Juliana's Favorite Math

Here are some of my favorite facts/theorems from studying math over the years.

• Pythagorean Theorem. Given a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(a^2 + b^2 = c^2\)
• Pythagorean Identity. \(\cos^2\theta + \sin^2\theta = 1\)
• Squeeze Theorem. Suppose that for all \(x\) in \([a,b]\) (except possibly at \(x=c\)) we have \(f(x) \le h(x) \le g(x)\). Also suppose that \(\lim_{x\to c}f(x) = \lim_{x\to c}g(x) = L\), where \(a\le c\le b\). Then \(\lim_{x\to c}h(x) = L\)
• Intermediate Value Theorem. Suppose that \(f(x)\) is continuous on \([a,b]\) and let \(M\) be any number between \(f(a)\) and \(f(b)\). Then there exists \(c\) such that \(a < b < c\) and \(f(c)=M\).
• Mean Value Theorem. Suppose \(f(x)\) is a function such where \(f(x)\) is continuous on \([a,b]\) and \(f(x)\) is differentiable on \(a,b\). Then there exists some \(c\) where \(a < c < b\) and \(f'(c)=\frac{f(b)-f(a)}{b-a}\).
• Differentiable \(\Rightarrow\) continuous, but continuous \(\not\Rightarrow\) differentiable
• Lagrange's Theorem. Given a finite group \(G\), if \(H\) is a subgroup of \(G\), then \(|H|\) divides \(|G|\)
• Cauchy's Theorem. if \(G\) is a finite group and \(p\) is a prime number dividing \(G\), then \(G\) contains an element of order \(p\)
• \(\mathbb{Z}\) is countable
• \(\mathbb{Q}\) is countable

Other things I enjoy:

• Homomorphisms and isomorphisms
• Euler's number
• Taylor series
• Eigenvectors and eigenvalues
• Set theory
• Graphs and trees
• Permutation groups
• All things automata-related (Turing machines, regular expressions, context-free grammars, etc.)