Favorite Math Results
A collection of nifty, interesting, and cool math results.
Calculus
Arc length. The length of a curve given by the function $f(x)$ on the interval $[a, b]$ is given by the integral \(\int_a^b \sqrt{1 + (f'(x))^2} \, dx\).
Bolzano-Weierstrass Theorem. \(S\) is compact iff every sequence of points in \(S\) has a convergent subsequence whose limit is in \(S\). Or, every bounded sequence has a convergent subsequence.
Cauchy Sequences. A sequence \(\{ \mathbf{x}_n \} \in \mathbb{R}^n\) is convergent iff it is Cauchy. A sequence is Cauchy if for every $\epsilon > 0$, there exists an $N$ such that for all $n, m > N$, \(\| \mathbf{x}_n - \mathbf{x}_m \| < \epsilon\).
Heine-Borel Theorem. If \(S\) is a subset of \(\mathbb{R}^n\), then \(S\) is compact iff \(S\) is closed and bounded.
Connectedness, Pathwise and Not. A set is connected iff it cannot be written as the union of two disjoint nonempty open sets. A set is path-wise connected iff for every pair of points in the set, there exists a continuous function that maps the interval $[0, 1]$ to the set and maps the endpoints to the given points. If a set is path connected, then it is connected. The converse is not true: consider the Topologist’s sine curve, defined as
\[S = \{ (x, \sin(1/x)) \mid 0 < x \leq 1 \} \cup \{ (0, y) \mid -1 \leq y \leq 1 \}\]Leibniz’s proof for the derivative of a quadratic. If \(x\) changes by \(\delta x\), then \(y = x^2\) should change by \(y + \delta y = (x + \delta x)^2 = x^2 + 2x\delta x + (\delta x)^2\). Relabel \(\delta x, \delta y\) as \(dx, dy\) and as they become ‘infinitely small’, then \((\delta x)^2\) gets infinitely smaller. Also subtract \(y = x^2\). So we get \(dy = 2x dx\), or \(dy/dx = 2x\).
Riemann integrability. \(f\) is integrable on \([a, b]\) if for every \(\epsilon > 0\) there is a partition \(P\) of \([a, b]\) such that \(U(f, P) - L(f, P) < \epsilon\), where \(U(f, P)\) is the upper sum and \(L(f, P)\) is the lower sum.
Zero-content. A bounded set is integrable even if it has a subset of zero content. For instance, the Cantor set defined by removing the middle third of the interval $[0, 1]$ and then removing the middle third of the remaining intervals, and so on until infinity.
Measurability. Any bounded set whose boundary is a finite union of pieces of smooth curves is measurable.
Linear Algebra
Rank-Nullity Theorem. For a linear transformation $T: V \to W$, the rank-nullity theorem states that \(\text{rank}(T) + \text{nullity}(T) = \dim(V)\).
Closed-form solution for the nth Fibonacci number using matrix decomposition. Begin with the following definition of the Fibonacci sequence, and then raise the given matrix to the power of n and decompose it to find a closed-form solution for the nth Fibonacci number. \(\begin{pmatrix} F_n \\ F_{n+1} \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} F_{n-1} \\ F_{n} \end{pmatrix}\)
Abstract Algebra
Isomorphisms.
- The field \(K\) of matrices with form \(\begin{pmatrix} a & b \\ -b & a \end{pmatrix}\) is isomorphic to the field \(\mathbb{C}\).
- The quotient group \(\mathbb{SL}_n(\mathbb{R}) / \mathbb{GL}_n(\mathbb{R})\) is isomorphic to the multiplicative group \(\mathbb{R}^+\).
- The special linear group is the set of \(n \times n\) matrices with determinant 1
- The general linear group is the set of \(n \times n\) matrices with non-zero determinant
The quotient group $$\mathbb{C}* / { x \in \mathbb{C} \mid x = 1 }\(is isomorphic to the multiplicative group\)\mathbb{R}^+$$. - “Taking the complex numbers modulo the unit circle is the same as taking the positive real numbers.”
Bezout’s Theorem. For any two integers \(a, b\), there exist integers \(x, y\) such that \(ax + by = \gcd(a, b)\).
Units in \(\mathbb{Z}_n\). The units in \(\mathbb{Z}_n\) are the integers \(a\) such that \(\gcd(a, n) = 1\). Note that units are elements with a multiplicative inverse.
Fields and Integral Domains. Every field is an integral domain, but not every integral domain is a field. (A field is a commutative ring with every element a unit. An integral domain is a commutative ring with no zero divisors – that is, if \(ab = 0\), then \(a = 0\) or \(b = 0\).)
Eisenstein’s Criterion. If a polynomial \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0\) has integer coefficients and there exists a prime \(p\) such that \(p \nmid a_n\), \(p \mid a_{n-1}, \ldots, a_0\), and \(p^2 \nmid a_0\), then \(f(x)\) is irreducible over the rationals.
Irreducibility of polynomials. \(p(x)\) is irreducible in \(F[x]\) iff \(F[x]/(p(x))\) is a field.
Extension field for polynomials. Let \(F\) be a field and \(p(x)\) an irreducible polynomial in \(F[x]\). Then the field \(F[x]/(p(x))\) is an extension field of \(F\) containing the root of \(p(x)\).
First Isomorphism Theorem for Rings. If \(\phi: R \to S\) is a ring homomorphism, then \(\ker(\phi)\) is an ideal of \(R\) and \(R/\ker(\phi) \cong \text{im}(\phi)\).
First Isomorphism Theorem for Groups. If \(\phi: G \to H\) is a group homomorphism, then \(\ker(\phi)\) is a normal subgroup of \(G\) and \(G/\ker(\phi) \cong \text{im}(\phi)\).
Prime Ideals. An ideal is prime if \(ab \in I\) implies \(a \in I\) or \(b \in I\). Let \(P\) be an ideal in a commutative ring \(R\) with identity. Then \(P\) is a prime ideal iff \(R/P\) is an integral domain.
Maximal Ideals. An ideal is maximal if it is not contained in any other proper ideal. Let \(M\) be an ideal in a commutative ring \(R\) with identity. Then \(M\) is a maximal ideal iff \(R/M\) is a field.
Relation between rings and groups. Every ring is a group under addition, and no ring is a group under multiplication.
Abelian groups. Every group of order 5 or less is abelian.
Cayley’s Theorem. Every group is isomorphic to a subgroup of the symmetric group on the group’s elements.
Cycles. Every permutation in \(S_n\) is a product of disjoint cycles.
Lagrange’s Theorem. If \(G\) is a finite group and \(H\) is a subgroup of \(G\), then the order of \(H\) divides the order of \(G\): \(|G| = |H| \cdot [G:H]\), where \([G:H]\) is the index of \(H\) in \(G\) (the number of left cosets of \(H\) in \(G\)).
Isomorphisms to prime modulo groups. Every group of prime order, with no subgroups, or simple and abelian is cyclic and isomorphic to \(\mathbb{Z}_p\).
Finite abelian groups. Every finite abelian group is isomorphic to a Cartesian product of cyclic groups of prime power order.