Python math.sqrt详解：平方根计算与勾股定理应用

import math
math.sqrt(x)

import math

# 常见数值的平方根
print(f"sqrt(0)    = {math.sqrt(0)}")
print(f"sqrt(1)    = {math.sqrt(1)}")
print(f"sqrt(2)    = {math.sqrt(2)}")
print(f"sqrt(4)    = {math.sqrt(4)}")
print(f"sqrt(9)    = {math.sqrt(9)}")
print(f"sqrt(100)  = {math.sqrt(100)}")

# 验证平方根的性质
print(f"\nsqrt(2) 的平方 = {math.sqrt(2) ** 2}")
print(f"sqrt(2) 精确值 = {2}")  # 注意浮点精度差异

# 大数的平方根
big = 10 ** 18
print(f"\nsqrt(10^18) = {math.sqrt(big)}")

# 输出：
# sqrt(0)    = 0.0
# sqrt(1)    = 1.0
# sqrt(2)    = 1.4142135623730951
# sqrt(4)    = 2.0
# sqrt(9)    = 3.0
# sqrt(100)  = 10.0
#
# sqrt(2) 的平方 = 2.0000000000000004
# sqrt(2) 精确值 = 2
#
# sqrt(10^18) = 1000000000.0

import math

# 勾股定理：已知两直角边求斜边
a, b = 3, 4
c = math.sqrt(a**2 + b**2)
print(f"直角边 {a} 和 {b}，斜边 = {c}")

# 计算两点之间的欧氏距离
p1 = (1, 2)
p2 = (4, 6)
distance = math.sqrt((p2[0] - p1[0])**2 + (p2[1] - p1[1])**2)
print(f"点 {p1} 和 {p2} 的距离 = {distance:.2f}")

# 三维空间距离
p3 = (1, 2, 3)
p4 = (4, 6, 9)
dist_3d = math.sqrt(sum((a - b)**2 for a, b in zip(p3, p4)))
print(f"3D距离 = {dist_3d:.2f}")

# 使用 hypot 简化（推荐）
print(f"\nhypot(3, 4) = {math.hypot(3, 4)}")
print(f"hypot(3, 4, 6) = {math.hypot(3, 4, 6):.2f}")  # Python 3.8+ 支持多维

# 输出：
# 直角边 3 和 4，斜边 = 5.0
# 点 (1, 2) 和 (4, 6) 的距离 = 5.00
# 3D距离 = 8.06
#
# hypot(3, 4) = 5.0
# hypot(3, 4, 6) = 7.81

import math

def std_deviation(data):
    """计算总体标准差"""
    n = len(data)
    mean = sum(data) / n
    variance = sum((x - mean) ** 2 for x in data) / n
    return math.sqrt(variance)

def rms(data):
    """计算均方根值（RMS）"""
    return math.sqrt(sum(x**2 for x in data) / len(data))

# 学生成绩标准差
scores = [85, 90, 78, 92, 88, 76, 95, 89]
std = std_deviation(scores)
mean = sum(scores) / len(scores)
print(f"成绩: {scores}")
print(f"平均分: {mean:.1f}, 标准差: {std:.2f}")

# 交流电的均方根值
voltage = [311, 280, 311, 280, 311, 280]  # 模拟交流电压采样
rms_value = rms(voltage)
print(f"\n电压采样: {voltage}")
print(f"RMS电压: {rms_value:.1f}V (约等于220V * sqrt(2)的峰值)")

# 一元二次方程求根
def quadratic(a, b, c):
    """求解 ax^2 + bx + c = 0"""
    discriminant = b**2 - 4*a*c
    if discriminant < 0:
        return None  # 无实数根
    sqrt_disc = math.sqrt(discriminant)
    x1 = (-b + sqrt_disc) / (2 * a)
    x2 = (-b - sqrt_disc) / (2 * a)
    return x1, x2

roots = quadratic(1, -5, 6)  # x^2 - 5x + 6 = 0
print(f"\nx^2 - 5x + 6 = 0 的根: {roots}")

# 输出：
# 成绩: [85, 90, 78, 92, 88, 76, 95, 89]
# 平均分: 86.6, 标准差: 6.13
#
# 电压采样: [311, 280, 311, 280, 311, 280]
# RMS电压: 296.5V (约等于220V * sqrt(2)的峰值)
#
# x^2 - 5x + 6 = 0 的根: (3.0, 2.0)