]
}
#=> [0, 0.0, 0.0]
#   [1, 1.0, 1.0]
#   [2, 1.4142135623731, 2.0]
#   [3, 1.73205080756888, 3.0]
#   [4, 2.0, 4.0]
#   [5, 2.23606797749979, 5.0]
#   [6, 2.44948974278318, 6.0]
#   [7, 2.64575131106459, 7.0]
#   [8, 2.82842712474619, 8.0]
#   [9, 3.0, 9.0]
#   [10, 3.16227766016838, 10.0]

Note that the limited precision of floating point arithmetic might lead to surprising results:

Math.sqrt(10**46).to_i  #=> 99999999999999991611392 (!)

See also BigDecimal#sqrt and Integer.sqrt.

Computes the tangent of x (expressed in radians).

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.tan(0) #=> 0.0

Computes the hyperbolic tangent of x (expressed in radians).

Domain: (-INFINITY, INFINITY)

Codomain: (-1, 1)

Math.tanh(0) #=> 0.0