>>>round(2.5)
2

>>>round(1.5)
2

• Truncating
• Rounding Up
• Rounding Down

def truncate(n, decimals=0):
    multiplier = 10**decimals
    return int(n * multiplier) / multiplier

>>> from rounding import truncate

>>> truncate(12.5)
12.0

>>> truncate(-5.963, 1)
-5.9

>>> truncate(1.625, 2)
1.62

>>> truncate(125.6, -1)
120.0

>>> truncate(-1374.25, -3)
-1000.0

The ceil() function gets its name from the mathematical term ceiling
ceiling describes the nearest integer which is greater than or equal to a given number

>>> import math

>>> math.ceil(3.7)
4

>>> math.ceil(2)
2

>>> math.ceil(-0.5)
0

import math

def round_up(n, decimals=0):
    multiplier = 10**decimals
    return math.ceil(n * multiplier) / multiplier

>>> round_up(22.45, -1)
30.0

>>> round_up(1352, -2)
1400.0

the math module has a floor() function

>>> import math

>>> math.floor(1.2)
1

>>> math.floor(-0.5)
-1

# ...

def round_down(n, decimals=0):
    multiplier = 10**decimals
    return math.floor(n * multiplier) / multiplier

>>> from rounding import round_down

>>> round_down(1.5)
1.0

>>> round_down(1.37, 1)
1.3

>>> round_down(-0.5)
-1.0

rounding bias describes how rounding affects numeric data in a dataset
the rounding up strategy has a round toward positive infinity bias
the rounding down strategy has a round toward negative infinity bias
the truncation strategy exhibits a round toward negative infinity bias on positive values and a round toward positive infinity for negative values
this behavior are said to have a round toward zero bias

import statistics
from rounding import truncate, round_up, round_down

numbers = [1.25, -2.67, 0.43, -1.79, 8.19, -4.32]

statistics.mean(numbers)
#0.18166666666666653

[truncate(n, 1) for n in numbers]
# [1.2, -2.6, 0.4, -1.7, 8.1, -4.3]

statistics.mean(truncate(n, 1) for n in numbers)
# 0.1833333333333333
statistics.mean(round_up(n, 1) for n in numbers)
# 0.23333333333333325
statistics.mean(round_down(n, 1) for n in numbers)
# 0.13333333333333316

• Rounding Half Up
• Rounding Half Down
• Rounding Half Away From Zero
• Rounding Half to Even

below is the implementation of the round_half_up() function

• if the digit in the first decimal place of the shifted value is less than 5, then adding 0.5 won't change the integer part of the shifted value
  the floor is equal to the integer part.
• if the first digit after the decimal place is greater than or equal to 5, then adding 0.5 will increase the integer part of the shifted value by 1
  the floor is equal to this larger integer.

# ...

def round_half_up(n, decimals=0):
    multiplier = 10**decimals
    return math.floor(n * multiplier + 0.5) / multiplier

very similar to the round_half_up() function
differences

• replacing math.floor() with math.ceil()
• subtracting 0.5 instead of adding 0.5

# ...

def round_half_down(n, decimals=0):
    multiplier = 10**decimals
    return math.ceil(n * multiplier - 0.5) / multiplier

>>> import statistics

>>> numbers = [-2.15, 1.45, -4.35, 12.75]
>>> statistics.mean(numbers)
1.925
>>> statistics.mean(round_half_up(n, 1) for n in numbers)
1.975
>>> statistics.mean(round_half_down(n, 1) for n in numbers)
1.8749999999999996

1. start as usual by shifting the decimal point to the right a given number of places
2. look at the digit d immediately to the right of the decimal place in this new number
   four cases to consider
   1. if n is positive and d >= 5, round up
   2. if n is positive and d < 5, round down
   3. if n is negative and d >= 5, round down
   4. if n is negative and d < 5, round up
3. shift the decimal place back to the left

if n >= 0:
    rounded = round_half_up(n, decimals)
else:
    rounded = round_half_down(n, decimals)

>>> import math

>>> math.copysign(1, -2)
-1.0

# ...

def round_half_away_from_zero(n, decimals=0):
    rounded_abs = round_half_up(abs(n), decimals)
    return math.copysign(rounded_abs, n)

>>> numbers = [-2.15, 1.45, -4.35, 12.75]

>>> import statistics
>>> statistics.mean(numbers)
1.925

>>> statistics.mean(round_half_away_from_zero(n, 1) for n in numbers)
1.925

the rounding half to even strategy is used by Python's built-in round() function

decimal.getcontext() returns a Context object representing the default context of the decimal module

>>> import decimal
>>> decimal.getcontext()
Context(
    prec=28,
    rounding=ROUND_HALF_EVEN,
    Emin=-999999,
    Emax=999999,
    capitals=1,
    clamp=0,
    flags=[],
    traps=[InvalidOperation, DivisionByZero, Overflow]
)

>>> from decimal import Decimal
>>> Decimal("0.1")
Decimal('0.1')

>>> Decimal(0.1)
Decimal('0.1000000000000000055511151231257827021181583404541015625')

>>> Decimal("1.65").quantize(Decimal("1.0"))
Decimal('1.6')

>>> round(Decimal("1.65"), 1)
Decimal('1.6')

decimal module automatically takes care of rounding after performing arithmetic preserving significant digits

>>> decimal.getcontext().prec = 2
>>> Decimal("1.23") + Decimal("2.32")
Decimal('3.6')

to change the default rounding strategy set the decimal.getcontext().rounding property to any one of several flags

the decimal.ROUND_CEILING strategy works just like the round_up() function
the results of decimal.ROUND_CEILING aren't symmetric around zero

>>> decimal.getcontext().rounding = decimal.ROUND_CEILING

>>> round(Decimal("1.32"), 1)
Decimal('1.4')

>>> round(Decimal("-1.32"), 1)
Decimal('-1.3')

the decimal.ROUND_FLOOR strategy works just like the round_down() function
the results of decimal.ROUND_FLOOR aren't symmetric around zero

>>> decimal.getcontext().rounding = decimal.ROUND_FLOOR

>>> round(Decimal("1.32"), 1)
Decimal('1.3')

>>> round(Decimal("-1.32"), 1)
Decimal('-1.4')

the decimal.ROUND_DOWN and decimal.ROUND_UP strategies have deceptive names
both ROUND_DOWN and ROUND_UP are symmetric around zero
the decimal.ROUND_DOWN strategy rounds numbers toward zero like the truncate() function
decimal.ROUND_UP rounds everything away from zero

>>> decimal.getcontext().rounding = decimal.ROUND_DOWN

>>> round(Decimal("1.32"), 1)
Decimal('1.3')

>>> round(Decimal("-1.32"), 1)
Decimal('-1.3')

>>> decimal.getcontext().rounding = decimal.ROUND_UP

>>> round(Decimal("1.32"), 1)
Decimal('1.4')

>>> round(Decimal("-1.32"), 1)
Decimal('-1.4')

the decimal.ROUND_HALF_UP method rounds everything to the nearest number
breaks ties by rounding away from zero
decimal.ROUND_HALF_UP works just like round_half_away_from_zero() and not like round_half_up()

>>> decimal.getcontext().rounding = decimal.ROUND_HALF_UP

>>> round(Decimal("1.35"), 1)
Decimal('1.4')

>>> round(Decimal("-1.35"), 1)
Decimal('-1.4')

decimal.ROUND_HALF_DOWN strategy that breaks ties by rounding toward zero

>>> decimal.getcontext().rounding = decimal.ROUND_HALF_DOWN

>>> round(Decimal("1.35"), 1)
Decimal('1.3')

>>> round(Decimal("-1.35"), 1)
Decimal('-1.3')

decimal.ROUND_05UP

>>> decimal.getcontext().rounding = decimal.ROUND_05UP

>>> round(Decimal("1.38"), 1)
Decimal('1.3')

>>> round(Decimal("1.35"), 1)
Decimal('1.3')

>>> round(Decimal("-1.35"), 1)
Decimal('-1.3')

>>> round(Decimal("1.49"), 1)
Decimal('1.4')

>>> round(Decimal("1.51"), 1)
Decimal('1.6')