# 3.2. Numeric Float

• Represents floating point number (vide IEEE-754)

• Could be both signed and unsigned

• Default float size is 64 bit

• Python automatically extends float when need bigger number

>>> data = 1.337
>>> data = +1.337
>>> data = -1.337

Floating-point numbers are not real numbers, so the result of 1.0/3.0 cannot be represented exactly without infinite precision. In the decimal (base 10) number system, one-third is a repeating fraction, so it has an infinite number of digits. Even simple non-repeating decimal numbers can be a problem. One-tenth (0.1) is obviously non-repeating, so we can express it exactly with a finite number of digits. As it turns out, since numbers within computers are stored in binary (base 2) form, even one-tenth cannot be represented exactly with floating-point numbers,

When should you use integers and when should you use floating-point numbers? A good rule of thumb is this: use integers to count things and use floating-point numbers for quantities obtained from a measuring device. As examples, we can measure length with a ruler or a laser range finder; we can measure volume with a graduated cylinder or a flow meter; we can measure mass with a spring scale or triple-beam balance. In all of these cases, the accuracy of the measured quantity is limited by the accuracy of the measuring device and the competence of the person or system performing the measurement. Environmental factors such as temperature or air density can affect some measurements. In general, the degree of inexactness of such measured quantities is far greater than that of the floating-point values that represent them.

Despite their inexactness, floating-point numbers are used every day throughout the world to solve sophisticated scientific and engineering problems. The limitations of floating-point numbers are unavoidable since values with infinite characteristics cannot be represented in a finite way. Floating-point numbers provide a good trade-off of precision for practicality.

Note

Source [1]

## 3.2.1. Without Zero Notation

• .44 - notation without leading zero

• 69. - notation without trailing zero

• Used by numpy

>>> data = .44
>>> print(data)
0.44

Trailing zero:

>>> data = 69.
>>> print(data)
69.0

## 3.2.2. Engineering Notation

• The exponential is a number divisible by 3

• Allows the numbers to explicitly match their corresponding SI prefixes

• The E (or e) should not be confused with the exponential e which holds a completely different significance

Table 3.1. Engineering notation

Name

Symbol

Base

Value

yotta

Y

1e24

1000000000000000000000000.0

zetta

Z

1e21

1000000000000000000000.0

exa

E

1e18

1000000000000000000.0

peta

P

1e15

1000000000000000.0

tera

T

1e12

1000000000000.0

giga

G

1e9

1000000000.0

mega

M

1e6

1000000.0

kilo

k

1e3

1000.0

1e0

1.0

milli

m

1e−3

0.001.0

micro

μ

1e−6

0.000001.0

nano

n

1e−9

0.000000001.0

pico

p

1e−12

0.000000000001.0

femto

f

1e−15

0.000000000000001.0

atto

a

1e−18

0.000000000000000001.0

zepto

z

1e−21

0.000000000000000000001.0

yocto

y

1e−24

0.000000000000000000000001.0

>>> x = 1e6
>>> print(x)
1000000.0
>>>
>>> x = 1E6
>>> print(x)
1000000.0
>>> x = +1e6
>>> print(x)
1000000.0
>>>
>>> x = -1e6
>>> print(x)
-1000000.0
>>> x = 1e-3
>>> print(x)
0.001
>>>
>>> x = 1e-6
>>> print(x)
1e-06

## 3.2.3. Scientific notation

• The E (or e) should not be confused with the exponential e which holds a completely different significance

>>> 1e1
10.0
>>>
>>> 1e2
100.0
>>>
>>> 1e3
1000.0
>>> 1e-3
0.001
>>>
>>> 1e-4
0.0001
>>>
>>> 1e-5
1e-05
>>>
>>> 1e-6
1e-06
>>> 1e3
1000.0
>>>
>>> -1e3
-1000.0
>>>
>>> 1e-3
0.001
>>>
>>> -1e-3
-0.001
>>> 1.337 * 1e3
1337.0
>>>
>>> 1.337 * 1e-3
0.001337
>>> 1.337e3
1337.0
>>>
>>> 1.337e-3
0.001337
>>>
>>> 1.337e-4
0.0001337
>>> 1.337e-5
1.337e-05

## 3.2.4. Type Conversion

Builtin function float() converts argument to float

>>> float(1)
1.0
>>>
>>> float(+1)
1.0
>>>
>>> float(-1)
-1.0
>>> float(1.337)
1.337
>>>
>>> float(+1.337)
1.337
>>>
>>> float(-1.337)
-1.337
>>> float('1.337')
1.337
>>>
>>> float('+1.337')
1.337
>>>
>>> float('-1.337')
-1.337
>>> float('1,337')
Traceback (most recent call last):
ValueError: could not convert string to float: '1,337'
>>> float('+1,337')
Traceback (most recent call last):
ValueError: could not convert string to float: '+1,337'
>>> float('-1,337')
Traceback (most recent call last):
ValueError: could not convert string to float: '-1,337'

## 3.2.5. Round Number

Rounding a number

>>> pi = 3.14159265359
>>>
>>>
>>> round(pi, 4)
3.1416
>>>
>>> round(pi, 2)
3.14
>>>
>>> round(pi)
3
>>>
>>> round(pi, 0)
3.0

Rounding a number in string formatting

>>> pi = 3.14159265359
>>>
>>>
>>> print(f'Pi number is {pi}')
Pi number is 3.14159265359
>>>
>>> print(f'Pi number is {pi:f}')
Pi number is 3.141593
>>>
>>> print(f'Pi number is {pi:.4f}')
Pi number is 3.1416
>>>
>>> print(f'Pi number is {pi:.2f}')
Pi number is 3.14
>>>
>>> print(f'Pi number is {pi:.0f}')
Pi number is 3
>>> round(10.5)
10
>>>
>>> round(10.51)
11

## 3.2.6. Type Checking

>>> x = 1.2
>>>
>>> type(x)
<class 'float'>
>>>
>>> x = 1.2
>>> type(x) is float
True
>>>
>>> type(x) in (int, float)
True
>>> x = 1.2
>>>
>>> isinstance(x, float)
True
>>>
>>> isinstance(x, (int,float))
True
>>>
>>> isinstance(x, int|float)  # since 3.10
True

## 3.2.8. Assignments

"""
* Assignment: Type Float Tax
* Type: class assignment
* Complexity: easy
* Lines of code: 3 lines
* Time: 3 min

English:
1. Cost of the service is 1000.00 PLN net
2. Service has value added tax (VAT) rate of 23%
3. Calculate gross values (with tax)
4. Run doctests - all must succeed

Polish:
1. Cena usługi wynosi 1000.00 PLN netto
2. Usługa objęta jest 23% stawką VAT
3. Oblicz cenę brutto (z podatkiem)
4. Uruchom doctesty - wszystkie muszą się powieść

Tests:
>>> import sys; sys.tracebacklimit = 0
>>> from pprint import pprint

>>> assert result is not Ellipsis, \
'Assign your result to variable result'
>>> assert type(result) is float, \
'Variable result has invalid type, should be float'

>>> pprint(result)
1230.0

"""

PLN = 1.00
VAT_23 = 23 / 100

PRICE = 1000.0 * PLN
TAX = PRICE * VAT_23

# Gross is net plus tax in PLN
# type: float
result = ...

"""
* Assignment: Type Float Tax
* Type: class assignment
* Complexity: easy
* Lines of code: 1 lines
* Time: 2 min

English:
1. Price of the service is 100.00 USD
2. Define result: float with price in cents
3. Run doctests - all must succeed

Polish:
1. Cena usługi wynosi 100.00 USD
2. Zdefiniuj result: float ceną w centach
3. Uruchom doctesty - wszystkie muszą się powieść

Tests:
>>> import sys; sys.tracebacklimit = 0
>>> from pprint import pprint

>>> assert result is not Ellipsis, \
'Assign your result to variable result'
>>> assert type(result) is float, \
'Variable result has invalid type, should be float'

>>> pprint(result)
10000.0
"""

DOLLAR = 1
CENT = DOLLAR / 100

PRICE = 100*DOLLAR

# PRICE in US cents
# type: float
result = ...

"""
* Assignment: Type Float Tax
* Type: class assignment
* Complexity: easy
* Lines of code: 4 lines
* Time: 3 min

English:
1. Cost of the service is 100.00 EUR
2. Calculate price in Polish Zloty PLN (PLN)
3. Calculate price in US Dollars (USD)
4. Calculate price in Australian Dollars (AUD)
3. Run doctests - all must succeed

Polish:
1. Cena usługi wynosi 100.00 EUR
2. Oblicz kwotę w polskich złotych (PLN)
3. Oblicz kwotę w dolarach amerykańskich (USD)
4. Oblicz kwotę w dolarach australijskich (AUD)
3. Uruchom doctesty - wszystkie muszą się powieść

Tests:
>>> import sys; sys.tracebacklimit = 0
>>> from pprint import pprint

>>> assert result_a is not Ellipsis, \
'Assign your result to variable result_a'
>>> assert type(result_a) is float, \
'Variable result_a has invalid type, should be float'

>>> assert result_b is not Ellipsis, \
'Assign your result to variable result_b'
>>> assert type(result_b) is float, \
'Variable result_b has invalid type, should be float'

>>> assert result_c is not Ellipsis, \
'Assign your result to variable result_c'
>>> assert type(result_c) is float, \
'Variable result_c has invalid type, should be float'

>>> assert result_d is not Ellipsis, \
'Assign your result to variable result_d'
>>> assert type(result_d) is float, \
'Variable result_d has invalid type, should be float'

>>> result = round(result_a, 1)
>>> pprint(result)
435.0

>>> result = round(result_b, 1)
>>> pprint(result)
110.0

>>> result = round(result_c, 1)
>>> pprint(result)
166.0

>>> result = round(result_d, 1)
>>> pprint(result)
149.0
"""

EUR = 1
PLN = EUR / 4.35
USD = EUR / 1.10
AUD = EUR / 1.66

PRICE = 100*EUR

# PRICE in Polish Zloty PLN (PLN)
# type: float
result_a = ...

# PRICE in US Dollars (USD)
# type: float
result_b = ...

# PRICE in Australian Dollars (AUD)
# type: float
result_c = ...

# type: float
result_d = ...

"""
* Assignment: Type Float Tax
* Type: class assignment
* Complexity: easy
* Lines of code: 1 lines
* Time: 3 min

English:
1. Cost of the service is 1000.00 PLN net
2. Service has value added tax (VAT) rate of 23%
3. Calculate VAT tax
4. Run doctests - all must succeed

Polish:
1. Cena usługi wynosi 1000.00 PLN netto
2. Usługa objęta jest 23% stawką VAT
3. Oblicz wartości podatku VAT
4. Uruchom doctesty - wszystkie muszą się powieść

Tests:
>>> import sys; sys.tracebacklimit = 0
>>> from pprint import pprint

>>> assert result is not Ellipsis, \
'Assign your result to variable result'
>>> assert type(result) is float, \
'Variable result has invalid type, should be float'

>>> pprint(result)
230.0
"""

PLN = 1.00
VAT_23 = 0.23

PRICE = 1000.00*PLN

# VAT tax for 1000.00 PLN net price
# type: float
result = ...

"""
* Assignment: Type Float Altitude
* Type: class assignment
* Complexity: easy
* Lines of code: 2 lines
* Time: 3 min

English:
1. Plane altitude is 10 km
2. Convert to feet (ft) in imperial system (US)
3. Convert to meters (m) in metric system (SI)
4. Run doctests - all must succeed

Polish:
1. Wysokość lotu samolotem wynosi 10 km
2. Przelicz je na stopy (ft) w systemie imperialnym (US)
3. Przelicz je na metry (m) w systie metrycznym (układ SI)
4. Uruchom doctesty - wszystkie muszą się powieść

Tests:
>>> import sys; sys.tracebacklimit = 0
>>> from pprint import pprint

>>> assert altitude_m is not Ellipsis, \
'Assign your result to variable altitude_m'
>>> assert altitude_ft is not Ellipsis, \
'Assign your result to variable altitude_ft'
>>> assert type(altitude_m) is float, \
'Variable altitude_m has invalid type, should be float'
>>> assert type(altitude_ft) is float, \
'Variable altitude_ft has invalid type, should be float'

>>> result = round(altitude_m, 1)
>>> pprint(result)
10000.0

>>> result = round(altitude_ft, 1)
>>> pprint(result)
32808.4
"""

m = 1
km = 1000 * m
ft = 0.3048 * m

ALTITUDE = 10*km

# ALTITUDE in meters
# type: float
altitude_m = ...

# ALTITUDE in feet
# type: float
altitude_ft = ...

## 3.2.9. Homework

"""
* Assignment: Type Float Volume
* Type: class assignment
* Complexity: easy
* Lines of code: 2 lines
* Time: 3 min

English:
1. Bottle volume is 500 mililiter
2. Convert to fluid unces (floz) in imperial system (US)
3. Convert to liters (liter) in metric system (SI)
4. Run doctests - all must succeed

Polish:
1. Objętość butelki wynosi 500 mililitrów
2. Przelicz je na uncje płynu (floz) w systemie imperialnym (US)
3. Przelicz je na litry (liter) w systie metrycznym (układ SI)
4. Uruchom doctesty - wszystkie muszą się powieść

Tests:
>>> import sys; sys.tracebacklimit = 0
>>> from pprint import pprint

>>> assert volume_floz is not Ellipsis, \
'Assign your result to variable volume_floz'
>>> assert volume_l is not Ellipsis, \
'Assign your result to variable volume_l'
>>> assert type(volume_floz) is float, \
'Variable volume_floz has invalid type, should be float'
>>> assert type(volume_l) is float, \
'Variable volume_l has invalid type, should be float'

>>> result = round(volume_floz, 1)
>>> pprint(result)
16.9

>>> result = round(volume_l, 1)
>>> pprint(result)
0.5
"""

liter = 1
milliliter = 0.001 * liter
floz = 0.02957344 * liter

VOLUME = 500 * milliliter

# VOLUME in fluid ounces
# type: float
volume_floz = ...

# VOLUME in liters
# type: float
volume_l = ...

"""
* Assignment: Type Float Round
* Type: class assignment
* Complexity: easy
* Lines of code: 4 lines
* Time: 5 min

English:
1. Euler's number is 2.71828
2. Round number using round() funtion
3. Run doctests - all must succeed

Polish:
1. Liczba Eulra to 2.71828
2. Zaokrąglij liczbę wykorzystując funkcję round()
3. Uruchom doctesty - wszystkie muszą się powieść

Tests:
>>> import sys; sys.tracebacklimit = 0
>>> from pprint import pprint

>>> assert a is not Ellipsis, \
'Assign your result to variable a'
>>> assert b is not Ellipsis, \
'Assign your result to variable b'
>>> assert c is not Ellipsis, \
'Assign your result to variable c'
>>> assert d is not Ellipsis, \
'Assign your result to variable d'
>>> assert e is not Ellipsis, \
'Assign your result to variable e'

>>> assert type(a) is float, \
'Variable a has invalid type, should be float'
>>> assert type(b) is float, \
'Variable b has invalid type, should be float'
>>> assert type(c) is float, \
'Variable c has invalid type, should be float'
>>> assert type(d) is float, \
'Variable d has invalid type, should be float'
>>> assert type(e) is int, \
'Variable e has invalid type, should be int'

>>> pprint(a)
2.718

>>> pprint(b)
2.72

>>> pprint(c)
2.7

>>> pprint(d)
3.0

>>> pprint(e)
3

"""

EULER = 2.71828

# Round Euler's number to 3 decimal places
# Use round()
# type: float
a = ...

# Round Euler's number to 2 decimal places
# Use round()
# type: float
b = ...

# Round Euler's number to 1 decimal places
# Use round()
# type: float
c = ...

# Round Euler's number to 0 decimal places
# Use round()
# type: float
d = ...

# Round Euler's number to nearest integer
# Use round()
# type: int
e = ...

"""
* Assignment: Type Float Round
* Type: class assignment
* Complexity: easy
* Lines of code: 4 lines
* Time: 5 min

English:
1. Euler's number is 2.71828
2. Round number using f-string formatting,
example: f'{number:.2f}'
3. Run doctests - all must succeed

Polish:
1. Liczba Eulra to 2.71828
2. Zaokrąglij liczbę wykorzystując formatowanie f-string,
3. Uruchom doctesty - wszystkie muszą się powieść

Tests:
>>> import sys; sys.tracebacklimit = 0
>>> from pprint import pprint

>>> assert a is not Ellipsis, \
'Assign your result to variable a'
>>> assert b is not Ellipsis, \
'Assign your result to variable b'
>>> assert c is not Ellipsis, \
'Assign your result to variable c'
>>> assert d is not Ellipsis, \
'Assign your result to variable d'

>>> assert type(a) is str, \
'Variable a has invalid type, should be str'
>>> assert type(b) is str, \
'Variable b has invalid type, should be str'
>>> assert type(c) is str, \
'Variable c has invalid type, should be str'
>>> assert type(d) is str, \
'Variable d has invalid type, should be str'

>>> pprint(a)
'Result: 2.718'

>>> pprint(b)
'Result: 2.72'

>>> pprint(c)
'Result: 2.7'

>>> pprint(d)
'Result: 3'

"""

EULER = 2.71828

# Round Euler's number to 3 decimal places
# Round number using f-string formatting
# type: str
a = f"Result: {EULER}"

# Round Euler's number to 2 decimal places
# Round number using f-string formatting
# type: str
b = f"Result: {EULER}"

# Round Euler's number to 1 decimal places
# Round number using f-string formatting
# type: str
c = f"Result: {EULER}"

# Round Euler's number to 0 decimal places
# Round number using f-string formatting
# type: str
d = f"Result: {EULER}"