Python内置模块之math数学运算全面解析

import math

# math模块提供：
# - 数学常量：pi, e, tau, inf, nan
# - 数值运算：取整、绝对值、阶乘、组合数
# - 幂和对数：sqrt, pow, exp, log, log2, log10
# - 三角函数：sin, cos, tan 及其反函数
# - 特殊函数：gcd, lcm, erf, gamma
# - 角度转换：degrees, radians

import math

# 圆周率 π
print(f"π = {math.pi}")          # 3.141592653589793
print(f"π/2 = {math.pi / 2}")    # 1.5707963267948966

# 自然常数 e
print(f"e = {math.e}")           # 2.718281828459045

# 圆的周长与面积
r = 5
circumference = 2 * math.pi * r
area = math.pi * r ** 2
print(f"半径{r}的圆: 周长={circumference:.2f}, 面积={area:.2f}")

# τ (tau) = 2π —— Python 3.6+
print(f"τ = {math.tau}")         # 6.283185307179586

# 无穷大和非数字
print(f"无穷大: {math.inf}")     # inf
print(f"负无穷: {-math.inf}")    # -inf
print(f"非数字: {math.nan}")     # nan

# 判断特殊值
print(math.isinf(math.inf))      # True
print(math.isnan(math.nan))      # True
print(math.isfinite(1.0))        # True
print(math.isfinite(math.inf))   # False

# 验证计算
print(1 / math.inf)              # 0.0
print(math.inf > 999999999)      # True
print(math.nan == math.nan)      # False! nan不等于任何值，包括自己

import math

# ceil(x) —— 向上取整（往正无穷方向）
print(f"ceil(3.2) = {math.ceil(3.2)}")     # 4
print(f"ceil(3.9) = {math.ceil(3.9)}")     # 4
print(f"ceil(-3.2) = {math.ceil(-3.2)}")   # -3

# floor(x) —— 向下取整（往负无穷方向）
print(f"floor(3.2) = {math.floor(3.2)}")   # 3
print(f"floor(3.9) = {math.floor(3.9)}")   # 3
print(f"floor(-3.2) = {math.floor(-3.2)}") # -4

# trunc(x) —— 截断小数部分（往0方向）
print(f"trunc(3.7) = {math.trunc(3.7)}")   # 3
print(f"trunc(-3.7) = {math.trunc(-3.7)}") # -3

# 💡 ceil vs floor vs trunc vs round 对比
x = -3.7
print(f"x={x}: ceil={math.ceil(x)}, floor={math.floor(x)}, "
      f"trunc={math.trunc(x)}, round={round(x)}")
# x=-3.7: ceil=-3, floor=-4, trunc=-3, round=-4

# fabs(x) —— 绝对值（返回float）
print(f"fabs(-5.5) = {math.fabs(-5.5)}")  # 5.5

# fmod(x, y) —— 浮点数取余（不同于%）
print(f"fmod(7.5, 2.0) = {math.fmod(7.5, 2.0)}")  # 1.5

import math

# factorial(n) —— n的阶乘（n >= 0）
for n in range(6):
    print(f"{n}! = {math.factorial(n)}")
# 0! = 1  1! = 1  2! = 2  3! = 6  4! = 24  5! = 120

# comb(n, k) —— 组合数 C(n,k)，从n个中选k个
print(f"C(5,2) = {math.comb(5, 2)}")   # 10
# 验证：C(5,2) = 5!/(2!*3!) = 120/(2*6) = 10

# perm(n, k) —— 排列数 P(n,k)，从n个中选k个并排序
print(f"P(5,2) = {math.perm(5, 2)}")   # 20
# 验证：P(5,2) = 5*4 = 20

# 实际应用：计算中奖概率
def lottery_probability(total, choose):
    """从total个号中选choose个，全中的概率"""
    return 1 / math.comb(total, choose)

print(f"双色球红球中奖概率: 1/{math.comb(33, 6)}")
# 双色球红球中奖概率: 1/17721088

import math

# sqrt(x) —— 平方根
print(f"sqrt(16) = {math.sqrt(16)}")     # 4.0
print(f"sqrt(2) = {math.sqrt(2):.10f}")  # 1.4142135624

# pow(x, y) —— x的y次方
print(f"pow(2, 10) = {math.pow(2, 10)}")  # 1024.0

# exp(x) —— e的x次方
print(f"exp(1) = {math.exp(1):.6f}")     # 2.718282
print(f"exp(2) = {math.exp(2):.6f}")     # 7.389056

# 对数
# log(x) —— 自然对数（以e为底）
print(f"log(e) = {math.log(math.e)}")     # 1.0

# log2(x) —— 以2为底
print(f"log2(8) = {math.log2(8)}")        # 3.0

# log10(x) —— 以10为底
print(f"log10(1000) = {math.log10(1000)}") # 3.0

# log(x, base) —— 以base为底的对数
print(f"log(8, 2) = {math.log(8, 2)}")    # 3.0
print(f"log(100, 10) = {math.log(100, 10)}") # 2.0

# 实用函数：计算信息熵（比特）
def entropy_bits(probabilities):
    """计算以2为底的熵"""
    return -sum(p * math.log2(p) for p in probabilities if p > 0)

import math

# 基本三角函数（参数是弧度）
angle = math.pi / 4  # 45度

print(f"sin(45°) = {math.sin(angle):.6f}")   # 0.707107
print(f"cos(45°) = {math.cos(angle):.6f}")   # 0.707107
print(f"tan(45°) = {math.tan(angle):.6f}")   # 1.000000

# 反三角函数
x = 0.5
print(f"asin(0.5) = {math.asin(x):.6f} rad = {math.degrees(math.asin(x)):.1f}°")
# asin(0.5) = 0.523599 rad = 30.0°

# atan2(y, x) —— 计算角度（考虑象限）
print(f"atan2(1, 1) = {math.atan2(1, 1):.6f} rad")   # 45°
print(f"atan2(1, -1) = {math.atan2(1, -1):.6f} rad")  # 135°
print(f"atan2(-1, -1) = {math.atan2(-1, -1):.6f} rad") # -135°

# 角度弧度转换
print(f"180° = {math.radians(180):.6f} rad")    # 3.141593
print(f"π rad = {math.degrees(math.pi):.1f}°")   # 180.0°

# 计算两点距离
def distance(x1, y1, x2, y2):
    """计算两点之间的欧几里得距离"""
    return math.sqrt((x2 - x1)**2 + (y2 - y1)**2)

print(f"距离: {distance(0, 0, 3, 4)}")  # 5.0

import math

# gcd(a, b) —— 最大公约数
print(f"gcd(48, 18) = {math.gcd(48, 18)}")     # 6
print(f"gcd(17, 31) = {math.gcd(17, 31)}")     # 1（互质）

# lcm(a, b) —— 最小公倍数（Python 3.9+）
print(f"lcm(4, 6) = {math.lcm(4, 6)}")         # 12
print(f"lcm(12, 18) = {math.lcm(12, 18)}")     # 36

# 应用：分数化简
def simplify_fraction(num, den):
    """化简分数"""
    g = math.gcd(num, den)
    return num // g, den // g

print(f"24/36 化简: {simplify_fraction(24, 36)}")  # (2, 3)

# 判断是否为完全平方数
def is_perfect_square(n):
    return n >= 0 and math.isqrt(n) ** 2 == n

print(is_perfect_square(25))   # True
print(is_perfect_square(26))   # False

# isclose —— 判断两个数是否接近
print(math.isclose(0.1 + 0.2, 0.3))  # True
print(0.1 + 0.2 == 0.3)               # False! 浮点精度问题