Math Results

A collection of nifty, interesting, elegant, cool math results.

Reals as a vector space over the rationals. The reals can be defined as the set of all Cauchy sequences of rationals, with addition and multiplication defined component-wise. This makes the reals a vector space (of infinite dimension) over the rationals. If the axiom of choice is true, then there exists a basis for this vector space.

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 \(dy/dx = 2x\).

Cool isomorphisms.

  • \(\mathbb{R}[x]/(x^2 + 1) \cong \mathbb{C}\).
  • \(\mathbb{H} \cong \mathbb{C} \times \mathbb{C}\) (where \(\mathbb{H}\) is the set of quaternions).
  • \(\mathbb{R} / \mathbb{Z} \cong \{ z \in \mathbb{C} : \vert z \vert = 1 \}\).
  • The ring of \(n \times n\) matrices over a field \(k\) is isomorphic to the ring of endomorphisms of a finite-dimensional vector space \(V\) over \(k\) with dimension \(n\).
  • 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 \(\mathbb{C}* / \{ x \in \mathbb{C} : \vert x \vert = 1 \}\) 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.”

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.

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.

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.

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.

Sylow’s Theorems. A Sylow \(p\)-subgroup of a group \(G\) is a maximal \(p\)-subgroup of \(G\). If \(G\) is a finite group and \(p\) is a prime dividing the order of \(G\), then

  • There exists a Sylow \(p\)-subgroup of \(G\).
  • All Sylow \(p\)-subgroups are conjugate.
  • The number of Sylow \(p\)-subgroups is congruent to 1 modulo \(p\) and divides the order of \(G\).

Branching Programs. Every function \(f : \{0, 1\}^n \to \{0, 1\}\) can be computed by a branching program of width 5 and length \(O(4^d)\).

Time Hierarchy. If \(r, t\) are time-constructible functions satisfying \(r(n) \log r(n) = o(t(n))\), then \(\text{DTIME}(r(n)) \subsetneq \text{DTIME}(t(n))\).

Space Hierarchy. If \(q, s\) are space-constructible functions satisfying \(q(n) = o(s(n))\), then \(\text{DSPACE}(q(n)) \subsetneq \text{DSPACE}(s(n))\).

Matrix product checking. To check if \(A \times B = C\) for three \(n \times n\) matrices \(A, B, C\), randomly sample \(r \in \{0, 1\}^n\) and check if \(ABr = Cr\) repeatedly. This randomized algorithm takes only \(O(n^2)\) time and has exponentially decreasing error rate.

A complication for proving \(P ?= NP\). There exists an oracle \(A\) such that \(P^A = NP^A\), and an oracle \(B\) such that \(P^B \neq NP^B\). Therefore any proof on \(P ?= NP\) may not work in relativized worlds where access to both is permitted, whereas other familiar proofs to relativize.

The blind man. A blind man is trying to buy a red rag and a blue rag. But he is not sure if the salesman is trying to trick him or not. The salesman could, for instance, give him two red rags or two blue rags. The blind man shuffles rags behind his back, but keeps track of the identities of each (so at first he doesn’t know the true colors, but he knows which rag is which). He randomly pulls out a rag and asks the salesman what color it is. Then, he puts it behind his back and shuffles again, repeating this several times. If the salesman gives two conflicting answers for the same rag (e.g., calling it red one time and blue another time), then the blind man knows that he is lying. If, on the other hand, the salesman consistently calls one rag blue and the other rag red, then the blind man is sure that he indeed has one blue and one red rag. The moral of this story is that even a “handicapped” verifier can compute powerful functions with randomness and interaction.