# 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

**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

*n*th 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.