欧拉常数 γ 大约是几？它从哪来？
About what is Euler's constant γ, and where does it come from?
γ≈0.5772，是调和级数 H_n 与 ln n 的差的极限：γ=lim(H_n−ln n)。它出现在数论和积分中，但是否是有理数至今未知。
γ≈0.5772 is the limit of H_n−ln n (harmonic minus log). It appears in number theory and integrals, yet whether it is rational remains unknown.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#欧拉 #常数
欧拉的多面体公式 V−E+F=2 对哪些形状会失效？
For which shapes does Euler's formula V−E+F=2 fail?
它对同胚于球面的凸多面体成立；一旦有多个洞（如环面，结果为0）或自交、非连通，公式需改成 V−E+F=2−2g，否则不成立。
It holds for sphere-like convex polyhedra; with holes (a torus gives 0) or self-intersections/disconnected parts, it becomes V−E+F=2−2g and otherwise fails.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#欧拉 #拓扑
欧拉公式 e^{iπ}+1=0 为什么被称为‘最美公式’？
Why is Euler's formula e^{iπ}+1=0 called the 'most beautiful'?
它在一行里把五个最基本的数（e、i、π、1、0）和加法、乘法、指数联系起来，把指数函数与三角函数统一为 e^{iθ}=cosθ+i sinθ。
In one line it links five fundamental constants (e, i, π, 1, 0) with addition, multiplication and exponentiation, unifying exponential and trigonometric functions via e^{iθ}=cosθ+i sinθ.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#欧拉 #公式
柯尼斯堡七桥问题想问什么？欧拉怎么解决的？
What did the Königsberg bridges problem ask, and how did Euler solve it?
问题是能否不重复地走遍七座桥。欧拉把陆地化为点、桥化为线，发现若存在这样的走法，每个点必须经偶数次桥；而七桥图中奇点数为4，故不可能。这开创了图论。
The question was whether one could cross all seven bridges without repeating any. Euler reduced land to vertices and bridges to edges, noting a required walk needs every vertex of even degree; the graph had four odd vertices, so impossible. This founded graph theory.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#欧拉 #图论