math moduli: ildiz, π va trigonometriya

4-bob 35 kod bloki 3158 so'z

Boshlanishiga: Toshkentdan Samarqandgacha necha km?

Ikki shaharning xaritadagi koordinatalarini (kenglik va uzunlik) bilsak (Toshkent: 41.3111, 69.2797; Samarqand: 39.6270, 66.9750), ular orasidagi to'g'ri chiziqli masofani hisoblay olamiz. Yer shar shaklida bo'lgani uchun bu yerda trigonometriya kerak bo'ladi. Formulani hozir yodlash shart emas, bob oxirida undagi har bir funksiyani tushunasiz.

import math

f1 = math.radians(41.3111)
l1 = math.radians(69.2797)
f2 = math.radians(39.6270)
l2 = math.radians(66.9750)

h = math.sin((f2 - f1) / 2) ** 2 + math.cos(f1) * math.cos(f2) * math.sin((l2 - l1) / 2) ** 2
masofa = 2 * 6371 * math.asin(math.sqrt(h))
print(f"Toshkent - Samarqand: {masofa:.0f} km (to'g'ri chiziq bo'yicha)")

Toshkent - Samarqand: 270 km (to'g'ri chiziq bo'yicha)

Yo'l bo'yicha masofa ko'proq (taxminan 300 km), chunki yo'l to'g'ri emas. Bu formula haversine deyiladi: navigatorlar, taksi va yetkazib berish ilovalari aynan shunday hisoblaydi.

Modul nima va uni qanday ulash kerak?

Modul — tayyor funksiyalar to'plami. Python bilan birga yuzlab modul keladi, math — shulardan biri. Uni ishlatishdan oldin import qilish kerak. Uch xil usul bor:

import math
print(math.sqrt(16))

4.0

from math import sqrt, pi
print(sqrt(16), pi)

4.0 3.141592653589793

from math import *
print(sqrt(16), floor(2.7))

4.0 2

print(mat.sqrt(25))

NameError: name 'mat' is not defined

Ildiz va daraja

import math
print(math.sqrt(49))
print(math.sqrt(2))
print(49 ** 0.5)

7.0
1.4142135623730951
7.0

math.sqrt() har doim float qaytaradi, hatto ildiz butun chiqsa ham (7.0). Butun ildiz kerak bo'lsa: math.isqrt(50) → 7 (kasrini tashlaydi).

Manfiy sondan ildiz olib bo'lmaydi:

import math
print(math.sqrt(-9))

ValueError: math domain error

math domain error — "matematik aniqlanish sohasidan tashqari". Xuddi shunday xato math.log(0) va math.log(-1) da ham chiqadi.

π, e va boshqa konstantalar

import math
print(math.pi)
print(math.e)
print(math.tau)

3.141592653589793
2.718281828459045
6.283185307179586

math.tau = 2π. Mashqlarda π = 3.14 deb hisoblash so'ralsa, 3.14 yozing. Aks holda doim math.pi ishlating, u ancha aniq.

import math
r = 5
print("3.14 bilan:", 2 * 3.14 * r)
print("math.pi bilan:", 2 * math.pi * r)

3.14 bilan: 31.400000000000002
math.pi bilan: 31.41592653589793

Yaxlitlashning to'rt yo'li

Bu jadval ko'p savolga javob beradi. Ikki son: 2.7 va -2.7.

import math
x = 2.7
print(math.floor(x), math.ceil(x), math.trunc(x), int(x), round(x))
x = -2.7
print(math.floor(x), math.ceil(x), math.trunc(x), int(x), round(x))

2 3 2 2 3
-3 -2 -2 -2 -3

Chiqish tartibi: floor, ceil, trunc, int, round.

import math
talaba = 45
print(talaba / 20)
print(math.ceil(talaba / 20), "ta avtobus")

2.25
3 ta avtobus

Modul, faktorial, EKUB

import math
print(abs(-7.5))
print(math.factorial(5))
print(math.gcd(48, 36))
print(math.lcm(4, 6))

7.5
120
12
12

• abs() — modul, math siz ham ishlaydi.
• factorial(5) = 1·2·3·4·5 = 120.
• gcd — eng katta umumiy bo'luvchi (EKUB), lcm — eng kichik umumiy karrali (EKUK).

Masofa va gipotenuza

To'g'ri burchakli uchburchakning katetlari 3 va 4 bo'lsa, gipotenuza √(3² + 4²) = 5. math.hypot() buni bitta chaqiruvda qiladi:

import math
print(math.sqrt(3 ** 2 + 4 ** 2))
print(math.hypot(3, 4))

5.0
5.0

Tekislikdagi ikki nuqta orasidagi masofa — xuddi shu formula: √((x₂−x₁)² + (y₂−y₁)²). Uni math.dist() ham hisoblaydi:

import math
print(math.hypot(4 - 1, 6 - 2))
print(math.dist((1, 2), (4, 6)))

5.0
5.0

Trigonometriya: radian va gradus

math.sin(), math.cos(), math.tan() burchakni radianda kutadi, gradusda emas. Bu eng ko'p uchraydigan xato:

import math
print(math.sin(30))
print(math.sin(math.radians(30)))

-0.9880316240928618
0.49999999999999994

sin(30°) = 0.5 bo'lishi kerak. Birinchi qatorda Python 30 radian ning sinusini hisobladi va -0.988 chiqdi. Ikkinchi qatorda burchak avval math.radians() bilan radianga o'girildi va 0.49999999999999994 chiqdi — bu 0.5, faqat kasr sonlarning kichik xatosi bilan (pastdagi eslatmaga qarang).

Teskarisi — math.degrees():

import math
print(math.radians(180))
print(math.degrees(math.pi / 2))

3.141592653589793
90.0

import math
print(math.sin(math.pi))
print(round(math.sin(math.pi), 10))

1.2246467991473532e-16
0.0

Logarifm va eksponenta

import math
print(math.log10(1000))
print(math.log2(1024))
print(math.log(math.e))
print(math.exp(1))

3.0
10.0
1.0
2.718281828459045

math.log(x) — natural logarifm (asosi e). Boshqa asos uchun ikkinchi qiymat beriladi: math.log(81, 3) = 4.0.

Kasrlarni to'g'ri solishtirish: isclose

3-bobda abs(x - 0.3) < 1e-9 deb yozgan edik. math da buning tayyor funksiyasi bor:

import math
print(0.1 + 0.2 == 0.3)
print(math.isclose(0.1 + 0.2, 0.3))

False
True

min, max, sum

Bular math ga kirmaydi, Pythonning o'zida bor, lekin hisob-kitobda ko'p kerak:

print(min(7, 3, 9))
print(max(7, 3, 9))
print(sum((7, 3, 9)))

3
9
19

sum() ga sonlarni qavs ichida to'plam sifatida beramiz. Ro'yxatlarni 9-bobda o'rganamiz.

random: tasodifiy sonlar

O'yinlar, test savollarini aralashtirish, tasodifiy g'olib tanlash uchun random moduli bor:

import random
random.seed(7)
print(random.randint(1, 6))
print(random.randint(1, 6))
print(random.random())
print(random.choice(("olma", "anor", "behi")))

3
2
0.3948234964231735
olma

• randint(1, 6) — 1 dan 6 gacha (6 ham kiradi) butun son, xuddi o'yin kubigi.
• random() — 0 va 1 orasida kasr son.
• choice(...) — berilganlardan bittasini tanlaydi.
• seed(7) — tasodifni takrorlanadigan qiladi. Kitobda natija har safar bir xil chiqishi uchun qo'ydik. O'zingiz sinab ko'rganda seed qatorini olib tashlang, har safar boshqacha chiqadi.

Kichik loyiha: kvadrat tenglama

ax² + bx + c = 0 tenglamani yechamiz. Diskriminant D = b² − 4ac, ildizlar x = (−b ± √D) / 2a.

import math

a = float(input("a = "))
b = float(input("b = "))
c = float(input("c = "))

d = b ** 2 - 4 * a * c
x1 = (-b + math.sqrt(d)) / (2 * a)
x2 = (-b - math.sqrt(d)) / (2 * a)
print(f"D = {d}")
print(f"x1 = {x1}, x2 = {x2}")

a = 1
b = -5
c = 6
D = 1.0
x1 = 3.0, x2 = 2.0

Tekshiramiz: x² − 5x + 6 = (x − 2)(x − 3), ildizlar 3 va 2. To'g'ri.

(2 * a) dagi qavsga e'tibor bering. Usiz / 2 * a bo'lib qoladi: avval 2 ga bo'linadi, keyin a ga ko'paytiriladi.

Endi D manfiy bo'lgan tenglamani beraylik:

import math

a = float(input("a = "))
b = float(input("b = "))
c = float(input("c = "))

d = b ** 2 - 4 * a * c
x1 = (-b + math.sqrt(d)) / (2 * a)

a = 1
b = 2
c = 5

ValueError: math domain error

D = 4 − 20 = −16, manfiy sondan ildiz yo'q. Dastur yiqildi. To'g'ri dastur avval D ni tekshirib, "haqiqiy ildiz yo'q" deb javob berishi kerak. Buning uchun if kerak — 6-bobda shu dasturni tugatamiz.

Xulosa

• import math, keyin math.funksiya(). Bu eng aniq usul.
• sqrt, pi, hypot, dist — geometriya; sin, cos — radianda, radians() bilan o'giring.
• floor pastga, ceil yuqoriga, trunc/int nol tomonga, round eng yaqiniga.
• math domain error — manfiy sondan ildiz yoki nol/manfiy sondan logarifm.
• math.isclose() — kasrlarni solishtirish uchun.
• random.randint(a, b) — tasodifiy butun son.

Mashqlar

Bu bobda π uchun math.pi ishlating (varaqdagi π = 3.14 o'rniga). Tekshiruvchi aniq qiymat bilan solishtiradi.

import math
a = float(input())
b = float(input())
print(math.sqrt(a * b))

import math
a = float(input())
b = float(input())
c = math.hypot(a, b)
print(c, a + b + c)

import math
r1 = float(input())
r2 = float(input())
s1 = math.pi * r1 ** 2
s2 = math.pi * r2 ** 2
print(s1, s2, s1 - s2)

import math
l = float(input())
r = l / (2 * math.pi)
print(r, math.pi * r ** 2)

import math
s = float(input())
r = math.sqrt(s / math.pi)
print(2 * r, r)

import math
x1 = float(input())
y1 = float(input())
x2 = float(input())
y2 = float(input())
print(math.dist((x1, y1), (x2, y2)))

import math
x1 = float(input())
y1 = float(input())
x2 = float(input())
y2 = float(input())
x3 = float(input())
y3 = float(input())
a = math.dist((x1, y1), (x2, y2))
b = math.dist((x2, y2), (x3, y3))
c = math.dist((x1, y1), (x3, y3))
p = (a + b + c) / 2
print(math.sqrt(p * (p - a) * (p - b) * (p - c)), a + b + c)

a = float(input())
b = float(input())
print(-b / a)

import math
a = float(input())
b = float(input())
c = float(input())
d = b ** 2 - 4 * a * c
x1 = (-b + math.sqrt(d)) / (2 * a)
x2 = (-b - math.sqrt(d)) / (2 * a)
print(min(x1, x2), max(x1, x2))

a1 = float(input())
b1 = float(input())
c1 = float(input())
a2 = float(input())
b2 = float(input())
c2 = float(input())
d = a1 * b2 - a2 * b1
print((c1 * b2 - c2 * b1) / d, (a1 * c2 - a2 * c1) / d)

import math
n = int(input())
k = int(input())
print(math.ceil(n / k))

import math
g = float(input())
r = math.radians(g)
print(f"{math.sin(r):.4f} {math.cos(r):.4f}")