3.2. Math Stdlib¶
3.2.1. Constans¶
inforInfinity-infor-Infinity1e6or1e-4
3.2.2. Functions¶
abs()round()pow()sum()min()max()divmod()complex()
3.2.3. math¶
3.2.4. Constants¶
import math
math.pi # The mathematical constant π = 3.141592..., to available precision.
math.e # The mathematical constant e = 2.718281..., to available precision.
math.tau # The mathematical constant τ = 6.283185..., to available precision.
3.2.5. Degree/Radians Conversion¶
import math
math.degrees(x) # Convert angle x from radians to degrees.
math.radians(x) # Convert angle x from degrees to radians.
3.2.6. Rounding to lower¶
import math
math.floor(3.14) # 3
math.floor(3.00000000000000) # 3
math.floor(3.00000000000001) # 3
math.floor(3.99999999999999) # 3
3.2.7. Rounding to higher¶
import math
math.ceil(3.14) # 4
math.ceil(3.00000000000000) # 3
math.ceil(3.00000000000001) # 4
math.ceil(3.99999999999999) # 4
3.2.8. Logarithms¶
import math
math.log(x) # if base is not set, then ``e``
math.log(x, base=2)
math.log(x, base=10)
math.log10() # Return the base-10 logarithm of x. This is usually more accurate than log(x, 10).
math.log2(x) # Return the base-2 logarithm of x. This is usually more accurate than log(x, 2).
math.exp(x)
3.2.9. Linear Algebra¶
import math
math.sqrt(x) # Return the square root of x.
math.pow(x, y) # Return x raised to the power y.
import math
math.hypot(*coordinates) # 2D, since Python 3.8 also multiple dimensions
math.dist(p, q) # Euclidean distance, since Python 3.8
math.gcd(*integers) # Greatest common divisor
math.lcm(*integers) # Least common multiple, since Python 3.9
math.perm(n, k=None) # Return the number of ways to choose k items from n items without repetition and with order.
math.prod(iterable, *, start=1) # Calculate the product of all the elements in the input iterable. The default start value for the product is 1., since Python 3.8
math.remainder(x, y) # Return the IEEE 754-style remainder of x with respect to y.
3.2.10. Trigonometry¶
import math
math.sin()
math.cos()
math.tan()
math.asin(x)
math.acos(x)
math.atan(x)
math.atan2(x)
Hyperbolic functions:
import math
math.sinh() # Return the hyperbolic sine of x.
math.cosh() # Return the hyperbolic cosine of x.
math.tanh() # Return the hyperbolic tangent of x.
math.asinh(x) # Return the inverse hyperbolic sine of x.
math.acosh(x) # Return the inverse hyperbolic cosine of x.
math.atanh(x) # Return the inverse hyperbolic tangent of x.
3.2.11. Infinity¶
float('inf') # inf
float('-inf') # -inf
float('Infinity') # inf
float('-Infinity') # -inf
from math import isinf
isinf(float('inf')) # True
isinf(float('Infinity')) # True
isinf(float('-inf')) # True
isinf(float('-Infinity')) # True
isinf(1e308) # False
isinf(1e309) # True
isinf(1e-9999999999999999) # False
3.2.12. Absolute value¶
abs(1) # 1
abs(-1) # 1
abs(1.2) # 1.2
abs(-1.2) # 1.2
from math import fabs
fabs(1) # 1.0
fabs(-1) # 1.0
fabs(1.2) # 1.2
fabs(-1.2) # 1.2
from math import fabs
vector = [1, 0, 1]
abs(vector)
# Traceback (most recent call last):
# TypeError: bad operand type for abs(): 'list'
fabs(vector)
# Traceback (most recent call last):
# TypeError: must be real number, not list
from math import sqrt
def vector_abs(vector):
return sqrt(sum(n**2 for n in vector))
vector = [1, 0, 1]
vector_abs(vector)
# 1.4142135623730951
from math import sqrt
class Vector:
def __init__(self, x, y, z):
self.x = x
self.y = y
self.z = z
def __abs__(self):
return sqrt(self.x**2 + self.y**2 + self.z**2)
vector = Vector(x=1, y=0, z=1)
abs(vector)
# 1.4142135623730951
3.2.13. Assignments¶
"""
* Assignment: Math Trigonometry Deg2Rad
* Complexity: easy
* Lines of code: 10 lines
* Time: 5 min
English:
1. Read input (angle in degrees) from user
2. User will type `int` or `float`
3. Print all trigonometric functions (sin, cos, tg, ctg)
4. If there is no value for this angle, raise an exception
5. Round results to two decimal places
6. Run doctests - all must succeed
Polish:
1. Program wczytuje od użytkownika wielkość kąta w stopniach
2. Użytkownik zawsze podaje `int` albo `float`
3. Wyświetl wartość funkcji trygonometrycznych (sin, cos, tg, ctg)
4. Jeżeli funkcja trygonometryczna nie istnieje dla danego kąta podnieś
stosowny wyjątek
5. Wyniki zaokrąglij do dwóch miejsc po przecinku
6. Uruchom doctesty - wszystkie muszą się powieść
Tests:
>>> import sys; sys.tracebacklimit = 0
>>> result_sin
0.02
>>> result_cos
1.0
>>> result_tg
0.02
>>> result_ctg
57.29
>>> result_pi
3.14
"""
import math
from unittest.mock import MagicMock
PRECISION = 2
# Simulate user input (for test automation)
input = MagicMock(side_effect=['1'])
degrees = input('What is the angle [deg]?: ')
result_sin = ...
result_cos = ...
result_tg = ...
result_ctg = ...
result_pi = ...
"""
* Assignment: Math Algebra Distance2D
* Complexity: easy
* Lines of code: 5 lines
* Time: 8 min
English:
1. Given are two points `A: tuple[int, int]` and `B: tuple[int, int]`
2. Coordinates are in cartesian system
3. Points `A` and `B` are in two dimensional space
4. Calculate distance between points using Euclidean algorithm
5. Run doctests - all must succeed
Polish:
1. Dane są dwa punkty `A: tuple[int, int]` i `B: tuple[int, int]`
2. Koordynaty są w systemie kartezjańskim
3. Punkty `A` i `B` są w dwuwymiarowej przestrzeni
4. Oblicz odległość między nimi wykorzystując algorytm Euklidesa
5. Uruchom doctesty - wszystkie muszą się powieść
Tests:
>>> A = (1, 0)
>>> B = (0, 1)
>>> result(A, B)
1.4142135623730951
>>> result((0,0), (1,0))
1.0
>>> result((0,0), (1,1))
1.4142135623730951
>>> result((0,1), (1,1))
1.0
>>> result((0,10), (1,1))
9.055385138137417
"""
from math import sqrt
# type: point = tuple[int,...]
# type: Callable[[point, point], float]
def result(A, B):
...
"""
* Assignment: Math Algebra DistanceND
* Complexity: easy
* Lines of code: 10 lines
* Time: 5 min
English:
1. Given are two points `A: Sequence[int]` and `B: Sequence[int]`
2. Coordinates are in cartesian system
3. Points `A` and `B` are in `N`-dimensional space
4. Points `A` and `B` must be in the same space
5. Calculate distance between points using Euclidean algorithm
6. Run doctests - all must succeed
Polish:
1. Dane są dwa punkty `A: Sequence[int]` i `B: Sequence[int]`
2. Koordynaty są w systemie kartezjańskim
3. Punkty `A` i `B` są w `N`-wymiarowej przestrzeni
4. Punkty `A` i `B` muszą być w tej samej przestrzeni
5. Oblicz odległość między nimi wykorzystując algorytm Euklidesa
6. Uruchom doctesty - wszystkie muszą się powieść
Hints:
* `for n1, n2 in zip(A, B)`
Tests:
>>> result((0,0,0), (0,0,0))
0.0
>>> result((0,0,0), (1,1,1))
1.7320508075688772
>>> result((0,1,0,1), (1,1,0,0))
1.4142135623730951
>>> result((0,0,1,0,1), (1,1,0,0,1))
1.7320508075688772
>>> result((0,0,1,0,1), (1,1))
Traceback (most recent call last):
ValueError: Points must be in the same dimensions
"""
from math import sqrt
# type: point = tuple[int,...]
# type: Callable[[point, point], float]
def result(A, B):
...
Figure 3.19. Calculate Euclidean distance in Cartesian coordinate system¶
- Hints:
\(distance(a, b) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2\)
\(distance(a, b) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + ... + (n_2 - n_1)^2}\)
"""
* Assignment: Math Algebra Matmul
* Complexity: hard
* Lines of code: 13 lines
* Time: 21 min
English:
1. Multiply matrices using nested `for` loops
2. Do not use any library, such as: `numpy`, `pandas`, itp
3. Run doctests - all must succeed
Polish:
1. Pomnóż macierze wykorzystując zagnieżdżone pętle `for`
2. Nie wykorzystuj żadnej biblioteki, tj.: `numpy`, `pandas`, itp
3. Uruchom doctesty - wszystkie muszą się powieść
Hints:
* Zero matrix
* Three nested `for` loops
Tests:
>>> A = [[1, 0],
... [0, 1]]
>>>
>>> B = [[4, 1],
... [2, 2]]
>>>
>>> result(A, B)
[[4, 1], [2, 2]]
>>> A = [[1,0,1,0],
... [0,1,1,0],
... [3,2,1,0],
... [4,1,2,0]]
>>>
>>> B = [[4,1],
... [2,2],
... [5,1],
... [2,3]]
>>>
>>> result(A, B)
[[9, 2], [7, 3], [21, 8], [28, 8]]
"""
# type: matrix = list[list[int]]
# type: Callable[[matrix, matrix], matrix]
def result(A, B):
...