Acquiring a good understanding of the Python Simple Mathematical Operators is another important thing to accomplish while learning programming.

## Chapter 9: Python Simple Mathematical Operators

#### Python Simple Mathematical Operators: Numerical types and their metaclasses

The numbers module contains the abstract metaclasses for the numerical types:

`subclasses numbers.Number numbers.Integral numbers.Rational numbers.Real numbers.Complex`

bool

✓ ✓ ✓ ✓ ✓

int

✓ ✓ ✓ ✓ ✓

fractions.Fraction ✓

― ✓ ✓ ✓

float

✓ ― ― ✓ ✓

✓ ― ― ― ✓

complex

decimal.Decimal

✓ ― ― ― ―

Python does common mathematical operators on its own, including integer and float division, multiplication, exponentiation, addition, and subtraction. The math module (included in all standard Python versions) oﬀers expanded functionality like trigonometric functions, root operations, logarithms, and many more.

## Python Simple Mathematical Operators: Division

Python does integer division when both operands are integers. The behavior of Python’s division operators have changed from Python 2.x and 3.x (see also Integer Division ).

a, b, c, d, e = 3, 2, 2.0, -3, 10

Python 2.x Version ≤ 2.7

In Python 2 the result of the ' / ' operator depends on the type of the numerator and denominator.

a / b # = 1

a / c #=1.5

d / b #=-2

b / a # = 0

d / e # = -1

Note that because both a and b are ints, the result is an int.

The result is always rounded down (floored).

Because c is a float, the result of a / c is a float.

You can also use the operator module:

import operator # the operator module provides 2-argument arithmetic functions

operator.div(a, b) # = 1

operator.div(a, b) # = 1

Python 2.x Version ≥ 2.2

What if you want float division:

Recommended:

fromfutureimport division # applies Python 3 style division to the entire module

a / b # = 1.5

a // b # = 1

Okay (if you don't want to apply to the whole module):

a / (b * 1.0) #=1.5

1.0 * a / b #=1.5

a / b * 1.0 # = 1.0 (careful with order of operations)

from operator import truediv

truediv(a, b) # = 1.5

Not recommended (may raise TypeError, eg if argument is complex):

float(a) / b # = 1.5

a / float(b) # = 1.5

Python 2.x Version ≥ 2.2

The ' // ' operator in Python 2 forces floored division regardless of type.

a // b # = 1

a // c # = 1.0

Python 3.x Version ≥ 3.0

In Python 3 the / operator performs 'true' division regardless of types. The // operator performs floor division and maintains type.

a / b #=1.5

e / b #=5.0

a // b # = 1

a // c #=1.0

import operator # the operator module provides 2-argument arithmetic functions

operator.truediv(a, b) #=1.5

operator.floordiv(a, b) # = 1

operator.floordiv(a, c) #=1.0

Possible combinations (builtin types):

int and int (gives an int in Python 2 and a float in Python 3)

int and float (gives a float)

int and complex (gives a complex)

float and float (gives a float)

float and complex (gives a complex)

complex and complex (gives a complex)

See PEP 238 for more information.

## Python Simple Mathematical Operators: Addition

a, b = 1, 2

## Using the “+” operator:

a + b # = 3

## Using the “in-place” “+=” operator to add and assign:

a += b # a = 3 (equivalent to a = a + b)

import operator

## contains 2 argument arithmetic functions for the examples

operator.add(a, b) # = 5 since a is set to 3 right before this line

### The “+=” operator is equivalent to:

a = operator.iadd(a, b) # a = 5 since a is set to 3 right before this line

Possible combinations (builtin types):

int and int (gives an int)

int and float (gives a float)

int and complex (gives a complex)

float and float (gives a float)

float and complex (gives a complex)

complex and complex (gives a complex)

Note: the + operator is also used for concatenating strings, lists and tuples:

"first string " + "second string" # = 'first string second string'

[1, 2, 3] + [4, 5, 6] #=[1,2,3,4,5,6]

## Python Simple Mathematical Operators: Exponentiation

a, b = 2, 3

(a ** b) # = 8

pow(a, b) # = 8

import math

math.pow(a, b) # = 8.0 (always float; does not allow complex results)

import operator

operator.pow(a, b) # = 8

Another diﬀerence between the built-in pow and math.pow is that the built-in pow can accept three arguments:

a, b, c = 2, 3, 2

pow(2, 3, 2) # 0, calculates (2 ** 3) % 2, but as per Python docs,

# does so more efficiently

## Python Simple Mathematical Operators: Special functions

The function math.sqrt(x) calculates the square root of x.

import math

import cmath

c = 4

math.sqrt(c)

cmath.sqrt(c)

#### = 2.0 (always float; does not allow complex results)

#### = (2+0j) (always complex)

To compute other roots, such as a cube root, raise the number to the reciprocal of the degree of the root. This could be done with any of the exponential functions or operator.

import math

x = 8

math.pow(x, 1/3) # evaluates to 2.0

x**(1/3) # evaluates to 2.0

The function math.exp(x) computes e ** x.

math.exp(0) # 1.0

math.exp(1) # 2.718281828459045 (e)

The function math.expm1(x) computes e ** x - 1. When x is small, this gives significantly better precision than math.exp(x) - 1.

math.expm1(0) # 0.0

math.exp(1e-6) - 1 # 1.0000004999621837e-06

math.expm1(1e-6) # 1.0000005000001665e-06

#### exact result # 1.000000500000166666708333341666…

## Python Simple Mathematical Operators: Trigonometric Functions

a, b = 1, 2

import math

math.sin(a) # returns the sine of 'a' in radians

#### Out: 0.8414709848078965

math.cosh(b) # returns the inverse hyperbolic cosine of 'b' in radians # Out: 3.7621956910836314

math.atan(math.pi) # returns the arc tangent of 'pi' in radians # Out: 1.2626272556789115

math.hypot(a, b) # returns the Euclidean norm, same as math.sqrt(aa + bb)

#### Out: 2.23606797749979

Note that math.hypot(x, y) is also the length of the vector (or Euclidean distance) from the origin (0, 0) to the point (x, y).

To compute the Euclidean distance between two points (x1, y1) & (x2, y2) you can use math.hypot as follows

math.hypot(x2-x1, y2-y1)

To convert from radians -> degrees and degrees -> radians respectively use math.degrees and math.radians

math.degrees(a)

#### Out: 57.29577951308232

math.radians(57.29577951308232)

#### Out: 1.0

## Python Simple Mathematical Operators: Inplace Operations

It is common within applications to need to have code like this:

a = a + 1

or

a = a * 2

There is an eﬀective shortcut for these in place operations:

a += 1

and

a *= 2

Any mathematic operator can be used before the ‘=’ character to make an inplace operation:

-= decrement the variable in place

+= increment the variable in place

*= multiply the variable in place

/= divide the variable in place

//= floor divide the variable in place # Python 3

%= return the modulus of the variable in place

**= raise to a power in place

Other in place operators exist for the bitwise operators (^, | etc)

## Python Simple Mathematical Operators: Subtraction

a, b = 1, 2

#### Using the “-” operator:

b - a # = 1

import operator # contains 2 argument arithmetic functions

operator.sub(b, a) # = 1

Possible combinations (builtin types):

int and int (gives an int)

int and float (gives a float)

int and complex (gives a complex)

float and float (gives a float)

float and complex (gives a complex)

complex and complex (gives a complex)

## Python Simple Mathematical Operators: Multiplication

a, b = 2, 3

a * b # = 6

import operator

operator.mul(a, b) # = 6

Possible combinations (builtin types):

int and int (gives an int)

int and float (gives a float)

int and complex (gives a complex)

float and float (gives a float)

float and complex (gives a complex)

complex and complex (gives a complex)

Note: The * operator is also used for repeated concatenation of strings, lists, and tuples:

3 * 'ab' # = 'ababab'

3 * ('a', 'b') # = ('a', 'b', 'a', 'b', 'a', 'b')

## Logarithms

By default, the math.log function calculates the logarithm of a number, base e. You can optionally specify a base as the second argument.

import math

import cmath

math.log(5) # = 1.6094379124341003

optional base argument. Default is math.e math.log(5, math.e) # = 1.6094379124341003

cmath.log(5) # = (1.6094379124341003+0j) math.log(1000, 10) # 3.0 (always returns float) cmath.log(1000, 10) # (3+0j)

Special variations of the math.log function exist for diﬀerent bases.

`Logarithm base e - 1 (higher precision for low values)`

math.log1p(5)# = 1.791759469228055

Logarithm base 2

math.log2(8) #=3.0

### Logarithm base 10

math.log10(100) # = 2.0

cmath.log10(100) # = (2+0j)

Section 9.9: Modulus

Like in many other languages, Python uses the % operator for calculating modulus.

3 % 4 # 3

10%2 # 0

6 % 4 # 2

Or by using the operator module:

import operator operator.mod(3 , 4) # 3

operator.mod(10 , 2) # 0

operator.mod(6 , 4) # 2

You can also use negative numbers.

-9 % 7 # 5

9%-7 # -5

-9 % -7 # -2

If you need to find the result of integer division and modulus, you can use the divmod function as a shortcut:

quotient, remainder = divmod(9, 4)