Numeric and Mathematical Modules

The modules described in this chapter provide numeric and math-related functions and data types. The numbers module defines an abstract hierarchy of numeric types. The math and cmath modules contain various mathematical functions for floating-point and complex numbers. The decimal module supports exact representations of decimal numbers, using arbitrary precision arithmetic.

The following modules are documented in this chapter:

• numbers — Numeric abstract base classes
  • Number
  • The numeric tower
    • Complex
      • Complex.real
      • Complex.imag
      • Complex.conjugate()
    • Real
    • Rational
      • Rational.numerator
      • Rational.denominator
    • Integral
  • Notes for type implementors
    • Adding More Numeric ABCs
    • Implementing the arithmetic operations
• math — Mathematical functions
  • Number-theoretic and representation functions
    • ceil()
    • comb()
    • copysign()
    • fabs()
    • factorial()
    • floor()
    • fmod()
    • frexp()
    • fsum()
    • gcd()
    • isclose()
    • isfinite()
    • isinf()
    • isnan()
    • isqrt()
    • lcm()
    • ldexp()
    • modf()
    • nextafter()
    • perm()
    • prod()
    • remainder()
    • sumprod()
    • trunc()
    • ulp()
  • Power and logarithmic functions
    • cbrt()
    • exp()
    • exp2()
    • expm1()
    • log()
    • log1p()
    • log2()
    • log10()
    • pow()
    • sqrt()
  • Trigonometric functions
    • acos()
    • asin()
    • atan()
    • atan2()
    • cos()
    • dist()
    • hypot()
    • sin()
    • tan()
  • Angular conversion
    • degrees()
    • radians()
  • Hyperbolic functions
    • acosh()
    • asinh()
    • atanh()
    • cosh()
    • sinh()
    • tanh()
  • Special functions
    • erf()
    • erfc()
    • gamma()
    • lgamma()
  • Constants
    • pi
    • e
    • tau
    • inf
    • nan
• cmath — Mathematical functions for complex numbers
  • Conversions to and from polar coordinates
    • phase()
    • polar()
    • rect()
  • Power and logarithmic functions
    • exp()
    • log()
    • log10()
    • sqrt()
  • Trigonometric functions
    • acos()
    • asin()
    • atan()
    • cos()
    • sin()
    • tan()
  • Hyperbolic functions
    • acosh()
    • asinh()
    • atanh()
    • cosh()
    • sinh()
    • tanh()
  • Classification functions
    • isfinite()
    • isinf()
    • isnan()
    • isclose()
  • Constants
    • pi
    • e
    • tau
    • inf
    • infj
    • nan
    • nanj
• decimal — Decimal fixed point and floating point arithmetic
  • Quick-start Tutorial
  • Decimal objects
    • Decimal
      • Decimal.adjusted()
      • Decimal.as_integer_ratio()
      • Decimal.as_tuple()
      • Decimal.canonical()
      • Decimal.compare()
      • Decimal.compare_signal()
      • Decimal.compare_total()
      • Decimal.compare_total_mag()
      • Decimal.conjugate()
      • Decimal.copy_abs()
      • Decimal.copy_negate()
      • Decimal.copy_sign()
      • Decimal.exp()
      • Decimal.from_float()
      • Decimal.fma()
      • Decimal.is_canonical()
      • Decimal.is_finite()
      • Decimal.is_infinite()
      • Decimal.is_nan()
      • Decimal.is_normal()
      • Decimal.is_qnan()
      • Decimal.is_signed()
      • Decimal.is_snan()
      • Decimal.is_subnormal()
      • Decimal.is_zero()
      • Decimal.ln()
      • Decimal.log10()
      • Decimal.logb()
      • Decimal.logical_and()
      • Decimal.logical_invert()
      • Decimal.logical_or()
      • Decimal.logical_xor()
      • Decimal.max()
      • Decimal.max_mag()
      • Decimal.min()
      • Decimal.min_mag()
      • Decimal.next_minus()
      • Decimal.next_plus()
      • Decimal.next_toward()
      • Decimal.normalize()
      • Decimal.number_class()
      • Decimal.quantize()
      • Decimal.radix()
      • Decimal.remainder_near()
      • Decimal.rotate()
      • Decimal.same_quantum()
      • Decimal.scaleb()
      • Decimal.shift()
      • Decimal.sqrt()
      • Decimal.to_eng_string()
      • Decimal.to_integral()
      • Decimal.to_integral_exact()
      • Decimal.to_integral_value()
    • Logical operands
  • Context objects
    • getcontext()
    • setcontext()
    • localcontext()
    • BasicContext
    • ExtendedContext
    • DefaultContext
    • Context
      • Context.clear_flags()
      • Context.clear_traps()
      • Context.copy()
      • Context.copy_decimal()
      • Context.create_decimal()
      • Context.create_decimal_from_float()
      • Context.Etiny()
      • Context.Etop()
      • Context.abs()
      • Context.add()
      • Context.canonical()
      • Context.compare()
      • Context.compare_signal()
      • Context.compare_total()
      • Context.compare_total_mag()
      • Context.copy_abs()
      • Context.copy_negate()
      • Context.copy_sign()
      • Context.divide()
      • Context.divide_int()
      • Context.divmod()
      • Context.exp()
      • Context.fma()
      • Context.is_canonical()
      • Context.is_finite()
      • Context.is_infinite()
      • Context.is_nan()
      • Context.is_normal()
      • Context.is_qnan()
      • Context.is_signed()
      • Context.is_snan()
      • Context.is_subnormal()
      • Context.is_zero()
      • Context.ln()
      • Context.log10()
      • Context.logb()
      • Context.logical_and()
      • Context.logical_invert()
      • Context.logical_or()
      • Context.logical_xor()
      • Context.max()
      • Context.max_mag()
      • Context.min()
      • Context.min_mag()
      • Context.minus()
      • Context.multiply()
      • Context.next_minus()
      • Context.next_plus()
      • Context.next_toward()
      • Context.normalize()
      • Context.number_class()
      • Context.plus()
      • Context.power()
      • Context.quantize()
      • Context.radix()
      • Context.remainder()
      • Context.remainder_near()
      • Context.rotate()
      • Context.same_quantum()
      • Context.scaleb()
      • Context.shift()
      • Context.sqrt()
      • Context.subtract()
      • Context.to_eng_string()
      • Context.to_integral_exact()
      • Context.to_sci_string()
  • Constants
    • MAX_PREC
    • MAX_EMAX
    • MIN_EMIN
    • MIN_ETINY
    • HAVE_THREADS
    • HAVE_CONTEXTVAR
  • Rounding modes
    • ROUND_CEILING
    • ROUND_DOWN
    • ROUND_FLOOR
    • ROUND_HALF_DOWN
    • ROUND_HALF_EVEN
    • ROUND_HALF_UP
    • ROUND_UP
    • ROUND_05UP
  • Signals
    • Clamped
    • DecimalException
    • DivisionByZero
    • Inexact
    • InvalidOperation
    • Overflow
    • Rounded
    • Subnormal
    • Underflow
    • FloatOperation
  • Floating Point Notes
    • Mitigating round-off error with increased precision
    • Special values
  • Working with threads
  • Recipes
  • Decimal FAQ
• fractions — Rational numbers
  • Fraction
    • Fraction.numerator
    • Fraction.denominator
    • Fraction.as_integer_ratio()
    • Fraction.is_integer()
    • Fraction.from_float()
    • Fraction.from_decimal()
    • Fraction.limit_denominator()
    • Fraction.__floor__()
    • Fraction.__ceil__()
    • Fraction.__round__()
    • Fraction.__format__()
• random — Generate pseudo-random numbers
  • Bookkeeping functions
    • seed()
    • getstate()
    • setstate()
  • Functions for bytes
    • randbytes()
  • Functions for integers
    • randrange()
    • randint()
    • getrandbits()
  • Functions for sequences
    • choice()
    • choices()
    • shuffle()
    • sample()
  • Discrete distributions
    • binomialvariate()
  • Real-valued distributions
    • random()
    • uniform()
    • triangular()
    • betavariate()
    • expovariate()
    • gammavariate()
    • gauss()
    • lognormvariate()
    • normalvariate()
    • vonmisesvariate()
    • paretovariate()
    • weibullvariate()
  • Alternative Generator
    • Random
    • SystemRandom
  • Notes on Reproducibility
  • Examples
  • Recipes
• statistics — Mathematical statistics functions
  • Averages and measures of central location
  • Measures of spread
  • Statistics for relations between two inputs
  • Function details
    • mean()
    • fmean()
    • geometric_mean()
    • harmonic_mean()
    • median()
    • median_low()
    • median_high()
    • median_grouped()
    • mode()
    • multimode()
    • pstdev()
    • pvariance()
    • stdev()
    • variance()
    • quantiles()
    • covariance()
    • correlation()
    • linear_regression()
  • Exceptions
    • StatisticsError
  • NormalDist objects
    • NormalDist
      • NormalDist.mean
      • NormalDist.median
      • NormalDist.mode
      • NormalDist.stdev
      • NormalDist.variance
      • NormalDist.from_samples()
      • NormalDist.samples()
      • NormalDist.pdf()
      • NormalDist.cdf()
      • NormalDist.inv_cdf()
      • NormalDist.overlap()
      • NormalDist.quantiles()
      • NormalDist.zscore()
    • NormalDist Examples and Recipes