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Content Copyright by Pierian Data

Math and Random Modules¶

Python comes with a built in math module and random module. In this lecture we will give a brief tour of their capabilities. Usually you can simply look up the function call you are looking for in the online documentation.

We won't go through every function available in these modules since there are so many, but we will show some useful ones.

Useful Math Functions¶

In [3]:

import math

In [9]:

help(math)

Help on built-in module math:

NAME
    math

DESCRIPTION
    This module is always available.  It provides access to the
    mathematical functions defined by the C standard.

FUNCTIONS
    acos(...)
        acos(x)
        
        Return the arc cosine (measured in radians) of x.
    
    acosh(...)
        acosh(x)
        
        Return the inverse hyperbolic cosine of x.
    
    asin(...)
        asin(x)
        
        Return the arc sine (measured in radians) of x.
    
    asinh(...)
        asinh(x)
        
        Return the inverse hyperbolic sine of x.
    
    atan(...)
        atan(x)
        
        Return the arc tangent (measured in radians) of x.
    
    atan2(...)
        atan2(y, x)
        
        Return the arc tangent (measured in radians) of y/x.
        Unlike atan(y/x), the signs of both x and y are considered.
    
    atanh(...)
        atanh(x)
        
        Return the inverse hyperbolic tangent of x.
    
    ceil(...)
        ceil(x)
        
        Return the ceiling of x as an Integral.
        This is the smallest integer >= x.
    
    copysign(...)
        copysign(x, y)
        
        Return a float with the magnitude (absolute value) of x but the sign 
        of y. On platforms that support signed zeros, copysign(1.0, -0.0) 
        returns -1.0.
    
    cos(...)
        cos(x)
        
        Return the cosine of x (measured in radians).
    
    cosh(...)
        cosh(x)
        
        Return the hyperbolic cosine of x.
    
    degrees(...)
        degrees(x)
        
        Convert angle x from radians to degrees.
    
    erf(...)
        erf(x)
        
        Error function at x.
    
    erfc(...)
        erfc(x)
        
        Complementary error function at x.
    
    exp(...)
        exp(x)
        
        Return e raised to the power of x.
    
    expm1(...)
        expm1(x)
        
        Return exp(x)-1.
        This function avoids the loss of precision involved in the direct evaluation of exp(x)-1 for small x.
    
    fabs(...)
        fabs(x)
        
        Return the absolute value of the float x.
    
    factorial(...)
        factorial(x) -> Integral
        
        Find x!. Raise a ValueError if x is negative or non-integral.
    
    floor(...)
        floor(x)
        
        Return the floor of x as an Integral.
        This is the largest integer <= x.
    
    fmod(...)
        fmod(x, y)
        
        Return fmod(x, y), according to platform C.  x % y may differ.
    
    frexp(...)
        frexp(x)
        
        Return the mantissa and exponent of x, as pair (m, e).
        m is a float and e is an int, such that x = m * 2.**e.
        If x is 0, m and e are both 0.  Else 0.5 <= abs(m) < 1.0.
    
    fsum(...)
        fsum(iterable)
        
        Return an accurate floating point sum of values in the iterable.
        Assumes IEEE-754 floating point arithmetic.
    
    gamma(...)
        gamma(x)
        
        Gamma function at x.
    
    gcd(...)
        gcd(x, y) -> int
        greatest common divisor of x and y
    
    hypot(...)
        hypot(x, y)
        
        Return the Euclidean distance, sqrt(x*x + y*y).
    
    isclose(...)
        isclose(a, b, *, rel_tol=1e-09, abs_tol=0.0) -> bool
        
        Determine whether two floating point numbers are close in value.
        
           rel_tol
               maximum difference for being considered "close", relative to the
               magnitude of the input values
            abs_tol
               maximum difference for being considered "close", regardless of the
               magnitude of the input values
        
        Return True if a is close in value to b, and False otherwise.
        
        For the values to be considered close, the difference between them
        must be smaller than at least one of the tolerances.
        
        -inf, inf and NaN behave similarly to the IEEE 754 Standard.  That
        is, NaN is not close to anything, even itself.  inf and -inf are
        only close to themselves.
    
    isfinite(...)
        isfinite(x) -> bool
        
        Return True if x is neither an infinity nor a NaN, and False otherwise.
    
    isinf(...)
        isinf(x) -> bool
        
        Return True if x is a positive or negative infinity, and False otherwise.
    
    isnan(...)
        isnan(x) -> bool
        
        Return True if x is a NaN (not a number), and False otherwise.
    
    ldexp(...)
        ldexp(x, i)
        
        Return x * (2**i).
    
    lgamma(...)
        lgamma(x)
        
        Natural logarithm of absolute value of Gamma function at x.
    
    log(...)
        log(x[, base])
        
        Return the logarithm of x to the given base.
        If the base not specified, returns the natural logarithm (base e) of x.
    
    log10(...)
        log10(x)
        
        Return the base 10 logarithm of x.
    
    log1p(...)
        log1p(x)
        
        Return the natural logarithm of 1+x (base e).
        The result is computed in a way which is accurate for x near zero.
    
    log2(...)
        log2(x)
        
        Return the base 2 logarithm of x.
    
    modf(...)
        modf(x)
        
        Return the fractional and integer parts of x.  Both results carry the sign
        of x and are floats.
    
    pow(...)
        pow(x, y)
        
        Return x**y (x to the power of y).
    
    radians(...)
        radians(x)
        
        Convert angle x from degrees to radians.
    
    sin(...)
        sin(x)
        
        Return the sine of x (measured in radians).
    
    sinh(...)
        sinh(x)
        
        Return the hyperbolic sine of x.
    
    sqrt(...)
        sqrt(x)
        
        Return the square root of x.
    
    tan(...)
        tan(x)
        
        Return the tangent of x (measured in radians).
    
    tanh(...)
        tanh(x)
        
        Return the hyperbolic tangent of x.
    
    trunc(...)
        trunc(x:Real) -> Integral
        
        Truncates x to the nearest Integral toward 0. Uses the __trunc__ magic method.

DATA
    e = 2.718281828459045
    inf = inf
    nan = nan
    pi = 3.141592653589793
    tau = 6.283185307179586

FILE
    (built-in)

Rounding Numbers¶

In [5]:

value = 4.35

In [6]:

math.floor(value)

Out[6]:

In [7]:

math.ceil(value)

Out[7]:

In [8]:

round(value)

Out[8]:

Mathematical Constants¶

In [20]:

math.pi

Out[20]:

3.141592653589793

In [21]:

from math import pi

In [22]:

pi

Out[22]:

3.141592653589793

In [23]:

math.e

Out[23]:

2.718281828459045

In [24]:

math.tau

Out[24]:

6.283185307179586

In [25]:

math.inf

Out[25]:

inf

In [26]:

math.nan

Out[26]:

nan

Logarithmic Values¶

In [10]:

math.e

Out[10]:

2.718281828459045

In [15]:

# Log Base e
math.log(math.e)

Out[15]:

1.0

In [12]:

# Will produce an error if value does not exist mathmatically
math.log(0)

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-12-7563e0a48092> in <module>()
----> 1 math.log(0)

ValueError: math domain error

In [13]:

math.log(10)

Out[13]:

2.302585092994046

In [17]:

math.e ** 2.302585092994046

Out[17]:

10.000000000000002

Custom Base¶

In [18]:

# math.log(x,base)
math.log(100,10)

Out[18]:

2.0

In [19]:

10**2

Out[19]:

Trigonometrics Functions¶

In [30]:

# Radians
math.sin(10)

Out[30]:

-0.5440211108893698

In [31]:

math.degrees(pi/2)

Out[31]:

90.0

In [32]:

math.radians(180)

Out[32]:

3.141592653589793

Random Module¶

Random Module allows us to create random numbers. We can even set a seed to produce the same random set every time.

The explanation of how a computer attempts to generate random numbers is beyond the scope of this course since it involves higher level mathmatics. But if you are interested in this topic check out:

Understanding a seed¶

Setting a seed allows us to start from a seeded psuedorandom number generator, which means the same random numbers will show up in a series. Note, you need the seed to be in the same cell if your using jupyter to guarantee the same results each time. Getting a same set of random numbers can be important in situations where you will be trying different variations of functions and want to compare their performance on random values, but want to do it fairly (so you need the same set of random numbers each time).

In [34]:

import random

In [41]:

random.randint(0,100)

Out[41]:

In [42]:

random.randint(0,100)

Out[42]:

In [45]:

# The value 101 is completely arbitrary, you can pass in any number you want
random.seed(101)
# You can run this cell as many times as you want, it will always return the same number
random.randint(0,100)

Out[45]:

In [46]:

random.randint(0,100)

Out[46]:

In [48]:

# The value 101 is completely arbitrary, you can pass in any number you want
random.seed(101)
print(random.randint(0,100))
print(random.randint(0,100))
print(random.randint(0,100))
print(random.randint(0,100))
print(random.randint(0,100))

Random Integers¶

In [49]:

random.randint(0,100)

Out[49]:

Random with Sequences¶

Grab a random item from a list¶

In [70]:

mylist = list(range(0,20))

In [71]:

mylist

Out[71]:

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

In [72]:

random.choice(mylist)

Out[72]:

In [73]:

mylist

Out[73]:

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

Sample with Replacement¶

Take a sample size, allowing picking elements more than once. Imagine a bag of numbered lottery balls, you reach in to grab a random lotto ball, then after marking down the number, you place it back in the bag, then continue picking another one.

In [77]:

random.choices(population=mylist,k=10)

Out[77]:

[15, 14, 17, 8, 17, 2, 19, 17, 6, 1]

Sample without Replacement¶

Once an item has been randomly picked, it can't be picked again. Imagine a bag of numbered lottery balls, you reach in to grab a random lotto ball, then after marking down the number, you leave it out of the bag, then continue picking another one.

In [78]:

random.sample(population=mylist,k=10)

Out[78]:

[17, 19, 11, 14, 1, 3, 4, 10, 5, 15]

Shuffle a list¶

Note: This effects the object in place!

In [79]:

# Don't assign this to anything!
random.shuffle(mylist)

In [80]:

mylist

Out[80]:

[9, 11, 7, 12, 10, 16, 0, 2, 18, 13, 3, 5, 17, 1, 15, 6, 14, 19, 4, 8]

Random Distributions¶

Uniform Distribution ¶

In [82]:

# Continuous, random picks a value between a and b, each value has equal change of being picked.
random.uniform(a=0,b=100)

Out[82]:

23.852305703497635

Normal/Gaussian Distribution ¶

In [83]:

random.gauss(mu=0,sigma=1)

Out[83]:

-0.21390381464435643

Final Note: If you find yourself using these libraries a lot, take a look at the NumPy library for Python, covers all these capabilities with extreme efficiency. We cover this library and a lot more in our data science and machine learning courses.